This text covers most of the areas that I teach. It has a good table of contents and index, but no glossary.

Ecology heavily uses modeling to simulate population dynamics, [137] [138] study ecosystems such as the predator-prey model, measure pollution diffusion, [139] or to assess climate change. [140] The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations. [141] 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. [142] 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. [182] 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. [183] 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: [25] 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. [5] 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. [167]Wise, David. "Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion". jwilson.coe.uga.edu. Archived from the original on June 1, 2019 . Retrieved October 26, 2019. 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. [121] [122] The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics". [25] However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge. Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics An explanation of the sigma (Σ) summation notation a b c d e f Kleiner, Israel (December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". Mathematics Magazine. Taylor & Francis, Ltd. 64 (5): 291–314. doi: 10.1080/0025570X.1991.11977625. JSTOR 2690647. Biology uses probability extensively – for example, in ecology or neurobiology. [137] Most of the discussion of probability in biology, however, centers on the concept of evolutionary fitness. [137]

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. [90] 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. [60]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. [211] To date, only one of these problems, the Poincaré conjecture, has been solved. [212] See also Musielak, Dora (2022). Leonhard Euler and the Foundations of Celestial Mechanics. Springer International Publishing. pp.1–183. ISBN 978-3-031-12322-1 . Retrieved March 19, 2023.

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 (384–322BC) this meaning was fully established. [14]

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. [b] [31]

There are no apparent interface issues, but the visual alignment of graphics and text is very ordinary, sometimes a little awkward - there has been no special effort made to improve the visual appeal or readability of the interface. Language is neutral other than references to men and women without another option. Word problems are mostly academic oriented as opposed to real world problems so there is little cultural bias. There are not multicultural examples. Number theory also studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number, and there are many questions about primes that appear simple but whose resolution continues to elude mathematicians.

In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind. [59] Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle. [185] Integration, measure theory and potential theory, all strongly related with probability theory on a continuum; 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] [4] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. 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. [41] Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas. Logic is the foundation that underlies mathematical logic and the rest of mathematics. It tries to formalize valid reasoning. In particular, it attempts to define what constitutes a proof.