A journey through more than 43,000 years of mathematics — told era by era, with interactive animations you can play with.
Mathematics is older than writing. The Lebombo bone, found in the mountains between South Africa and Eswatini and dated to roughly 43,000 years ago, carries 29 deliberate notches. The Ishango bone from Central Africa, around 20,000 years old, has groups of notches that some researchers read as evidence of counting in tens or even an awareness of doubling.
Early people counted animals, days and the phases of the Moon with tally marks, pebbles and fingers. The idea that one mark can stand for one object — a one-to-one correspondence — is the seed from which all of arithmetic grew.
In the cities of Sumer and Babylon, scribes pressed wedge-shaped marks into wet clay. They built a place-value system in base 60 — the reason an hour still has 60 minutes and a circle 360 degrees. Thousands of tablets survive, full of multiplication tables, reciprocals, interest calculations and problems that we would now solve with quadratic equations.
The tablet YBC 7289 (around 1800–1600 BCE) shows a square with its diagonal and gives √2 as 1;24,51,10 in base 60 — about 1.41421296, correct to roughly six decimal places. Plimpton 322 lists number triples connected to right-angled triangles, more than a thousand years before Pythagoras.
Egyptian mathematics was practical: measuring fields after the Nile's flood, sharing bread among workers and designing pyramids. Two papyri tell us most of what we know. The Rhind papyrus, copied by the scribe Ahmes around 1550 BCE from an older text, contains 84 problems; the Moscow papyrus includes the correct formula for the volume of a truncated pyramid.
Egyptians multiplied by repeated doubling and wrote fractions as sums of different unit fractions, such as 2/5 = 1/3 + 1/15. Their method of multiplication is, in effect, binary arithmetic — the same idea modern computers use.
Greek thinkers changed the question from "how do we calculate?" to "why is it true?". Thales is traditionally credited with the first geometric proofs; the Pythagoreans studied numbers, music and the famous theorem about right triangles — and discovered, to their dismay, that √2 cannot be written as a fraction.
Around 300 BCE in Alexandria, Euclid organised geometry into the Elements: definitions, axioms and step-by-step proofs. It became the most widely studied mathematics book in history. Later, Apollonius classified the conic sections, Eratosthenes estimated the size of the Earth, Diophantus worked on equations, and Hypatia taught and wrote commentaries in Alexandria.
Archimedes of Syracuse is often called the greatest mathematician of the ancient world. To pin down π, he drew polygons inside and outside a circle and kept doubling their sides from 6 up to 96. The perimeters squeezed π between 223/71 and 22/7 — about 3.1408 and 3.1429.
His "method of exhaustion" — trapping a curved quantity between simpler shapes that get closer and closer — anticipated integral calculus by nearly 2,000 years. He also found the area of a parabolic segment, the volume and surface area of a sphere, and the law of the lever.
Chinese mathematicians calculated with counting rods laid out on a board, using a decimal place-value system. The Nine Chapters on the Mathematical Art, compiled over centuries and commented on by Liu Hui in 263 CE, solves systems of linear equations with a method essentially the same as Gaussian elimination — including negative numbers.
Liu Hui used polygons to approximate π as 3.1416, and in the 5th century Zu Chongzhi showed that π lies between 3.1415926 and 3.1415927, giving the remarkable fraction 355/113. The Sunzi Suanjing posed the problem that leads to the Chinese Remainder Theorem, used today in cryptography.
The decimal place-value system with a symbol for zero — the numerals used across the world today — was developed in India. Aryabhata (499 CE) gave π ≈ 3.1416 and early sine tables; Brahmagupta (628 CE) wrote the first rules for calculating with zero and negative numbers; Bhaskara II's Lilavati (1150) taught arithmetic and algebra through verse.
Around 1400, Madhava of Sangamagrama and the Kerala school discovered infinite series for π, sine and cosine — results Europe would reach only in the 17th century. One of them is π/4 = 1 − 1/3 + 1/5 − 1/7 + … , which you can watch converge on the right.
In Baghdad's House of Wisdom, scholars translated Greek and Indian works and pushed far beyond them. Around 820, Muhammad ibn Musa al-Khwarizmi wrote al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala — the book that gave us the word algebra — and his name gives us the word algorithm. He solved quadratic equations by completing the square, shown geometrically on the right.
Omar Khayyam solved cubic equations geometrically; al-Biruni measured the Earth's radius; Ibn al-Haytham advanced optics; Nasir al-Din al-Tusi made trigonometry a subject in its own right; and in 1424 al-Kashi calculated π to 16 decimal places. Indian numerals travelled west through this world, reaching Europe as "Arabic numerals".
In 1202, Leonardo of Pisa — Fibonacci — published Liber Abaci, persuading European merchants to adopt Hindu-Arabic numerals. One of its puzzles, about breeding rabbits, produces the sequence 1, 1, 2, 3, 5, 8, 13… whose ratios approach the golden ratio.
Printing spread mathematics quickly. In 1545 Gerolamo Cardano's Ars Magna published formulas for cubic and quartic equations (building on Scipione del Ferro, Tartaglia and Ferrari), and Rafael Bombelli began to calculate with square roots of negative numbers. François Viète introduced letters for unknowns and constants — the start of modern algebraic notation.
John Napier's logarithms (1614) turned multiplication into addition and saved astronomers years of work. In 1637 René Descartes joined algebra and geometry with coordinates, and Pierre de Fermat developed similar ideas — along with methods for maxima, minima and tangents.
Building on this, Isaac Newton (from the 1660s) and Gottfried Wilhelm Leibniz (published 1684) independently created calculus: the mathematics of change. The derivative is the slope of a tangent line — move the point on the curve to see it. Leibniz's notation, dy/dx and ∫, is the one we still use.
Leonhard Euler (1707–1783) wrote more mathematics than almost anyone in history. He gave us the notation f(x), e, i and Σ, founded graph theory with the Seven Bridges of Königsberg (1736), and connected the most important constants in e^{iπ} + 1 = 0.
The Bernoulli family, Lagrange and Laplace extended calculus to mechanics, astronomy and probability. The number e ≈ 2.71828, which appears whenever something grows continuously, can be found as the limit of (1 + 1/n)^n — try it on the right.
Carl Friedrich Gauss published Disquisitiones Arithmeticae in 1801 and made foundational advances across number theory, geometry and statistics. Cauchy and Weierstrass made calculus rigorous; Évariste Galois and Niels Abel showed why there is no general formula for equations of degree five; Lobachevsky and Bolyai revealed geometries where Euclid's parallel postulate fails; Riemann reshaped geometry and posed his famous hypothesis in 1859.
George Boole's Laws of Thought (1854) turned logic into algebra — the foundation of every digital circuit. Ada Lovelace's notes (1843) described an algorithm for Charles Babbage's Analytical Engine, and Georg Cantor (1870s) showed that some infinities are bigger than others.
In 1900 David Hilbert set out 23 problems that shaped the century. Emmy Noether transformed abstract algebra and linked symmetry to conservation laws in physics; Srinivasa Ramanujan produced astonishing formulas; Kurt Gödel's incompleteness theorems (1931) showed the limits of proof; and Alan Turing's 1936 paper defined computation itself.
Claude Shannon founded information theory in 1948, John von Neumann helped design early computers, and computers began to change mathematics: chaos theory (Lorenz, 1963), fractals (Mandelbrot), the computer-assisted proof of the four colour theorem (1976) and Andrew Wiles's proof of Fermat's Last Theorem (1994). The fractal on the right is generated from the simple rule z → z² + c.
Grigori Perelman proved the Poincaré conjecture in the early 2000s, and in 2014 Maryam Mirzakhani became the first woman to receive the Fields Medal. Mathematicians increasingly work with computers, using proof assistants such as Lean to check proofs line by line.
Mathematics now powers the modern world: linear algebra and calculus drive machine learning and AI, number theory secures online banking, statistics underpins medicine and research, and differential equations model climate, epidemics and engineering. The story that began with notches on a bone is still being written — by researchers, engineers and students like you.
"Mathematics is the language in which God has written the universe."
— attributed to Galileo Galilei
How Alan Turing's 1936 paper defined computation, how his work at Bletchley Park cracked Enigma, and how his ideas shaped AI, with worked examples.
Read article · 6 min →History of MathematicsLearn how ancient Egyptians multiplied by doubling, used unit fractions and calculated pyramid slopes, with worked problems from the famous Rhind Papyrus.
Read article · 5 min →History of MathematicsExplore the life and genius of Archimedes, from the Eureka story and his bounds on pi to spheres, levers and ideas that anticipated calculus by 1,800 years.
Read article · 6 min →History of MathematicsExplore Babylonian mathematics: the base-60 system, famous clay tablets like YBC 7289 and Plimpton 322, and how scribes solved quadratics 4,000 years ago.
Read article · 5 min →History of MathematicsThe life of Carl Friedrich Gauss, from adding 1 to 100 as a schoolboy to the 17-gon, modular arithmetic and finding Ceres, with his methods worked out.
Read article · 6 min →History of MathematicsDiscover ancient Chinese mathematics: counting rods, The Nine Chapters, early matrix elimination, Zu Chongzhi's pi and the Chinese Remainder Theorem.
Read article · 6 min →History of MathematicsHow Georg Cantor proved that some infinities are bigger than others, with his countability proofs, diagonal argument and the Cantor set worked step by step.
Read article · 6 min →History of MathematicsHow Greek thinkers from Thales and Pythagoras to Euclid turned mathematics into a science of proof, with worked examples of their methods and theorems.
Read article · 6 min →History of MathematicsDiscover how humans first counted, from notched bones and tally marks to clay tokens and written numerals, with worked examples of ancient number systems.
Read article · 7 min →History of MathematicsHow a gambler's question and a 1654 exchange of letters between Pascal and Fermat created probability theory, with their original methods worked out.
Read article · 6 min →History of MathematicsHow Indian mathematicians like Aryabhata and Brahmagupta gave the world zero, decimal place value and rules for negatives, with worked examples.
Read article · 5 min →History of MathematicsMeet Leonhard Euler, who wrote over 800 works even while blind, solved the Basel problem and the Königsberg bridges, and gave us e, i, f(x) and more.
Read article · 6 min →History of MathematicsHow 2,000 years of doubt about Euclid's parallel postulate led Gauss, Bolyai, Lobachevsky and Riemann to new geometries, with worked triangle examples.
Read article · 7 min →History of MathematicsMeet Omar Khayyam, Al-Biruni and Ibn al-Haytham, Islamic Golden Age scholars who solved cubics, measured the Earth and pioneered optics, with examples.
Read article · 6 min →History of MathematicsThe story of Srinivasa Ramanujan, the largely self-taught Indian genius who worked with Hardy at Cambridge, with his partitions, 1729 and pi formulas explained.
Read article · 6 min →History of MathematicsTrace algebra from Babylonian word problems and al-Khwarizmi's al-jabr to Cardano, Viete and Descartes, with worked examples of each historical method.
Read article · 6 min →History of MathematicsFollow pi from Babylonian clay tablets and Archimedes' polygons to infinite series and supercomputers, with the historical methods worked step by step.
Read article · 6 min →History of MathematicsHow Isaac Newton and Gottfried Leibniz independently invented calculus, why they fought over priority, and how their methods worked, with clear worked examples.
Read article · 6 min →History of MathematicsFrom a note in a book margin around 1637 to Andrew Wiles' proof in 1995, the story of Fermat's Last Theorem, with Pythagorean triples and key steps worked out.
Read article · 5 min →History of MathematicsThe lives and discoveries of Hypatia of Alexandria, Emmy Noether and Maryam Mirzakhani, three women who shaped mathematics across 1,600 years, with examples.
Read article · 6 min →