History of mathematics

History of mathematics

Number theory:

rational numbers, perfect numbers & prime numbers - used by Pythagoreans 6th-5thC BC
- numbers were regarded by Pythagoreans as patterns of dots which form characteristic figures, as on the sides of a dice; although we use Arabic symbols, which have no resemblance to these dot-patterns, we still call numbers "figures", ie. shapes.
- hence square numbers, triangular, cubic, pyramidal numbers, etc.
- addition of even numbers formed oblong numbers, where the ratio of the sides represented exactly the concordant intervals of the musical octave
law of signs (ie. multiply 2 negative numbers results in a positive) - known by Babylonians & Indians in 3rdC BC
- but negative numbers rejected by Greeks & Europeans until Renaissance when used to represent debt in businesses!
irrational numbers accepted after theory of Eudoxus (c. 370 BC)
prime numbers:
- 300BC: Euclid gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way
- Euclid also showed that if the number 2ⁿ - 1 is prime then the number 2^n-1(2ⁿ - 1) is a perfect number
- 1603: Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes)
- early 17thC, Fermat proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares. The number of the form 2ⁿ - 1 also attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbers M_n because Mersenne studied them.
- 18thC: Euler founds Analytic Number Theory
perfect numbers
- 100AD: Nicomachus goes on to describe certain results concerning perfect numbers, held as truths for many centuries, but were later proven fallacies.
- 1603: Cataldi finds the sixth perfect number, namely 2¹⁶(2¹⁷ - 1) = 8589869056 and the seventh perfect number, namely 2¹⁸(2¹⁹ - 1) = 137438691328
- 1732: Euler proves that the eighth perfect number was 2³⁰(2³¹ - 1) = 2305843008139952128
- 1876: Lucas shows that 2¹²⁶(2¹²⁷- 1) is indeed a perfect number
- 1883: Pervusin showed that 2⁶⁰(2⁶¹- 1) is a perfect number
- 1911: Powers showed that 2⁸⁸(2⁸⁹ - 1) was a perfect number, then a few years later he showed that 2¹⁰¹- 1 is a prime and so 2¹⁰⁰(2¹⁰¹- 1) is a perfect number
- today 39 perfect numbers are known, 2⁸⁸(2⁸⁹- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer
complex numbers:
- in order to solve polynomial equations, Bombelli (c 1530) defined square root of negative one & then the arithmetic of the imaginary numbers
- Hamilton (1805-1865) defined complex addition, from which closure, associativity, and commutativity can be deduced
hypercomplex numbers:
- Hamilton (c. 1843) kept the idea of ordered pairs, but he generalized from ordered pairs of reals to ordered pairs of complexes: , giving a quadruple of reals called a quaternion.
- in 1844, just one year after Hamilton finally formulated his quaternion algebra, Grassman (1809-1877) formulated general th-order hypercomplex algebras.
1850: primary numbers;
1874-99: Cantor develops set theory

Algebra:

Babylonians use simultaneous linear equations and early matrices
Chinese 200-100BC 1st to use matrices & use Gaussian elimination which would not become well known in the West until the early 19^th Century!
ALGEBRA comes from the title of a work written in Arabic c 825 by al-Khowarizmi, al-jabr w'al-muqâbalah, in which al-jabr means "the reunion of broken parts." When this was translated from Arabic into Latin four centuries later, the title emerged as Ludus algebrae et almucgrabalaeque.
- in 1140 Robert of Chester translated the Arabic title into Latin as Liber algebrae et almucabala.
- the earliest known use of the word algebra in English in its mathematical sense is by Robert Recorde in The Pathwaie to Knowledge in 1551
1629: Gerard uses brackets & other abbreviations in mathematics;
1631: Oughtred proposes symbol 'X' for multiplication;
1663: Newton discovers the binomial theorem;
1683: discovery of determinants in matrices simultaneously in Europe & in Japan.
1748: Euler's pure analytical mathematics;
1750: Cramer gave the general rule for n x n matrix systems in a paper Introduction to the analysis of algebraic curves
1770, Euler's introduction to algebra;
ABELIAN EQUATION. Leopold Kronecker (1823-1891) introduced the term Abelsche Gleichung in an 1853 paper on algebraically soluble equations.
1801: The term 'determinant' was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms.
1850: The first to use the term 'matrix' was Sylvester
1853-97, group theory comes of age with Burnside's Theory of groups of finite order published in 1897

Geometry:

Pythagorus' theorem
Euclidean geometry (c300BC):
- golden ratio
Ptolemy - trigonometric tables
1678: geometrical theorem on the nature of concurrency;
1826, Lobachevsky's non-Euclidean geometry;
1849-70: development of cubic surface geometry theorems
Analytical geometry:
- The term ANALYTIC GEOMETRY was apparently first used (as geometria analytica) in Geometria analytica sive specimina artis analyticae, published by Samuel Horsley (1733-1806) in volume 1 of Isaaci Newtoni opera quae exstant omnia. Commentariis illustrabat Samuel Horsley (1779).
1908: Minkowkski's 4D geometry;

Probability:

1654: Pascal & Fermat state the theory of probability;

Calculus:

1665: Newton experiments with gravity & invents differential calculus;
1675: Leibniz invents integral calculus;
1687: Newton "Philosophiae naturalis principia mathematica";
1655-99: study of elliptic integrals by Wallis, Newton & then Bernoulli.
1715, calculus of finite differences;
1864, Bertrand's treatise on differential & integral calculus;