Ancient Mathematics (3000–300 BC)
Babylonian mathematicians circa 1800 BC could solve quadratic equations using geometric methods. Egyptian mathematics focused on practical measurement — the Rhind Papyrus (1650 BC) shows methods for calculating areas and volumes. The Greeks transformed mathematics from calculation into proof: Pythagoras's theorem, Euclid's axiomatic geometry (Elements, ~300 BC), Archimedes's calculation of π.
The Islamic Golden Age (800–1200 AD)
Al-Khwarizmi's Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr (c. 830 AD) gave us algebra — the word itself derives from al-jabr. Omar Khayyam solved cubic equations geometrically. The Indian numeral system (including zero) was transmitted to Europe through Arabic mathematicians, replacing Roman numerals and enabling modern arithmetic.
The Scientific Revolution (1600–1800)
Descartes invented analytic geometry — the coordinate plane connecting algebra and geometry. Newton and Leibniz independently invented calculus. Euler standardized notation (e, i, π, f(x), Σ) and proved Euler's identity: e^(iπ) + 1 = 0. Gauss proved the fundamental theorem of algebra and made foundational contributions to number theory, statistics, and geometry.
The 19th and 20th Centuries
Riemann proposed the Riemann hypothesis (1859) — the most important unsolved problem in mathematics. Cantor discovered that infinities have different sizes. Gödel proved his incompleteness theorems (1931): any sufficiently powerful formal system contains true statements that cannot be proved within it. Turing defined computation. The Clay Mathematics Institute currently offers $1 million for each of seven Millennium Prize Problems, six of which remain unsolved.
Frequently Asked Questions
Who invented algebra?
Algebra was formalized by the Persian mathematician Al-Khwarizmi around 830 AD in his treatise Al-Kitab al-mukhtasar fi hisab al-jabr. The word 'algebra' derives from 'al-jabr' (the reunion of broken parts). Earlier Babylonian and Greek mathematicians solved algebraic problems without the general framework.
What is the Riemann Hypothesis?
The Riemann hypothesis, proposed by Bernhard Riemann in 1859, conjectures that all non-trivial zeros of the Riemann zeta function have real part equal to 1/2. It has deep implications for the distribution of prime numbers. It remains unproven and is one of the Clay Millennium Prize Problems with a $1 million prize.
What are the Millennium Prize Problems?
The Millennium Prize Problems are seven mathematical problems identified by the Clay Mathematics Institute in 2000, each carrying a $1 million prize for solution. They are: P vs NP, the Riemann Hypothesis, Hodge Conjecture, Yang-Mills Existence and Mass Gap, Navier-Stokes Existence and Smoothness, Birch and Swinnerton-Dyer Conjecture, and the Poincaré Conjecture (solved by Perelman in 2003).