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Mathematics
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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[1] algebra,[2] geometry,[1] and analysis,[3] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.[4]

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications.[5][6]

Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[7] Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.[8] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,[9] which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.
Etymology

The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt",[10] "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in Classical times.[b] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".[14] In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art".[10]

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.[15]

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.[16]

The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, inherited from Greek.[17] In English, the noun mathematics takes a singular verb. It is often shortened to maths[18] or, in North America, math.[19]
Areas of mathematics

Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes.[20] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.[21]

During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areas–arithmetic, geometry, algebra, calculus[22]–endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics.[23] The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.[24]

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.[25][9] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[26] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.[27]
Number theory
This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.

Number theory began with the manipulation of numbers, that is, natural numbers {\displaystyle (\mathbb {N} ),} and later expanded to integers {\displaystyle (\mathbb {Z} )} and rational numbers {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations.[28] Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria.[29] The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.[30]

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra.[31] Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort.[32]

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).[27]
Geometry
On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.[33]

A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.[34][35]

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space.[c][33]

Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.[36]

Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.[33]

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.[37][9] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space.[38]

Today's subareas of geometry include:[27]

    Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
    Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
    Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions.
    Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
    Riemannian geometry, the study of distance properties in curved spaces.
    Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.
    Topology, the study of properties that are kept under continuous deformations.
        Algebraic topology, the use in topology of algebraic methods, mainly homological algebra.
    Discrete geometry, the study of finite configurations in geometry.
    Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.
    Complex geometry, the geometry obtained by replacing real numbers with complex numbers.

Algebra
The quadratic formula, which concisely expresses the solutions of all quadratic equations
The Rubik's Cube group is a concrete application of group theory.[39]

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.[40][41] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts'[42] that he used for naming one of these methods in the title of his main treatise.

Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.[43] Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.[44] The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.[45] (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:[27]

    group theory;
    field theory;
    vector spaces, whose study is essentially the same as linear algebra;
    ring theory;
    commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
    homological algebra;
    Lie algebra and Lie group theory;
    Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.[46] The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.[47]
Calculus and analysis
A Cauchy sequence consists of elements such that all subsequent terms of a term become arbitrarily close to each other as the sequence progresses (from left to right).

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.[48] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results.[49] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:[27]

    Multivariable calculus
    Functional analysis, where variables represent varying functions;
    Integration, measure theory and potential theory, all strongly related with probability theory on a continuum;
    Ordinary differential equations;
    Partial differential equations;
    Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

Discrete mathematics
A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.

Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.[50] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.[d] Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.[51]

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century.[52] The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.[53]

Discrete mathematics includes:[27]

    Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes
    Graph theory and hypergraphs
    Coding theory, including error correcting codes and a part of cryptography
    Matroid theory
    Discrete geometry
    Discrete probability distributions
    Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
    Discrete optimization, including combinatorial optimization, integer programming, constraint programming

Mathematical logic and set theory
The Venn diagram is a commonly used method to illustrate the relations between sets.

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.[54][55] Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.[56]

Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets[57] but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory.[58]

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics.[59] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.[25] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning.[60] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.[61]

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.[62] This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.[63][64]

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory.[27] Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.[65]
Statistics and other decision sciences
Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory.[66]

The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments.[67] The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data.[e]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[68] Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.[69]
Computational mathematics

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity.[70][71] Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors.[72] Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.
History
Ancient

The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,[73] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.[74][75]
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[76] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.[77]

In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right.[78] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.[79] His book, Elements, is widely considered the most successful and influential textbook of all time.[80] The greatest mathematician of antiquity is often held to be Archimedes (c. 287 – c. 212 BC) of Syracuse.[81] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[82] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[83] trigonometry (Hipparchus of Nicaea, 2nd century BC),[84] and the beginnings of algebra (Diophantus, 3rd century AD).[85]
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics.[86] Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.[87][88]
Medieval and later
A page from al-Khwārizmī's Algebra

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of Algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.[89] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.[90] The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.[91]

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.
Carl Friedrich Gauss

Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics.[92] In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.[62]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[93]
Symbolic notation and terminology
An explanation of the sigma (Σ) summation notation

Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas.[94] More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs,[95] such as + (plus), × (multiplication), {\textstyle \int } (integral), = (equal), and < (less than).[96] All these symbols are generally grouped according to specific rules to form expressions and formulas.[97] Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses.

Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary.[98]

Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism.[99] Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or"). Finally, many mathematical terms are common words that are used with a completely different meaning.[100] This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring".
Relationship with sciences

Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws.[101] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.[102] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.[103] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.[104]

There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation.[105] In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).[106] However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.[107][108][109][110]
Pure and applied mathematics

Isaac Newton

Gottfried Wilhelm von Leibniz

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics.[111] For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.[112] However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece.[113] The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.[114]

In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics.[111][115] This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.[116]

The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere.[117][118] Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".[119][120]

An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis.[121] An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high.[122] For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.[123]

In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas.[124][125] The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".[27] However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.
Unreasonable effectiveness

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner.[6] It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.[126] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.[127] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.[128]

In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four.[129][130]

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon {\displaystyle \Omega ^{-}.} In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.[131][132][133]
Specific sciences
Physics
Diagram of a pendulum

Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,[134] and is also the motivation of major mathematical developments.[135]
Computing

The rise of technology in the 20th century opened the way to a new science: computing.[f] This field is closely related to mathematics in several ways. Theoretical computer science is essentially mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in cryptography and coding theory. Discrete mathematics is useful in many areas of computer science, such as complexity theory, information theory, graph theory, and so on.[citation needed]

In return, computing has also become essential for obtaining new results. This is a group of techniques known as experimental mathematics, which is the use of experimentation to discover mathematical insights.[136] The most well-known example is the four-color theorem, which was proven in 1976 with the help of a computer. This revolutionized traditional mathematics, where the rule was that the mathematician should verify each part of the proof. In 1998, the Kepler conjecture on sphere packing seemed to also be partially proven by computer. An international team had since worked on writing a formal proof; it was finished (and verified) in 2015.[137]

Once written formally, a proof can be verified using a program called a proof assistant.[138] These programs are useful in situations where one is uncertain about a proof's correctness.[138]

A major open problem in theoretical computer science is P versus NP. It is one of the seven Millennium Prize Problems.[139]
Biology and chemistry
The skin of this giant pufferfish exhibits a Turing pattern, which can be modeled by reaction–diffusion systems.

Biology uses probability extensively – for example, in ecology or neurobiology.[140] Most of the discussion of probability in biology, however, centers on the concept of evolutionary fitness.[140]

Ecology heavily uses modeling to simulate population dynamics,[140][141] study ecosystems such as the predator-prey model, measure pollution diffusion,[142] or to assess climate change.[143] The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations.[144] However, there is the problem of model validation. This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model.[145]

Genotype evolution can be modeled with the Hardy-Weinberg principle.[citation needed]

Phylogeography uses probabilistic models.[citation needed]

Medicine uses statistical hypothesis testing, run on data from clinical trials, to determine whether a new treatment works.[citation needed]

Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions. It turns out that the form of macromolecules in biology is variable and determines the action. Such modeling uses Euclidean geometry; neighboring atoms form a polyhedron whose distances and angles are fixed by the laws of interaction.[citation needed]
Earth sciences

Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes.[citation needed] Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models.[citation needed]

Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, economics, sociology,[146] and psychology.[147]
Supply and demand curves, like this one, are a staple of mathematical economics.

The fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus (lit. 'economic man').[148] In this model, the individual seeks to maximize their self-interest,[148] and always makes optimal choices using perfect information.[149][better source needed] This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual calculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms which would be difficult to discover by a "literary" analysis.[citation needed] For example, explanations of economic cycles are not trivial. Without mathematical modeling, it is hard to go beyond statistical observations or unproven speculation.[citation needed]

However, many people have rejected or criticized the concept of Homo economicus.[149][better source needed] Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.[149][better source needed]

At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which explains phases of economic growth or crisis.[150] Towards the end of the 19th century, Nicolas-Remi Brück [fr] and Charles Henri Lagrange [fr] extended their analysis into geopolitics.[151] Peter Turchin has worked on developing cliodynamics since the 1990s.[152]

Even so, mathematization of the social sciences is not without danger. In the controversial book Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.[153] The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.[citation needed]
Relationship with astrology and esotericism

Some renowned mathematicians have also been considered to be renowned astrologists; for example, Ptolemy, Arab astronomers, Regiomantus, Cardano, Kepler, or John Dee. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, Theodor Zwinger wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences [of stars]" to foresee their "conjunctions and oppositions".[154]

Astrology is no longer considered a science.[155]
Philosophy
Reality

The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.[156]

Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.[131]

    Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.[157] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...

Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.[158]
Proposed definitions

There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities.[159][160] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[159] There is not even consensus on whether mathematics is an art or a science.[160] Some just say, "mathematics is what mathematicians do".[159] This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.[161]

Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.[162] In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given.[163] With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.

Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction.[164] Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.[165]
Rigor

Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules,[g] without any use of empirical evidence and intuition.[h][166] Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' concision, rigorous proofs can require hundreds of pages to express, such as the 255-page Feit–Thompson theorem.[i] The emergence of computer-assisted proofs has allowed proof lengths to further expand,[j][167] The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.[9]

The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.[9]

At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.[9] It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.[168]

Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.[169]
Training and practice
Education

Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant.[170]

Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.[171] Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE.[172] The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt.[173] Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period (c. 1500 – c. 500 BCE).[174] In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy.[175]

Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was L'Enseignement Mathématique, which began publication in 1899.[176] The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component—initially for its military applications.[177] While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.[178]

During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.[179] Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as math anxiety or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.[180]
Psychology (aesthetic, creativity and intuition)

The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.[181][182] An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.[183]

Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles.[184] This aspect of mathematical activity is emphasized in recreational mathematics.

Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.[185] Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis.[186]

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.[187] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science).[131] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Cultural impact
Artistic expression

Notes that sound well together to a Western ear are sounds whose fundamental frequencies of vibration are in simple ratios. For example, an octave doubles the frequency and a perfect fifth multiplies it by {\displaystyle {\frac {3}{2}}}.[188][189]
Fractal with a scaling symmetry and a central symmetry

Humans, as well as some other animals, find symmetric patterns to be more beautiful.[190] Mathematically, the symmetries of an object form a group known as the symmetry group.[191]

For example, the group underlying mirror symmetry is the cyclic group of two elements, {\displaystyle \mathbb {Z} /2\mathbb {Z} }. A Rorschach test is a figure invariant by this symmetry,[192] as are butterfly and animal bodies more generally (at least on the surface).[193] Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea.[citation needed] Fractals possess self-similarity.[194][195]
Popularization

Popular mathematics is the act of presenting mathematics without technical terms.[196] Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract.[197] However, popular mathematics writing can overcome this by using applications or cultural links.[198] Despite this, mathematics is rarely the topic of popularization in printed or televised media.
Awards and prize problems
The front side of the Fields Medal with an illustration of the Greek polymath Archimedes

The most prestigious award in mathematics is the Fields Medal,[199][200] established in 1936 and awarded every four years (except around World War II) to up to four individuals.[201][202] It is considered the mathematical equivalent of the Nobel Prize.[202]

Other prestigious mathematics awards include:[203]

    The Abel Prize, instituted in 2002[204] and first awarded in 2003[205]
    The Chern Medal for lifetime achievement, introduced in 2009[206] and first awarded in 2010[207]
    The AMS Leroy P. Steele Prize, awarded since 1970[208]
    The Wolf Prize in Mathematics, also for lifetime achievement,[209] instituted in 1978[210]

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[211] This list has achieved great celebrity among mathematicians,[212] and, as of 2022, at least thirteen of the problems (depending how some are interpreted) have been solved.[211]

A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.[213] To date, only one of these problems, the Poincaré conjecture, has been solved.[214]
See also

    List of mathematical jargon
    Lists of mathematicians
    Lists of mathematics topics
    Mathematical constant
    Mathematical sciences
    Mathematics and art
    Mathematics education
    Outline of mathematics
    Philosophy of mathematics
    Relationship between mathematics and physics
    Science, technology, engineering, and mathematics

References
Notes

    ^ Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating formulas.
    ^ This meaning can be found in Plato's Republic, Book 6 Section 510c.[11] However, Plato did not use a math- word; Aristotle did, commenting on it.[12][better source needed][13][better source needed]
    ^ This includes conic sections, which are intersections of circular cylinders and planes.
    ^ However, some advanced methods of analysis are sometimes used; for example, methods of complex analysis applied to generating series.
    ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.
    ^ Ada Lovelace, in the 1840s, is known for having written the first computer program ever in collaboration with Charles Babbage
    ^ This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without computers and proof assistants. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.
    ^ This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.
    ^ This the length of the original paper that does not contain the proofs of some previously published auxiliary results. The book devoted to the complete proof has more than 1,000 pages.
    ^ For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software

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    ^ Brown, Ronald; Porter, Timothy (January 2000). "The Methodology of Mathematics". The Mathematical Gazette. 79 (485): 321–334. doi:10.2307/3618304. JSTOR 3618304. S2CID 178923299. Archived from the original on March 23, 2023. Retrieved November 25, 2022.
    ^ Strauss, Danie (2011). "Defining mathematics". Acta Academica. 43 (4): 1–28. Retrieved November 25, 2022.
    ^ Hamami, Yacin (June 2022). "Mathematical Rigor and Proof" (PDF). The Review of Symbolic Logic. 15 (2): 409–449. doi:10.1017/S1755020319000443. S2CID 209980693. Archived (PDF) from the original on December 5, 2022. Retrieved November 21, 2022.
    ^ Peterson 1988, p. 4: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Apple proof of the Four Color Theorem)
    ^ Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs". Philosophy of Mathematics. 42 (167 (4)). Revue Internationale de Philosophie: 500–508.
    ^ Davis, Jon D.; McDuffie, Amy Roth; Drake, Corey; Seiwell, Amanda L. (2019). "Teachers' perceptions of the official curriculum: Problem solving and rigor". International Journal of Educational Research. 93: 91–100. doi:10.1016/j.ijer.2018.10.002. S2CID 149753721.
    ^ Endsley, Kezia (2021). Mathematicians and Statisticians: A Practical Career Guide. Practical Career Guides. Rowman & Littlefield. pp. 1–3. ISBN 978-1-5381-4517-3. Retrieved November 29, 2022.
    ^ Robson, Eleanor (2009). "Mathematics education in an Old Babylonian scribal school". In Robson, Eleanor; Stedall, Jacqueline (eds.). The Oxford Handbook of the History of Mathematics. OUP Oxford. ISBN 978-0-19-921312-2. Retrieved November 24, 2022.
    ^ Bernard, Alain; Proust, Christine; Ross, Micah (2014). "Mathematics Education in Antiquity". In Karp, A.; Schubring, G. (eds.). Handbook on the History of Mathematics Education. New York: Springer. pp. 27–53. doi:10.1007/978-1-4614-9155-2_3. ISBN 978-1-4614-9154-5.
    ^ Dudley, Underwood (April 2002). "The World's First Mathematics Textbook". Math Horizons. 9 (4). Taylor & Francis, Ltd.: 8–11. doi:10.1080/10724117.2002.11975154. JSTOR 25678363. S2CID 126067145.
    ^ Subramarian, F. Indian pedagogy and problem solving in ancient Thamizhakam (PDF). History and Pedagogy of Mathematics conference, July 16–20, 2012. Archived (PDF) from the original on November 28, 2022. Retrieved November 29, 2022.
    ^ Siu, Man Keung (2004). "Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?". How Chinese Learn Mathematics (PDF). Series on Mathematics Education. Vol. 1. pp. 157–185. doi:10.1142/9789812562241_0006. ISBN 978-981-256-014-8. Retrieved November 26, 2022.
    ^ Jones, Phillip S. (1967). "The History of Mathematical Education". The American Mathematical Monthly. 74 (1). Taylor & Francis, Ltd.: 38–55. doi:10.2307/2314867. JSTOR 2314867.
    ^ Schubring, Gert; Furinghetti, Fulvia; Siu, Man Keung (August 2012). "Introduction: the history of mathematics teaching. Indicators for modernization processes in societies". ZDM Mathematics Education. 44 (4): 457–459. doi:10.1007/s11858-012-0445-7. S2CID 145507519.
    ^ von Davier, Matthias; Foy, Pierre; Martin, Michael O.; Mullis, Ina V.S. (2020). "Examining eTIMSS Country Differences Between eTIMSS Data and Bridge Data: A Look at Country-Level Mode of Administration Effects". TIMSS 2019 International Results in Mathematics and Science (PDF). TIMSS & PIRLS International Study Center, Lynch School of Education and Human Development and International Association for the Evaluation of Educational Achievement. p. 13.1. ISBN 978-1-889938-54-7. Archived (PDF) from the original on November 29, 2022. Retrieved November 29, 2022.
    ^ Rowan-Kenyon, Heather T.; Swan, Amy K.; Creager, Marie F. (March 2012). "Social Cognitive Factors, Support, and Engagement: Early Adolescents' Math Interests as Precursors to Choice of Career" (PDF). The Career Development Quarterly. 60 (1): 2–15. doi:10.1002/j.2161-0045.2012.00001.x. Archived (PDF) from the original on November 22, 2023. Retrieved November 29, 2022.
    ^ Luttenberger, Silke; Wimmer, Sigrid; Paechter, Manuela (2018). "Spotlight on math anxiety". Psychology Research and Behavior Management. 11: 311–322. doi:10.2147/PRBM.S141421. PMC 6087017. PMID 30123014.
    ^ Yaftian, Narges (June 2, 2015). "The Outlook of the Mathematicians' Creative Processes". Procedia - Social and Behavioral Sciences. 191: 2519–2525. doi:10.1016/j.sbspro.2015.04.617.
    ^ Nadjafikhah, Mehdi; Yaftian, Narges (October 10, 2013). "The Frontage of Creativity and Mathematical Creativity". Procedia - Social and Behavioral Sciences. 90: 344–350. doi:10.1016/j.sbspro.2013.07.101.
    ^ van der Poorten, A. (1979). "A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)" (PDF). The Mathematical Intelligencer. 1 (4): 195–203. doi:10.1007/BF03028234. S2CID 121589323. Archived (PDF) from the original on September 6, 2015. Retrieved November 22, 2022.
    ^ Petkovi, Miodrag (September 2, 2009). Famous Puzzles of Great Mathematicians. American Mathematical Society. pp. xiii–xiv. ISBN 978-0-8218-4814-2. Retrieved November 25, 2022.
    ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. ISBN 978-0-521-42706-7. Retrieved November 22, 2022. See also A Mathematician's Apology.
    ^ Alon, Noga; Goldston, Dan; Sárközy, András; Szabados, József; Tenenbaum, Gérald; Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015). Alladi, Krishnaswami; Krantz, Steven G. (eds.). "Reflections on Paul Erdős on His Birth Centenary, Part II". Notices of the American Mathematical Society. 62 (3): 226–247. doi:10.1090/noti1223.
    ^ See, for example Bertrand Russell's statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his History of Western Philosophy. 1919. p. 60.
    ^ Cazden, Norman (October 1959). "Musical intervals and simple number ratios". Journal of Research in Music Education. 7 (2): 197–220. doi:10.1177/002242945900700205. JSTOR 3344215. S2CID 220636812.
    ^ Budden, F. J. (October 1967). "Modern mathematics and music". The Mathematical Gazette. 51 (377). Cambridge University Press ({CUP}): 204–215. doi:10.2307/3613237. JSTOR 3613237. S2CID 126119711.
    ^ Enquist, Magnus; Arak, Anthony (November 1994). "Symmetry, beauty and evolution". Nature. 372 (6502): 169–172. Bibcode:1994Natur.372..169E. doi:10.1038/372169a0. ISSN 1476-4687. PMID 7969448. S2CID 4310147. Archived from the original on December 28, 2022. Retrieved December 29, 2022.
    ^ Hestenes, David (1999). "Symmetry Groups" (PDF). geocalc.clas.asu.edu. Archived (PDF) from the original on January 1, 2023. Retrieved December 29, 2022.
    ^ Bender, Sara (September 2020). "The Rorschach Test". In Carducci, Bernardo J.; Nave, Christopher S.; Mio, Jeffrey S.; Riggio, Ronald E. (eds.). The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment. Wiley. pp. 367–376. doi:10.1002/9781119547167.ch131. ISBN 978-1-119-05751-2.
    ^ Weyl, Hermann (2015). Symmetry. Princeton Science Library. Vol. 47. Princeton University Press. p. 4. ISBN 978-1-4008-7434-7.
    ^ Bradley, Larry (2010). "Fractals – Chaos & Fractals". www.stsci.edu. Archived from the original on March 7, 2023. Retrieved December 29, 2022.
    ^ "Self-similarity". math.bu.edu. Archived from the original on March 2, 2023. Retrieved December 29, 2022.
    ^ Kissane, Barry (July 2009). Popular mathematics. 22nd Biennial Conference of The Australian Association of Mathematics Teachers. Fremantle, Western Australia: Australian Association of Mathematics Teachers. pp. 125–126. Archived from the original on March 7, 2023. Retrieved December 29, 2022.
    ^ Steen, L. A. (2012). Mathematics Today Twelve Informal Essays. Springer Science & Business Media. p. 2. ISBN 978-1-4613-9435-8. Retrieved January 3, 2023.
    ^ Pitici, Mircea (2017). The Best Writing on Mathematics 2016. Princeton University Press. ISBN 978-1-4008-8560-2. Retrieved January 3, 2023.
    ^ Monastyrsky 2001, p. 1: "The Fields Medal is now indisputably the best known and most influential award in mathematics."
    ^ Riehm 2002, pp. 778–782.
    ^ "Fields Medal | International Mathematical Union (IMU)". www.mathunion.org. Archived from the original on December 26, 2018. Retrieved February 21, 2022.
    ^ Jump up to: a b "Fields Medal". Maths History. Archived from the original on March 22, 2019. Retrieved February 21, 2022.
    ^ "Honours/Prizes Index". MacTutor History of Mathematics Archive. Archived from the original on December 17, 2021. Retrieved February 20, 2023.
    ^ "About the Abel Prize". The Abel Prize. Archived from the original on April 14, 2022. Retrieved January 23, 2022.
    ^ "Abel Prize | mathematics award". Encyclopedia Britannica. Archived from the original on January 26, 2020. Retrieved January 23, 2022.
    ^ "Chern Medal Award" (PDF). www.mathunion.org. June 1, 2009. Archived (PDF) from the original on June 17, 2009. Retrieved February 21, 2022.
    ^ "Chern Medal Award". International Mathematical Union (IMU). Archived from the original on August 25, 2010. Retrieved January 23, 2022.
    ^ "The Leroy P Steele Prize of the AMS". School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on November 17, 2022. Retrieved November 17, 2022.
    ^ Chern, S. S.; Hirzebruch, F. (September 2000). Wolf Prize in Mathematics. doi:10.1142/4149. ISBN 978-981-02-3945-9. Archived from the original on February 21, 2022. Retrieved February 21, 2022.
    ^ "The Wolf Prize". Wolf Foundation. Archived from the original on January 12, 2020. Retrieved January 23, 2022.
    ^ Jump up to: a b "Hilbert's Problems: 23 and Math". Simons Foundation. May 6, 2020. Archived from the original on January 23, 2022. Retrieved January 23, 2022.
    ^ Feferman, Solomon (1998). "Deciding the undecidable: Wrestling with Hilbert's problems" (PDF). In the Light of Logic. Logic and Computation in Philosophy series. Oxford University Press. pp. 3–27. ISBN 978-0-19-508030-8. Retrieved November 29, 2022.
    ^ "The Millennium Prize Problems". Clay Mathematics Institute. Archived from the original on July 3, 2015. Retrieved January 23, 2022.
    ^ "Millennium Problems". Clay Mathematics Institute. Archived from the original on December 20, 2018. Retrieved January 23, 2022.

Sources

    Bouleau, Nicolas (1999). Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur. L'Harmattan. ISBN 978-2-7384-8125-2.
    Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). New York: Wiley. ISBN 978-0-471-54397-8.
    Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Saunders. ISBN 978-0-03-029558-4.
    Kleiner, Israel (2007). Kleiner, Israel (ed.). A History of Abstract Algebra. Springer Science & Business Media. doi:10.1007/978-0-8176-4685-1. ISBN 978-0-8176-4684-4. LCCN 2007932362. OCLC 76935733. S2CID 117392219. Retrieved February 8, 2024.
    Kline, Morris (1990). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. ISBN 978-0-19-506135-2.
    Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal" (PDF). CMS – Notes – de la SMC. 33 (2–3). Canadian Mathematical Society. Archived (PDF) from the original on August 13, 2006. Retrieved July 28, 2006.
    Oakley, Barbara (2014). A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra). New York: Penguin Random House. ISBN 978-0-399-16524-5. "A Mind for Numbers."
    Peirce, Benjamin (1881). Peirce, Charles Sanders (ed.). "Linear associative algebra". American Journal of Mathematics. 4 (1–4) (Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.): 97–229. doi:10.2307/2369153. hdl:2027/hvd.32044030622997. JSTOR 2369153. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. Google Eprint and as an extract, D. Van Nostrand, 1882, Google Eprint. Retrieved November 17, 2020..
    Peterson, Ivars (1988). The Mathematical Tourist: Snapshots of Modern Mathematics. W. H. Freeman and Company. ISBN 0-7167-1953-3. LCCN 87033078. OCLC 17202382.
    Popper, Karl R. (1995). "On knowledge". In Search of a Better World: Lectures and Essays from Thirty Years. New York: Routledge. Bibcode:1992sbwl.book.....P. ISBN 978-0-415-13548-1.
    Riehm, Carl (August 2002). "The Early History of the Fields Medal" (PDF). Notices of the AMS. 49 (7): 778–782. Archived (PDF) from the original on October 26, 2006. Retrieved October 2, 2006.
    Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society. 43 (1): 101–109. doi:10.1090/S0273-0979-05-01069-4. Archived (PDF) from the original on July 23, 2006. Retrieved June 24, 2006.
    Whittle, Peter (1994). "Almost home". In Kelly, F.P. (ed.). Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. ISBN 978-0-471-94829-2. Archived from the original on December 19, 2013.

Further reading

    Benson, Donald C. (1999). The Moment of Proof: Mathematical Epiphanies. Oxford University Press. ISBN 978-0-19-513919-8.
    Davis, Philip J.; Hersh, Reuben (1999). The Mathematical Experience (Reprint ed.). Boston; New York: Mariner Books. ISBN 978-0-395-92968-1. Available online (registration required).
    Courant, Richard; Robbins, Herbert (1996). What Is Mathematics?: An Elementary Approach to Ideas and Methods (2nd ed.). New York: Oxford University Press. ISBN 978-0-19-510519-3.
    Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. W.W. Norton & Company. ISBN 978-0-393-04002-9.
    Hazewinkel, Michiel, ed. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers. – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and online. Archived December 20, 2012, at archive.today.
    Hodgkin, Luke Howard (2005). A History of Mathematics: From Mesopotamia to Modernity. Oxford University Press. ISBN 978-0-19-152383-0.
    Jourdain, Philip E. B. (2003). "The Nature of Mathematics". In James R. Newman (ed.). The World of Mathematics. Dover Publications. ISBN 978-0-486-43268-7.
    Pappas, Theoni (1986). The Joy Of Mathematics. San Carlos, California: Wide World Publishing. ISBN 978-0-933174-65-8.
    Waltershausen, Wolfgang Sartorius von (1965) [1856]. Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 978-3-253-01702-5.


en.wikipedia.org
Biology
Contributors to Wikimedia projects
85–109 minutes
Biology is the science of life. It spans multiple levels from biomolecules and cells to organisms and populations.

Biology is the scientific study of life.[1][2][3] It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field.[1][2][3] For instance, all organisms are made up of cells that process hereditary information encoded in genes, which can be transmitted to future generations. Another major theme is evolution, which explains the unity and diversity of life.[1][2][3] Energy processing is also important to life as it allows organisms to move, grow, and reproduce.[1][2][3] Finally, all organisms are able to regulate their own internal environments.[1][2][3][4][5]

Biologists are able to study life at multiple levels of organization,[1] from the molecular biology of a cell to the anatomy and physiology of plants and animals, and evolution of populations.[1][6] Hence, there are multiple subdisciplines within biology, each defined by the nature of their research questions and the tools that they use.[7][8][9] Like other scientists, biologists use the scientific method to make observations, pose questions, generate hypotheses, perform experiments, and form conclusions about the world around them.[1]

Life on Earth, which emerged more than 3.7 billion years ago,[10] is immensely diverse. Biologists have sought to study and classify the various forms of life, from prokaryotic organisms such as archaea and bacteria to eukaryotic organisms such as protists, fungi, plants, and animals. These various organisms contribute to the biodiversity of an ecosystem, where they play specialized roles in the cycling of nutrients and energy through their biophysical environment.
History
A drawing of a fly from facing up, with wing detail
Diagram of a fly from Robert Hooke's innovative Micrographia, 1665

The earliest of roots of science, which included medicine, can be traced to ancient Egypt and Mesopotamia in around 3000 to 1200 BCE.[11][12] Their contributions shaped ancient Greek natural philosophy.[11][12][13][14] Ancient Greek philosophers such as Aristotle (384–322 BCE) contributed extensively to the development of biological knowledge. He explored biological causation and the diversity of life. His successor, Theophrastus, began the scientific study of plants.[15] Scholars of the medieval Islamic world who wrote on biology included al-Jahiz (781–869), Al-Dīnawarī (828–896), who wrote on botany,[16] and Rhazes (865–925) who wrote on anatomy and physiology. Medicine was especially well studied by Islamic scholars working in Greek philosopher traditions, while natural history drew heavily on Aristotelian thought.

Biology began to quickly develop with Anton van Leeuwenhoek's dramatic improvement of the microscope. It was then that scholars discovered spermatozoa, bacteria, infusoria and the diversity of microscopic life. Investigations by Jan Swammerdam led to new interest in entomology and helped to develop techniques of microscopic dissection and staining.[17] Advances in microscopy had a profound impact on biological thinking. In the early 19th century, biologists pointed to the central importance of the cell. In 1838, Schleiden and Schwann began promoting the now universal ideas that (1) the basic unit of organisms is the cell and (2) that individual cells have all the characteristics of life, although they opposed the idea that (3) all cells come from the division of other cells, continuing to support spontaneous generation. However, Robert Remak and Rudolf Virchow were able to reify the third tenet, and by the 1860s most biologists accepted all three tenets which consolidated into cell theory.[18][19]

Meanwhile, taxonomy and classification became the focus of natural historians. Carl Linnaeus published a basic taxonomy for the natural world in 1735, and in the 1750s introduced scientific names for all his species.[20] Georges-Louis Leclerc, Comte de Buffon, treated species as artificial categories and living forms as malleable—even suggesting the possibility of common descent.[21]
In 1842, Charles Darwin penned his first sketch of On the Origin of Species.[22]

Serious evolutionary thinking originated with the works of Jean-Baptiste Lamarck, who presented a coherent theory of evolution.[23] The British naturalist Charles Darwin, combining the biogeographical approach of Humboldt, the uniformitarian geology of Lyell, Malthus's writings on population growth, and his own morphological expertise and extensive natural observations, forged a more successful evolutionary theory based on natural selection; similar reasoning and evidence led Alfred Russel Wallace to independently reach the same conclusions.[24][25]

The basis for modern genetics began with the work of Gregor Mendel in 1865.[26] This outlined the principles of biological inheritance.[27] However, the significance of his work was not realized until the early 20th century when evolution became a unified theory as the modern synthesis reconciled Darwinian evolution with classical genetics.[28] In the 1940s and early 1950s, a series of experiments by Alfred Hershey and Martha Chase pointed to DNA as the component of chromosomes that held the trait-carrying units that had become known as genes. A focus on new kinds of model organisms such as viruses and bacteria, along with the discovery of the double-helical structure of DNA by James Watson and Francis Crick in 1953, marked the transition to the era of molecular genetics. From the 1950s onwards, biology has been vastly extended in the molecular domain. The genetic code was cracked by Har Gobind Khorana, Robert W. Holley and Marshall Warren Nirenberg after DNA was understood to contain codons. The Human Genome Project was launched in 1990 to map the human genome.[29]
Chemical basis
Atoms and molecules

All organisms are made up of chemical elements;[30] oxygen, carbon, hydrogen, and nitrogen account for most (96%) of the mass of all organisms, with calcium, phosphorus, sulfur, sodium, chlorine, and magnesium constituting essentially all the remainder. Different elements can combine to form compounds such as water, which is fundamental to life.[30] Biochemistry is the study of chemical processes within and relating to living organisms. Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including molecular synthesis, modification, mechanisms, and interactions.
Water
Model of hydrogen bonds (1) between molecules of water

Life arose from the Earth's first ocean, which formed some 3.8 billion years ago.[31] Since then, water continues to be the most abundant molecule in every organism. Water is important to life because it is an effective solvent, capable of dissolving solutes such as sodium and chloride ions or other small molecules to form an aqueous solution. Once dissolved in water, these solutes are more likely to come in contact with one another and therefore take part in chemical reactions that sustain life.[31] In terms of its molecular structure, water is a small polar molecule with a bent shape formed by the polar covalent bonds of two hydrogen (H) atoms to one oxygen (O) atom (H2O).[31] Because the O–H bonds are polar, the oxygen atom has a slight negative charge and the two hydrogen atoms have a slight positive charge.[31] This polar property of water allows it to attract other water molecules via hydrogen bonds, which makes water cohesive.[31] Surface tension results from the cohesive force due to the attraction between molecules at the surface of the liquid.[31] Water is also adhesive as it is able to adhere to the surface of any polar or charged non-water molecules.[31] Water is denser as a liquid than it is as a solid (or ice).[31] This unique property of water allows ice to float above liquid water such as ponds, lakes, and oceans, thereby insulating the liquid below from the cold air above.[31] Water has the capacity to absorb energy, giving it a higher specific heat capacity than other solvents such as ethanol.[31] Thus, a large amount of energy is needed to break the hydrogen bonds between water molecules to convert liquid water into water vapor.[31] As a molecule, water is not completely stable as each water molecule continuously dissociates into hydrogen and hydroxyl ions before reforming into a water molecule again.[31] In pure water, the number of hydrogen ions balances (or equals) the number of hydroxyl ions, resulting in a pH that is neutral.
Organic compounds
Organic compounds such as glucose are vital to organisms.

Organic compounds are molecules that contain carbon bonded to another element such as hydrogen.[31] With the exception of water, nearly all the molecules that make up each organism contain carbon.[31][32] Carbon can form covalent bonds with up to four other atoms, enabling it to form diverse, large, and complex molecules.[31][32] For example, a single carbon atom can form four single covalent bonds such as in methane, two double covalent bonds such as in carbon dioxide (CO2), or a triple covalent bond such as in carbon monoxide (CO). Moreover, carbon can form very long chains of interconnecting carbon–carbon bonds such as octane or ring-like structures such as glucose.

The simplest form of an organic molecule is the hydrocarbon, which is a large family of organic compounds that are composed of hydrogen atoms bonded to a chain of carbon atoms. A hydrocarbon backbone can be substituted by other elements such as oxygen (O), hydrogen (H), phosphorus (P), and sulfur (S), which can change the chemical behavior of that compound.[31] Groups of atoms that contain these elements (O-, H-, P-, and S-) and are bonded to a central carbon atom or skeleton are called functional groups.[31] There are six prominent functional groups that can be found in organisms: amino group, carboxyl group, carbonyl group, hydroxyl group, phosphate group, and sulfhydryl group.[31]

In 1953, the Miller–Urey experiment showed that organic compounds could be synthesized abiotically within a closed system mimicking the conditions of early Earth, thus suggesting that complex organic molecules could have arisen spontaneously in early Earth (see abiogenesis).[33][31]
Macromolecules
The (a) primary, (b) secondary, (c) tertiary, and (d) quaternary structures of a hemoglobin protein

Macromolecules are large molecules made up of smaller subunits or monomers.[34] Monomers include sugars, amino acids, and nucleotides.[35] Carbohydrates include monomers and polymers of sugars.[36] Lipids are the only class of macromolecules that are not made up of polymers. They include steroids, phospholipids, and fats,[35] largely nonpolar and hydrophobic (water-repelling) substances.[37] Proteins are the most diverse of the macromolecules. They include enzymes, transport proteins, large signaling molecules, antibodies, and structural proteins. The basic unit (or monomer) of a protein is an amino acid.[34] Twenty amino acids are used in proteins.[34] Nucleic acids are polymers of nucleotides.[38] Their function is to store, transmit, and express hereditary information.[35]
Cells

Cell theory states that cells are the fundamental units of life, it a unity of Protoplasm made of cytoplasm and nucleus surrounded by a cell membrane.[39] that all living things are composed of one or more cells, and that all cells arise from preexisting cells through cell division.[40] Most cells are very small, with diameters ranging from 1 to 100 micrometers and are therefore only visible under a light or electron microscope.[41] There are generally two types of cells: eukaryotic cells, which contain a nucleus, and prokaryotic cells, which do not. Prokaryotes are single-celled organisms such as bacteria, whereas eukaryotes can be single-celled or multicellular. In multicellular organisms, every cell in the organism's body is derived ultimately from a single cell in a fertilized egg.
Cell structure
Structure of an animal cell depicting various organelles

Every cell is enclosed within a cell membrane that separates its cytoplasm from the extracellular space.[42] A cell membrane consists of a lipid bilayer, including cholesterols that sit between phospholipids to maintain their fluidity at various temperatures. Cell membranes are semipermeable, allowing small molecules such as oxygen, carbon dioxide, and water to pass through while restricting the movement of larger molecules and charged particles such as ions.[43] Cell membranes also contains membrane proteins, including integral membrane proteins that go across the membrane serving as membrane transporters, and peripheral proteins that loosely attach to the outer side of the cell membrane, acting as enzymes shaping the cell.[44] Cell membranes are involved in various cellular processes such as cell adhesion, storing electrical energy, and cell signalling and serve as the attachment surface for several extracellular structures such as a cell wall, glycocalyx, and cytoskeleton.
Structure of a plant cell

Within the cytoplasm of a cell, there are many biomolecules such as proteins and nucleic acids.[45] In addition to biomolecules, eukaryotic cells have specialized structures called organelles that have their own lipid bilayers or are spatially units.[46] These organelles include the cell nucleus, which contains most of the cell's DNA, or mitochondria, which generates adenosine triphosphate (ATP) to power cellular processes. Other organelles such as endoplasmic reticulum and Golgi apparatus play a role in the synthesis and packaging of proteins, respectively. Biomolecules such as proteins can be engulfed by lysosomes, another specialized organelle. Plant cells have additional organelles that distinguish them from animal cells such as a cell wall that provides support for the plant cell, chloroplasts that harvest sunlight energy to produce sugar, and vacuoles that provide storage and structural support as well as being involved in reproduction and breakdown of plant seeds.[46] Eukaryotic cells also have cytoskeleton that is made up of microtubules, intermediate filaments, and microfilaments, all of which provide support for the cell and are involved in the movement of the cell and its organelles.[46] In terms of their structural composition, the microtubules are made up of tubulin (e.g., α-tubulin and β-tubulin whereas intermediate filaments are made up of fibrous proteins.[46] Microfilaments are made up of actin molecules that interact with other strands of proteins.[46]
Metabolism
Example of an enzyme-catalysed exothermic reaction

All cells require energy to sustain cellular processes. Metabolism is the set of chemical reactions in an organism. The three main purposes of metabolism are: the conversion of food to energy to run cellular processes; the conversion of food/fuel to monomer building blocks; and the elimination of metabolic wastes. These enzyme-catalyzed reactions allow organisms to grow and reproduce, maintain their structures, and respond to their environments. Metabolic reactions may be categorized as catabolic—the breaking down of compounds (for example, the breaking down of glucose to pyruvate by cellular respiration); or anabolic—the building up (synthesis) of compounds (such as proteins, carbohydrates, lipids, and nucleic acids). Usually, catabolism releases energy, and anabolism consumes energy. The chemical reactions of metabolism are organized into metabolic pathways, in which one chemical is transformed through a series of steps into another chemical, each step being facilitated by a specific enzyme. Enzymes are crucial to metabolism because they allow organisms to drive desirable reactions that require energy that will not occur by themselves, by coupling them to spontaneous reactions that release energy. Enzymes act as catalysts—they allow a reaction to proceed more rapidly without being consumed by it—by reducing the amount of activation energy needed to convert reactants into products. Enzymes also allow the regulation of the rate of a metabolic reaction, for example in response to changes in the cell's environment or to signals from other cells.
Cellular respiration
Respiration in a eukaryotic cell

Cellular respiration is a set of metabolic reactions and processes that take place in cells to convert chemical energy from nutrients into adenosine triphosphate (ATP), and then release waste products.[47] The reactions involved in respiration are catabolic reactions, which break large molecules into smaller ones, releasing energy. Respiration is one of the key ways a cell releases chemical energy to fuel cellular activity. The overall reaction occurs in a series of biochemical steps, some of which are redox reactions. Although cellular respiration is technically a combustion reaction, it clearly does not resemble one when it occurs in a cell because of the slow, controlled release of energy from the series of reactions.

Sugar in the form of glucose is the main nutrient used by animal and plant cells in respiration. Cellular respiration involving oxygen is called aerobic respiration, which has four stages: glycolysis, citric acid cycle (or Krebs cycle), electron transport chain, and oxidative phosphorylation.[48] Glycolysis is a metabolic process that occurs in the cytoplasm whereby glucose is converted into two pyruvates, with two net molecules of ATP being produced at the same time.[48] Each pyruvate is then oxidized into acetyl-CoA by the pyruvate dehydrogenase complex, which also generates NADH and carbon dioxide. Acetyl-Coa enters the citric acid cycle, which takes places inside the mitochondrial matrix. At the end of the cycle, the total yield from 1 glucose (or 2 pyruvates) is 6 NADH, 2 FADH2, and 2 ATP molecules. Finally, the next stage is oxidative phosphorylation, which in eukaryotes, occurs in the mitochondrial cristae. Oxidative phosphorylation comprises the electron transport chain, which is a series of four protein complexes that transfer electrons from one complex to another, thereby releasing energy from NADH and FADH2 that is coupled to the pumping of protons (hydrogen ions) across the inner mitochondrial membrane (chemiosmosis), which generates a proton motive force.[48] Energy from the proton motive force drives the enzyme ATP synthase to synthesize more ATPs by phosphorylating ADPs. The transfer of electrons terminates with molecular oxygen being the final electron acceptor.

If oxygen were not present, pyruvate would not be metabolized by cellular respiration but undergoes a process of fermentation. The pyruvate is not transported into the mitochondrion but remains in the cytoplasm, where it is converted to waste products that may be removed from the cell. This serves the purpose of oxidizing the electron carriers so that they can perform glycolysis again and removing the excess pyruvate. Fermentation oxidizes NADH to NAD+ so it can be re-used in glycolysis. In the absence of oxygen, fermentation prevents the buildup of NADH in the cytoplasm and provides NAD+ for glycolysis. This waste product varies depending on the organism. In skeletal muscles, the waste product is lactic acid. This type of fermentation is called lactic acid fermentation. In strenuous exercise, when energy demands exceed energy supply, the respiratory chain cannot process all of the hydrogen atoms joined by NADH. During anaerobic glycolysis, NAD+ regenerates when pairs of hydrogen combine with pyruvate to form lactate. Lactate formation is catalyzed by lactate dehydrogenase in a reversible reaction. Lactate can also be used as an indirect precursor for liver glycogen. During recovery, when oxygen becomes available, NAD+ attaches to hydrogen from lactate to form ATP. In yeast, the waste products are ethanol and carbon dioxide. This type of fermentation is known as alcoholic or ethanol fermentation. The ATP generated in this process is made by substrate-level phosphorylation, which does not require oxygen.
Photosynthesis
Photosynthesis changes sunlight into chemical energy, splits water to liberate O2, and fixes CO2 into sugar.

Photosynthesis is a process used by plants and other organisms to convert light energy into chemical energy that can later be released to fuel the organism's metabolic activities via cellular respiration. This chemical energy is stored in carbohydrate molecules, such as sugars, which are synthesized from carbon dioxide and water.[49][50][51] In most cases, oxygen is released as a waste product. Most plants, algae, and cyanobacteria perform photosynthesis, which is largely responsible for producing and maintaining the oxygen content of the Earth's atmosphere, and supplies most of the energy necessary for life on Earth.[52]

Photosynthesis has four stages: Light absorption, electron transport, ATP synthesis, and carbon fixation.[48] Light absorption is the initial step of photosynthesis whereby light energy is absorbed by chlorophyll pigments attached to proteins in the thylakoid membranes. The absorbed light energy is used to remove electrons from a donor (water) to a primary electron acceptor, a quinone designated as Q. In the second stage, electrons move from the quinone primary electron acceptor through a series of electron carriers until they reach a final electron acceptor, which is usually the oxidized form of NADP+, which is reduced to NADPH, a process that takes place in a protein complex called photosystem I (PSI). The transport of electrons is coupled to the movement of protons (or hydrogen) from the stroma to the thylakoid membrane, which forms a pH gradient across the membrane as hydrogen becomes more concentrated in the lumen than in the stroma. This is analogous to the proton-motive force generated across the inner mitochondrial membrane in aerobic respiration.[48]

During the third stage of photosynthesis, the movement of protons down their concentration gradients from the thylakoid lumen to the stroma through the ATP synthase is coupled to the synthesis of ATP by that same ATP synthase.[48] The NADPH and ATPs generated by the light-dependent reactions in the second and third stages, respectively, provide the energy and electrons to drive the synthesis of glucose by fixing atmospheric carbon dioxide into existing organic carbon compounds, such as ribulose bisphosphate (RuBP) in a sequence of light-independent (or dark) reactions called the Calvin cycle.[53]
Cell signaling

Cell signaling (or communication) is the ability of cells to receive, process, and transmit signals with its environment and with itself.[54][55] Signals can be non-chemical such as light, electrical impulses, and heat, or chemical signals (or ligands) that interact with receptors, which can be found embedded in the cell membrane of another cell or located deep inside a cell.[56][55] There are generally four types of chemical signals: autocrine, paracrine, juxtacrine, and hormones.[56] In autocrine signaling, the ligand affects the same cell that releases it. Tumor cells, for example, can reproduce uncontrollably because they release signals that initiate their own self-division. In paracrine signaling, the ligand diffuses to nearby cells and affects them. For example, brain cells called neurons release ligands called neurotransmitters that diffuse across a synaptic cleft to bind with a receptor on an adjacent cell such as another neuron or muscle cell. In juxtacrine signaling, there is direct contact between the signaling and responding cells. Finally, hormones are ligands that travel through the circulatory systems of animals or vascular systems of plants to reach their target cells. Once a ligand binds with a receptor, it can influence the behavior of another cell, depending on the type of receptor. For instance, neurotransmitters that bind with an inotropic receptor can alter the excitability of a target cell. Other types of receptors include protein kinase receptors (e.g., receptor for the hormone insulin) and G protein-coupled receptors. Activation of G protein-coupled receptors can initiate second messenger cascades. The process by which a chemical or physical signal is transmitted through a cell as a series of molecular events is called signal transduction
Cell cycle
In meiosis, the chromosomes duplicate and the homologous chromosomes exchange genetic information during meiosis I. The daughter cells divide again in meiosis II to form haploid gametes.

The cell cycle is a series of events that take place in a cell that cause it to divide into two daughter cells. These events include the duplication of its DNA and some of its organelles, and the subsequent partitioning of its cytoplasm into two daughter cells in a process called cell division.[57] In eukaryotes (i.e., animal, plant, fungal, and protist cells), there are two distinct types of cell division: mitosis and meiosis.[58] Mitosis is part of the cell cycle, in which replicated chromosomes are separated into two new nuclei. Cell division gives rise to genetically identical cells in which the total number of chromosomes is maintained. In general, mitosis (division of the nucleus) is preceded by the S stage of interphase (during which the DNA is replicated) and is often followed by telophase and cytokinesis; which divides the cytoplasm, organelles and cell membrane of one cell into two new cells containing roughly equal shares of these cellular components. The different stages of mitosis all together define the mitotic phase of an animal cell cycle—the division of the mother cell into two genetically identical daughter cells.[59] The cell cycle is a vital process by which a single-celled fertilized egg develops into a mature organism, as well as the process by which hair, skin, blood cells, and some internal organs are renewed. After cell division, each of the daughter cells begin the interphase of a new cycle. In contrast to mitosis, meiosis results in four haploid daughter cells by undergoing one round of DNA replication followed by two divisions.[60] Homologous chromosomes are separated in the first division (meiosis I), and sister chromatids are separated in the second division (meiosis II). Both of these cell division cycles are used in the process of sexual reproduction at some point in their life cycle. Both are believed to be present in the last eukaryotic common ancestor.

Prokaryotes (i.e., archaea and bacteria) can also undergo cell division (or binary fission). Unlike the processes of mitosis and meiosis in eukaryotes, binary fission takes in prokaryotes takes place without the formation of a spindle apparatus on the cell. Before binary fission, DNA in the bacterium is tightly coiled. After it has uncoiled and duplicated, it is pulled to the separate poles of the bacterium as it increases the size to prepare for splitting. Growth of a new cell wall begins to separate the bacterium (triggered by FtsZ polymerization and "Z-ring" formation)[61] The new cell wall (septum) fully develops, resulting in the complete split of the bacterium. The new daughter cells have tightly coiled DNA rods, ribosomes, and plasmids.
Sexual reproduction and meiosis

Meiosis is a central feature of sexual reproduction in eukaryotes, and the most fundamental function of meiosis appears to be conservation of the integrity of the genome that is passed on to progeny by parents.[62][63] Two aspects of sexual reproduction, meiotic recombination and outcrossing, are likely maintained respectively by the adaptive advantages of recombinational repair of genomic DNA damage and genetic complementation which masks the expression of deleterious recessive mutations.[64]

The beneficial effect of genetic complementation, derived from outcrossing (cross-fertilization) is also referred to as hybrid vigor or heterosis. Charles Darwin in his 1878 book The Effects of Cross and Self-Fertilization in the Vegetable Kingdom[65] at the start of chapter XII noted “The first and most important of the conclusions which may be drawn from the observations given in this volume, is that generally cross-fertilisation is beneficial and self-fertilisation often injurious, at least with the plants on which I experimented.” Genetic variation, often produced as a byproduct of sexual reproduction, may provide long-term advantages to those sexual lineages that engage in outcrossing.[64]
Genetics
Inheritance
Punnett square depicting a cross between two pea plants heterozygous for purple (B) and white (b) blossoms

Genetics is the scientific study of inheritance.[66][67][68] Mendelian inheritance, specifically, is the process by which genes and traits are passed on from parents to offspring.[27] It has several principles. The first is that genetic characteristics, alleles, are discrete and have alternate forms (e.g., purple vs. white or tall vs. dwarf), each inherited from one of two parents. Based on the law of dominance and uniformity, which states that some alleles are dominant while others are recessive; an organism with at least one dominant allele will display the phenotype of that dominant allele. During gamete formation, the alleles for each gene segregate, so that each gamete carries only one allele for each gene. Heterozygotic individuals produce gametes with an equal frequency of two alleles. Finally, the law of independent assortment, states that genes of different traits can segregate independently during the formation of gametes, i.e., genes are unlinked. An exception to this rule would include traits that are sex-linked. Test crosses can be performed to experimentally determine the underlying genotype of an organism with a dominant phenotype.[69] A Punnett square can be used to predict the results of a test cross. The chromosome theory of inheritance, which states that genes are found on chromosomes, was supported by Thomas Morgans's experiments with fruit flies, which established the sex linkage between eye color and sex in these insects.[70]
Genes and DNA

Further information: Gene and DNA
Bases lie between two spiraling DNA strands.

A gene is a unit of heredity that corresponds to a region of deoxyribonucleic acid (DNA) that carries genetic information that controls form or function of an organism. DNA is composed of two polynucleotide chains that coil around each other to form a double helix.[71] It is found as linear chromosomes in eukaryotes, and circular chromosomes in prokaryotes. The set of chromosomes in a cell is collectively known as its genome. In eukaryotes, DNA is mainly in the cell nucleus.[72] In prokaryotes, the DNA is held within the nucleoid.[73] The genetic information is held within genes, and the complete assemblage in an organism is called its genotype.[74] DNA replication is a semiconservative process whereby each strand serves as a template for a new strand of DNA.[71] Mutations are heritable changes in DNA.[71] They can arise spontaneously as a result of replication errors that were not corrected by proofreading or can be induced by an environmental mutagen such as a chemical (e.g., nitrous acid, benzopyrene) or radiation (e.g., x-ray, gamma ray, ultraviolet radiation, particles emitted by unstable isotopes).[71] Mutations can lead to phenotypic effects such as loss-of-function, gain-of-function, and conditional mutations.[71] Some mutations are beneficial, as they are a source of genetic variation for evolution.[71] Others are harmful if they were to result in a loss of function of genes needed for survival.[71]
Gene expression
The extended central dogma of molecular biology includes all the processes involved in the flow of genetic information.

Gene expression is the molecular process by which a genotype encoded in DNA gives rise to an observable phenotype in the proteins of an organism's body. This process is summarized by the central dogma of molecular biology, which was formulated by Francis Crick in 1958.[75][76][77] According to the Central Dogma, genetic information flows from DNA to RNA to protein. There are two gene expression processes: transcription (DNA to RNA) and translation (RNA to protein).[78]
Gene regulation

The regulation of gene expression by environmental factors and during different stages of development can occur at each step of the process such as transcription, RNA splicing, translation, and post-translational modification of a protein.[79] Gene expression can be influenced by positive or negative regulation, depending on which of the two types of regulatory proteins called transcription factors bind to the DNA sequence close to or at a promoter.[79] A cluster of genes that share the same promoter is called an operon, found mainly in prokaryotes and some lower eukaryotes (e.g., Caenorhabditis elegans).[79][80] In positive regulation of gene expression, the activator is the transcription factor that stimulates transcription when it binds to the sequence near or at the promoter. Negative regulation occurs when another transcription factor called a repressor binds to a DNA sequence called an operator, which is part of an operon, to prevent transcription. Repressors can be inhibited by compounds called inducers (e.g., allolactose), thereby allowing transcription to occur.[79] Specific genes that can be activated by inducers are called inducible genes, in contrast to constitutive genes that are almost constantly active.[79] In contrast to both, structural genes encode proteins that are not involved in gene regulation.[79] In addition to regulatory events involving the promoter, gene expression can also be regulated by epigenetic changes to chromatin, which is a complex of DNA and protein found in eukaryotic cells.[79]
Genes, development, and evolution

Development is the process by which a multicellular organism (plant or animal) goes through a series of changes, starting from a single cell, and taking on various forms that are characteristic of its life cycle.[81] There are four key processes that underlie development: Determination, differentiation, morphogenesis, and growth. Determination sets the developmental fate of a cell, which becomes more restrictive during development. Differentiation is the process by which specialized cells from less specialized cells such as stem cells.[82][83] Stem cells are undifferentiated or partially differentiated cells that can differentiate into various types of cells and proliferate indefinitely to produce more of the same stem cell.[84] Cellular differentiation dramatically changes a cell's size, shape, membrane potential, metabolic activity, and responsiveness to signals, which are largely due to highly controlled modifications in gene expression and epigenetics. With a few exceptions, cellular differentiation almost never involves a change in the DNA sequence itself.[85] Thus, different cells can have very different physical characteristics despite having the same genome. Morphogenesis, or the development of body form, is the result of spatial differences in gene expression.[81] A small fraction of the genes in an organism's genome called the developmental-genetic toolkit control the development of that organism. These toolkit genes are highly conserved among phyla, meaning that they are ancient and very similar in widely separated groups of animals. Differences in deployment of toolkit genes affect the body plan and the number, identity, and pattern of body parts. Among the most important toolkit genes are the Hox genes. Hox genes determine where repeating parts, such as the many vertebrae of snakes, will grow in a developing embryo or larva.[86]
Evolution
Evolutionary processes
Natural selection for darker traits

Evolution is a central organizing concept in biology. It is the change in heritable characteristics of populations over successive generations.[87][88] In artificial selection, animals were selectively bred for specific traits. [89] Given that traits are inherited, populations contain a varied mix of traits, and reproduction is able to increase any population, Darwin argued that in the natural world, it was nature that played the role of humans in selecting for specific traits.[89] Darwin inferred that individuals who possessed heritable traits better adapted to their environments are more likely to survive and produce more offspring than other individuals.[89] He further inferred that this would lead to the accumulation of favorable traits over successive generations, thereby increasing the match between the organisms and their environment.[90][91][92][89][93]
Speciation

A species is a group of organisms that mate with one another and speciation is the process by which one lineage splits into two lineages as a result of having evolved independently from each other.[94] For speciation to occur, there has to be reproductive isolation.[94] Reproductive isolation can result from incompatibilities between genes as described by Bateson–Dobzhansky–Muller model. Reproductive isolation also tends to increase with genetic divergence. Speciation can occur when there are physical barriers that divide an ancestral species, a process known as allopatric speciation.[94]
Phylogeny
Phylogenetic tree showing the domains of bacteria, archaea, and eukaryotes


A phylogeny is an evolutionary history of a specific group of organisms or their genes.[95] It can be represented using a phylogenetic tree, a diagram showing lines of descent among organisms or their genes. Each line drawn on the time axis of a tree represents a lineage of descendants of a particular species or population. When a lineage divides into two, it is represented as a fork or split on the phylogenetic tree.[95] Phylogenetic trees are the basis for comparing and grouping different species.[95] Different species that share a feature inherited from a common ancestor are described as having homologous features (or synapomorphy).[96][97][95] Phylogeny provides the basis of biological classification.[95] This classification system is rank-based, with the highest rank being the domain followed by kingdom, phylum, class, order, family, genus, and species.[95] All organisms can be classified as belonging to one of three domains: Archaea (originally Archaebacteria); bacteria (originally eubacteria), or eukarya (includes the protist, fungi, plant, and animal kingdoms).[98]
History of life

The history of life on Earth traces how organisms have evolved from the earliest emergence of life to present day. Earth formed about 4.5 billion years ago and all life on Earth, both living and extinct, descended from a last universal common ancestor that lived about 3.5 billion years ago.[99][100] Geologists have developed a geologic time scale that divides the history of the Earth into major divisions, starting with four eons (Hadean, Archean, Proterozoic, and Phanerozoic), the first three of which are collectively known as the Precambrian, which lasted approximately 4 billion years.[101] Each eon can be divided into eras, with the Phanerozoic eon that began 539 million years ago[102] being subdivided into Paleozoic, Mesozoic, and Cenozoic eras.[101] These three eras together comprise eleven periods (Cambrian, Ordovician, Silurian, Devonian, Carboniferous, Permian, Triassic, Jurassic, Cretaceous, Tertiary, and Quaternary).[101]

The similarities among all known present-day species indicate that they have diverged through the process of evolution from their common ancestor.[103] Biologists regard the ubiquity of the genetic code as evidence of universal common descent for all bacteria, archaea, and eukaryotes.[104][10][105][106] Microbial mats of coexisting bacteria and archaea were the dominant form of life in the early Archean epoch and many of the major steps in early evolution are thought to have taken place in this environment.[107] The earliest evidence of eukaryotes dates from 1.85 billion years ago,[108][109] and while they may have been present earlier, their diversification accelerated when they started using oxygen in their metabolism. Later, around 1.7 billion years ago, multicellular organisms began to appear, with differentiated cells performing specialised functions.[110]

Algae-like multicellular land plants are dated back even to about 1 billion years ago,[111] although evidence suggests that microorganisms formed the earliest terrestrial ecosystems, at least 2.7 billion years ago.[112] Microorganisms are thought to have paved the way for the inception of land plants in the Ordovician period. Land plants were so successful that they are thought to have contributed to the Late Devonian extinction event.[113]

Ediacara biota appear during the Ediacaran period,[114] while vertebrates, along with most other modern phyla originated about 525 million years ago during the Cambrian explosion.[115] During the Permian period, synapsids, including the ancestors of mammals, dominated the land,[116] but most of this group became extinct in the Permian–Triassic extinction event 252 million years ago.[117] During the recovery from this catastrophe, archosaurs became the most abundant land vertebrates;[118] one archosaur group, the dinosaurs, dominated the Jurassic and Cretaceous periods.[119] After the Cretaceous–Paleogene extinction event 66 million years ago killed off the non-avian dinosaurs,[120] mammals increased rapidly in size and diversity.[121] Such mass extinctions may have accelerated evolution by providing opportunities for new groups of organisms to diversify.[122]
Diversity
Bacteria and Archaea
Bacteria – Gemmatimonas aurantiaca (-=1 Micrometer)

Bacteria are a type of cell that constitute a large domain of prokaryotic microorganisms. Typically a few micrometers in length, bacteria have a number of shapes, ranging from spheres to rods and spirals. Bacteria were among the first life forms to appear on Earth, and are present in most of its habitats. Bacteria inhabit soil, water, acidic hot springs, radioactive waste,[123] and the deep biosphere of the Earth's crust. Bacteria also live in symbiotic and parasitic relationships with plants and animals. Most bacteria have not been characterised, and only about 27 percent of the bacterial phyla have species that can be grown in the laboratory.[124]
Archaea – Halobacteria

Archaea constitute the other domain of prokaryotic cells and were initially classified as bacteria, receiving the name archaebacteria (in the Archaebacteria kingdom), a term that has fallen out of use.[125] Archaeal cells have unique properties separating them from the other two domains, Bacteria and Eukaryota. Archaea are further divided into multiple recognized phyla. Archaea and bacteria are generally similar in size and shape, although a few archaea have very different shapes, such as the flat and square cells of Haloquadratum walsbyi.[126] Despite this morphological similarity to bacteria, archaea possess genes and several metabolic pathways that are more closely related to those of eukaryotes, notably for the enzymes involved in transcription and translation. Other aspects of archaeal biochemistry are unique, such as their reliance on ether lipids in their cell membranes,[127] including archaeols. Archaea use more energy sources than eukaryotes: these range from organic compounds, such as sugars, to ammonia, metal ions or even hydrogen gas. Salt-tolerant archaea (the Haloarchaea) use sunlight as an energy source, and other species of archaea fix carbon, but unlike plants and cyanobacteria, no known species of archaea does both. Archaea reproduce asexually by binary fission, fragmentation, or budding; unlike bacteria, no known species of Archaea form endospores.

The first observed archaea were extremophiles, living in extreme environments, such as hot springs and salt lakes with no other organisms. Improved molecular detection tools led to the discovery of archaea in almost every habitat, including soil, oceans, and marshlands. Archaea are particularly numerous in the oceans, and the archaea in plankton may be one of the most abundant groups of organisms on the planet.

Archaea are a major part of Earth's life. They are part of the microbiota of all organisms. In the human microbiome, they are important in the gut, mouth, and on the skin.[128] Their morphological, metabolic, and geographical diversity permits them to play multiple ecological roles: carbon fixation; nitrogen cycling; organic compound turnover; and maintaining microbial symbiotic and syntrophic communities, for example.[129]
Eukaryotes
Euglena, a single-celled eukaryote that can both move and photosynthesize

Eukaryotes are hypothesized to have split from archaea, which was followed by their endosymbioses with bacteria (or symbiogenesis) that gave rise to mitochondria and chloroplasts, both of which are now part of modern-day eukaryotic cells.[130] The major lineages of eukaryotes diversified in the Precambrian about 1.5 billion years ago and can be classified into eight major clades: alveolates, excavates, stramenopiles, plants, rhizarians, amoebozoans, fungi, and animals.[130] Five of these clades are collectively known as protists, which are mostly microscopic eukaryotic organisms that are not plants, fungi, or animals.[130] While it is likely that protists share a common ancestor (the last eukaryotic common ancestor),[131] protists by themselves do not constitute a separate clade as some protists may be more closely related to plants, fungi, or animals than they are to other protists. Like groupings such as algae, invertebrates, or protozoans, the protist grouping is not a formal taxonomic group but is used for convenience.[130][132] Most protists are unicellular; these are called microbial eukaryotes.[130]

Plants are mainly multicellular organisms, predominantly photosynthetic eukaryotes of the kingdom Plantae, which would exclude fungi and some algae. Plant cells were derived by endosymbiosis of a cyanobacterium into an early eukaryote about one billion years ago, which gave rise to chloroplasts.[133] The first several clades that emerged following primary endosymbiosis were aquatic and most of the aquatic photosynthetic eukaryotic organisms are collectively described as algae, which is a term of convenience as not all algae are closely related.[133] Algae comprise several distinct clades such as glaucophytes, which are microscopic freshwater algae that may have resembled in form to the early unicellular ancestor of Plantae.[133] Unlike glaucophytes, the other algal clades such as red and green algae are multicellular. Green algae comprise three major clades: chlorophytes, coleochaetophytes, and stoneworts.[133]

Fungi are eukaryotes that digest foods outside their bodies,[134] secreting digestive enzymes that break down large food molecules before absorbing them through their cell membranes. Many fungi are also saprobes, feeding on dead organic matter, making them important decomposers in ecological systems.[134]

Animals are multicellular eukaryotes. With few exceptions, animals consume organic material, breathe oxygen, are able to move, can reproduce sexually, and grow from a hollow sphere of cells, the blastula, during embryonic development. Over 1.5 million living animal species have been described—of which around 1 million are insects—but it has been estimated there are over 7 million animal species in total. They have complex interactions with each other and their environments, forming intricate food webs.[135]
Viruses
Bacteriophages attached to a bacterial cell wall

Viruses are submicroscopic infectious agents that replicate inside the cells of organisms.[136] Viruses infect all types of life forms, from animals and plants to microorganisms, including bacteria and archaea.[137][138] More than 6,000 virus species have been described in detail.[139] Viruses are found in almost every ecosystem on Earth and are the most numerous type of biological entity.[140][141]

The origins of viruses in the evolutionary history of life are unclear: some may have evolved from plasmids—pieces of DNA that can move between cells—while others may have evolved from bacteria. In evolution, viruses are an important means of horizontal gene transfer, which increases genetic diversity in a way analogous to sexual reproduction.[142] Because viruses possess some but not all characteristics of life, they have been described as "organisms at the edge of life",[143] and as self-replicators.[144]
Ecology

Ecology is the study of the distribution and abundance of life, the interaction between organisms and their environment.[145]
Ecosystems

The community of living (biotic) organisms in conjunction with the nonliving (abiotic) components (e.g., water, light, radiation, temperature, humidity, atmosphere, acidity, and soil) of their environment is called an ecosystem.[146][147][148] These biotic and abiotic components are linked together through nutrient cycles and energy flows.[149] Energy from the sun enters the system through photosynthesis and is incorporated into plant tissue. By feeding on plants and on one another, animals move matter and energy through the system. They also influence the quantity of plant and microbial biomass present. By breaking down dead organic matter, decomposers release carbon back to the atmosphere and facilitate nutrient cycling by converting nutrients stored in dead biomass back to a form that can be readily used by plants and other microbes.[150]
Populations
Reaching carrying capacity through a logistic growth curve

A population is the group of organisms of the same species that occupies an area and reproduce from generation to generation.[151][152][153][154][155] Population size can be estimated by multiplying population density by the area or volume. The carrying capacity of an environment is the maximum population size of a species that can be sustained by that specific environment, given the food, habitat, water, and other resources that are available.[156] The carrying capacity of a population can be affected by changing environmental conditions such as changes in the availability resources and the cost of maintaining them. In human populations, new technologies such as the Green revolution have helped increase the Earth's carrying capacity for humans over time, which has stymied the attempted predictions of impending population decline, the most famous of which was by Thomas Malthus in the 18th century.[151]
Communities
A (a) trophic pyramid and a (b) simplified food web. The trophic pyramid represents the biomass at each level.[157]

A community is a group of populations of species occupying the same geographical area at the same time. A biological interaction is the effect that a pair of organisms living together in a community have on each other. They can be either of the same species (intraspecific interactions), or of different species (interspecific interactions). These effects may be short-term, like pollination and predation, or long-term; both often strongly influence the evolution of the species involved. A long-term interaction is called a symbiosis. Symbioses range from mutualism, beneficial to both partners, to competition, harmful to both partners.[158] Every species participates as a consumer, resource, or both in consumer–resource interactions, which form the core of food chains or food webs.[159] There are different trophic levels within any food web, with the lowest level being the primary producers (or autotrophs) such as plants and algae that convert energy and inorganic material into organic compounds, which can then be used by the rest of the community.[52][160][161] At the next level are the heterotrophs, which are the species that obtain energy by breaking apart organic compounds from other organisms.[159] Heterotrophs that consume plants are primary consumers (or herbivores) whereas heterotrophs that consume herbivores are secondary consumers (or carnivores). And those that eat secondary consumers are tertiary consumers and so on. Omnivorous heterotrophs are able to consume at multiple levels. Finally, there are decomposers that feed on the waste products or dead bodies of organisms.[159] On average, the total amount of energy incorporated into the biomass of a trophic level per unit of time is about one-tenth of the energy of the trophic level that it consumes. Waste and dead material used by decomposers as well as heat lost from metabolism make up the other ninety percent of energy that is not consumed by the next trophic level.[162]
Biosphere
Fast carbon cycle showing the movement of carbon between land, atmosphere, and oceans in billions of tons per year. Yellow numbers are natural fluxes, red are human contributions, white are stored carbon. Effects of the slow carbon cycle, such as volcanic and tectonic activity, are not included.[163]

In the global ecosystem or biosphere, matter exists as different interacting compartments, which can be biotic or abiotic as well as accessible or inaccessible, depending on their forms and locations.[164] For example, matter from terrestrial autotrophs are both biotic and accessible to other organisms whereas the matter in rocks and minerals are abiotic and inaccessible. A biogeochemical cycle is a pathway by which specific elements of matter are turned over or moved through the biotic (biosphere) and the abiotic (lithosphere, atmosphere, and hydrosphere) compartments of Earth. There are biogeochemical cycles for nitrogen, carbon, and water.
Conservation

Conservation biology is the study of the conservation of Earth's biodiversity with the aim of protecting species, their habitats, and ecosystems from excessive rates of extinction and the erosion of biotic interactions.[165][166][167] It is concerned with factors that influence the maintenance, loss, and restoration of biodiversity and the science of sustaining evolutionary processes that engender genetic, population, species, and ecosystem diversity.[168][169][170][171] The concern stems from estimates suggesting that up to 50% of all species on the planet will disappear within the next 50 years,[172] which has contributed to poverty, starvation, and will reset the course of evolution on this planet.[173][174] Biodiversity affects the functioning of ecosystems, which provide a variety of services upon which people depend. Conservation biologists research and educate on the trends of biodiversity loss, species extinctions, and the negative effect these are having on our capabilities to sustain the well-being of human society. Organizations and citizens are responding to the current biodiversity crisis through conservation action plans that direct research, monitoring, and education programs that engage concerns at local through global scales.[175][168][169][170]
See also

    Biology in fiction
    Glossary of biology
    List of biological websites
    List of biologists
    List of biology journals
    List of biology topics
    List of life sciences
    List of omics topics in biology
    National Association of Biology Teachers
    Outline of biology
    Periodic table of life sciences in Tinbergen's four questions
    Science tourism
    Terminology of biology

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Further reading

    Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P. (2002). Molecular Biology of the Cell (4th ed.). Garland. ISBN 978-0-8153-3218-3. OCLC 145080076.
    Begon, M.; Townsend, C. R.; Harper, J. L. (2005). Ecology: From Individuals to Ecosystems (4th ed.). Blackwell Publishing Limited. ISBN 978-1-4051-1117-1. OCLC 57639896.
    Campbell, Neil (2004). Biology (7th ed.). Benjamin-Cummings Publishing Company. ISBN 978-0-8053-7146-8. OCLC 71890442.
    Colinvaux, Paul (1979). Why Big Fierce Animals are Rare: An Ecologist's Perspective (reissue ed.). Princeton University Press. ISBN 978-0-691-02364-9. OCLC 10081738.
    Mayr, Ernst (1982). The Growth of Biological Thought: Diversity, Evolution, and Inheritance. Harvard University Press. ISBN 978-0-674-36446-2. Archived from the original on 2015-10-03. Retrieved 2015-06-27.
    Hoagland, Mahlon (2001). The Way Life Works. Jones and Bartlett Publishers inc. ISBN 978-0-7637-1688-2. OCLC 223090105.
    Janovy, John (2004). On Becoming a Biologist (2nd ed.). Bison Books. ISBN 978-0-8032-7620-8. OCLC 55138571.
    Johnson, George B. (2005). Biology, Visualizing Life. Holt, Rinehart, and Winston. ISBN 978-0-03-016723-2. OCLC 36306648.
    Tobin, Allan; Dusheck, Jennie (2005). Asking About Life (3rd ed.). Belmont, California: Wadsworth. ISBN 978-0-534-40653-0.

External links

    Biology at Curlie
    OSU's Phylocode
    Biology Online – Wiki Dictionary
    MIT video lecture series on biology
    OneZoom Tree of Life

Journal links

    PLOS Biology A peer-reviewed, open-access journal published by the Public Library of Science
    Current Biology: General journal publishing original research from all areas of biology
    Biology Letters: A high-impact Royal Society journal publishing peer-reviewed biology papers of general interest
    Science: Internationally renowned AAAS science journal – see sections of the life sciences
    International Journal of Biological Sciences: A biological journal publishing significant peer-reviewed scientific papers
    Perspectives in Biology and Medicine: An interdisciplinary scholarly journal publishing essays of broad relevance