Mathematics is an ongoing journey of thousands of years and millions of minds. The list below is by no means complete, but here are ten great mathematicians whose work forever changed not only math but the way in which the world itself is understood.

## Pythagoras (c. 500 b.c.)

Perhaps the world’s first great mathematician, and credited as inventor of the Pythagorean theorem (a2 + b2 = c2), Pythagoras lived so long ago that the details of his life and work are sketchy. His actual writings have not survived, and most of what is known about him comes through the later Greeks, such as Plato and Aristotle.

The work of Pythagoras is more accurately attributed to the Pythagoreans, the composite work of himself and his followers. But this work stands as an original cornerstone of mathematics.

## Euclid (c. 300 b.c.)

Euclid is commonly known as the “Father of Geometry.” Unlike the work of Pythagoras, Euclid’s written work survives to this day. Foremost among these, his Elements formalizes the study of geometry based upon five postulates, from which all subsequent theorems are derived.

Euclid’s work was the foundation for hundreds of years of Greek mathematics to follow. And his method of formalization became the basis for the work of later mathematicians, especially David Hilbert (see below), who attempted to derive all mathematics from a finite set of similar axioms.

## Muhammed ibn Musa al-Khwarismi (c. 780–850)

Although the Greeks are credited with the first great strides in mathematics, most of their efforts were based in geometry rather than abstraction. Their concept of numbers, unfortunately, lacked a symbol for 0, which limited their ability to develop more sophisticated methods of calculation.

Al-Khwarismi is widely considered the inventor of algebra. His book, The Compendious Book on Calculation by Completion and Balancing (in Arabic, al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala) is the first work to standardize methods of solving classes of equations (such as linear and quadratic equations.)

The Arabic word al-jabr — which refers to Al-Khwarismi’s method of completion by subtracting equal numbers from both sides of an equation — is adapted into English and other European languages as the word algebra.

## Rene Descartes (1596–1650)

Descartes is noted for his pivotal achievements as both philosopher and mathematician. If Al-Khwarismi made his mark by distinguishing algebra as a separate area of study from geometry, Descartes made his own mark by merging the two, unifying over 2,000 years of mathematical progress.

Descartes invented analytic geometry, defining lines and shapes on a pair of axes known as the Cartesian graph or, more simply, the xy-graph. This innovation enables the use of algebra as a tool for the study and systemization of geometry. It’s also the foundation of Isaac Newton’s calculus, which became the indispensable instrument of modern physics.

## Isaac Newton (1642–1727)

The father of modern physics and the inventor of calculus, Isaac Newton stands as perhaps the greatest scientist of all time. His vision of the universe redefined science for the next two centuries. And his method of calculus — which he originated in his early 20s — permitted the equations generated by his new physics to be calculated.

Calculus permits the computation of infinitely long lists of numbers, provided that those numbers become smaller and smaller and eventually approach 0. Although this insight originated with the Greeks, Newton originated a generalized method for making such calculations that continue to be used and perfected to this day.

## Bernhard Riemann (1826–1866)

In his relatively short life, Bernhard Riemann solved some of the most difficult problems of his time and opened new frontiers that still remain relevant to this day.

By demonstrating the Fundamental Theorem of Calculus, Riemann unified the two branches of calculus (differential and integral), solving a nearly two-centuries-old problem that had remained unsolved since Newton’s day. His version of non-Euclidean geometry (geometry based on a set of postulates different from Euclid’s) proved to be a more accurate representation of the geometry of our universe — so much so that Albert Einstein used it as the mathematical basis for his General Theory of Relativity.

Even after almost a century and a half his death, his famous Riemann’s Hypothesis remains the greatest unsolved problem in number theory.

## Georg Cantor (1845–1918)

A mathematical innovator like no other, Georg Cantor created the basis for a new understanding not only of the infinite, but also what lies beyond it.

His formulation of varying levels of infinity — called the transfinite numbers — enables sets that contain infinitely many elements to be compared on the basis of size. His ingenious diagonalization proof shows that the number of points on any line segment is actually greater than infinity, requiring a separate classification.

One of Cantor’s most surprising results shows that the many levels of infinity are, themselves, infinite. That is, given any set, no matter how large, Cantor showed how to construct an even larger set.

## David Hilbert (1862–1943)

Throughout his long life, David Hilbert changed not only virtually every branch of mathematics but the very nature of how math is done. His influential Hilbert Project sought to create a logical foundation rooting all of math in a common set of axioms, much as Euclid had done for geometry.

In 1900, Hilbert listed 23 important and unsolved problems of his day. More than a century later, though many of these problems are solved, some remain open. Interestingly, several of these problems have been solved by methods that are not universally accepted by mathematicians (for example, proofs generated by computer). Notably, the Riemann Hypothesis — considered by Hilbert himself to be the most important — remains unsolved.

## Srinivasa Ramanujan (1887–1920)

In his short life and with virtually no formal training as a mathematician, Ramanujan proved thousands of results, primarily in analysis and number theory.

Beginning as a child in India, far from the European hub of mathematical knowledge, Ramanujan derived much of what was already known in mathematics on his own. By the time he was in his teens, he was already moving beyond the edges of mathematical frontiers with proofs of original theorems. Discovered by noted mathematician G. H. Hardy, Ramanujan was brought to Cambridge, England, and continued his prolific work there until his untimely death at 32.

## Kurt Gödel (1906–1978)

Widely considered among math scholars to be the greatest mathematician of the 20th century, Kurt Gödel was a close friend of Albert Einstein and stood shoulder to shoulder with Einstein in his genius.

As a logician, his early work was part of David Hilbert’s project to create a logical foundation in which all mathematics — now and forever — could be rooted. Gödel’s great insight — rigorously proven in a seminal paper of 1931 — is that any set of axioms, no matter how well chosen, invariably lead to “undecideable statements” — that is, statements that may be true or false, but cannot be proven as such within the confines of the axioms that have been defined. Thus, Gödel demonstrates any formulation of mathematics must be incomplete.

The philosophical implications of Gödel’s work — which seems to say that even the subtlest mathematics is inherently incapable of describing the whole of scientific truth — are still disputed with great interest.