Great Mathematicians Who Changed the World of Numbers

Mathematics has been shaped over centuries by brilliant thinkers whose contributions transformed our understanding of the world. Here is an in-depth look at eleven of the most influential mathematicians in history — their lives, their discoveries, and their lasting impact on mathematics and beyond.

Table of Contents

• 1. Pythagoras of Samos
• 2. Euclid of Alexandria
• 3. Archimedes of Syracuse
• 4. René Descartes
• 5. Pierre de Fermat
• 6. Blaise Pascal
• 7. Isaac Newton
• 8. Leonhard Euler
• 9. Carl Friedrich Gauss
• 10. David Hilbert
• 11. Srinivasa Ramanujan
• Glossary of Terms

1. Pythagoras of Samos (c. 570 – c. 495 BCE)

• Born: c. 570 BCE, Samos, Greece
• Died: c. 495 BCE
• Nationality: Greek
• Known for: The Pythagorean Theorem, the Pythagorean Brotherhood, number mysticism, the mathematics of musical harmony

Early Life

Pythagoras was born on the Greek island of Samos around 570 BCE. As a young man, he travelled extensively. He studied under the priests of Egypt and possibly under Babylonian scholars. These travels shaped his thinking deeply. He came to see mathematics, philosophy, and religion as closely connected.

He eventually settled in Croton, Italy. There he founded a school called the Pythagorean Brotherhood. This school combined the study of mathematics with a unique philosophy about the mystical nature of numbers.

Key Discoveries

Pythagoras is most famous for the Pythagorean Theorem. It states that in a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. This theorem is still fundamental in geometry today. It has applications in physics, engineering, architecture, and many other fields.

His school explored several important ideas:

• The properties of odd and even numbers
• Perfect numbers — numbers that equal the sum of their own divisors, such as 6 (because 1 + 2 + 3 = 6)
• The mystical qualities of specific numbers
• The harmonic properties of sound — they discovered that musical intervals are based on simple ratios between numbers, which laid the foundation for the study of acoustics (the science of sound)

Pythagoras believed that numbers were the essence of all things. He held that the universe itself could be understood through mathematical relationships.

Legacy

Though little of Pythagoras’ original work survives, his influence is undeniable. His ideas shaped the thinking of later Greek philosophers including Plato and Aristotle. His school’s work helped form Western philosophy and mathematics. The Pythagorean Theorem is still taught to students around the world.

2. Euclid of Alexandria (c. 300 BCE)

• Active around: c. 300 BCE
• Worked in: Alexandria, Egypt
• Nationality: Greek
• Known for: Elements (the most influential maths textbook ever written), the Euclidean algorithm, proving that prime numbers are infinite

Background

Euclid is often called the "Father of Geometry." He was an ancient Greek mathematician who worked in Alexandria, Egypt around 300 BCE. Very little is known about his personal life. But his mathematical writings changed the world permanently.

Key Works

His most famous work is Elements. It is one of the most influential textbooks in history. For over two thousand years, it was the definitive guide to geometry. Elements is a collection of definitions, postulates (starting assumptions used to build a mathematical system), and proofs. It systematically covers the geometry and number theory known in Euclid’s time.

In Elements, Euclid presents five postulates as the foundation of Euclidean geometry. The most debated is the parallel postulate. He used these axioms (basic rules assumed to be true without needing proof) to build a complete system of deductive reasoning (deriving complex results logically from simple starting points). This set a new standard for mathematical rigour. Its influence reached far beyond mathematics, into philosophy and science.

Euclid’s contributions to number theory include:

• The Euclidean algorithm — a method for finding the greatest common divisor of two numbers (the largest number that divides both exactly), which remains relevant in modern mathematics
• A proof that there are infinitely many prime numbers — still celebrated as a beautiful example of mathematical reasoning

Beyond Elements, Euclid also wrote:

• Optics — exploring the geometry of vision
• Data — covering geometric problem-solving

Legacy

Elements served as the primary geometry textbook in Europe and the Islamic world until the 19th century. That is over two thousand years of continuous use. The rigour and systematic structure of Elements continue to be a model for mathematical proofs. Euclid remains a cornerstone of mathematics.

3. Archimedes of Syracuse (c. 287 – 212 BCE)

• Born: c. 287 BCE, Syracuse, Sicily
• Died: c. 212 BCE
• Nationality: Greek
• Known for: Archimedes’ Principle (buoyancy), On the Sphere and Cylinder, the Archimedean screw, methods that anticipated calculus

Early Life

Archimedes was born in the city of Syracuse on the island of Sicily. He was a Greek mathematician, engineer, and inventor. His influence on mathematics and science is unmatched in the ancient world.

Key Discoveries

Archimedes’ work on three-dimensional shapes is among his greatest mathematical achievements. His treatise On the Sphere and Cylinder presents his discoveries about the volumes and surface areas of spheres, cylinders, and related objects. His approach anticipated integral calculus (a branch of mathematics used to calculate areas, volumes, and totals) — nearly two thousand years before Newton and Leibniz formally developed it.

He also made important discoveries through his methods of exhaustion (a technique for calculating areas under curves by approximating with many small shapes). He used these methods to prove geometric theorems with remarkable precision.

One of his most famous contributions is Archimedes’ Principle. It describes buoyancy (the upward force that a fluid exerts on any object placed in it). According to legend, he discovered this while taking a bath. He realised that an object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces. This principle remains fundamental in engineering and fluid mechanics today.

Inventions and Legacy

Archimedes was also a prolific inventor. His creations included:

• The Archimedean screw — a device designed to lift water from a lower level to a higher one, still used in parts of the world today
• War machines — including catapults and grappling hooks designed to defend Syracuse from Roman invaders

These inventions showed an advanced understanding of leverage, hydraulics, and other physical principles.

Despite his technical achievements, Archimedes considered himself primarily a mathematician. His work inspired future thinkers including Galileo and Newton. His combination of theoretical mathematics and practical invention represents a rare and lasting blend of genius.

4. René Descartes (1596 – 1650)

• Born: 1596, France
• Died: 1650
• Nationality: French
• Known for: The Cartesian coordinate system, analytical geometry, the use of variables in algebra, "Cogito, ergo sum"

Background

René Descartes was a French philosopher and mathematician. He is often called the father of analytical geometry (a branch of mathematics that uses algebra to study geometric shapes). His work connected two areas that had previously been treated as completely separate: algebra and geometry.

Key Discoveries

Descartes’ most important mathematical contribution was establishing the link between algebra and geometry. He showed that geometric shapes could be represented by algebraic equations. This allowed mathematicians to study geometry using algebraic techniques. It paved the way directly for calculus and modern mathematics.

Descartes introduced the Cartesian coordinate system — named after him. It allows any point in space to be described as a pair of numbers, such as (3, 5). This idea is used in every graph and map drawn today. It fundamentally changed how we describe position and shape.

He also introduced the use of variables in algebra, which revolutionised how equations are written and solved.

His philosophical works are equally significant. His famous declaration — "Cogito, ergo sum" ("I think, therefore I am") — is one of the most recognised phrases in Western philosophy. His major work, Meditations on First Philosophy, emphasised doubt and reason as the foundations of knowledge.

Legacy

Descartes’ method of deductive reasoning and his insistence on clarity and rigour influenced both philosophy and mathematics. His work is foundational to modern science and logic.

5. Pierre de Fermat (1607 – 1665)

• Born: 1607, Beaumont-de-Lomagne, France
• Died: 1665
• Nationality: French
• Known for: Fermat’s Last Theorem, co-founding probability theory with Pascal, contributions to analytical geometry and calculus

Background

Pierre de Fermat was a French lawyer who taught himself mathematics. He was born in 1607 in Beaumont-de-Lomagne, France. He pursued a career in law, but maintained a deep passion for mathematics throughout his life. Despite being an amateur mathematician, his innovative ideas in number theory, geometry, calculus, and probability earned him a reputation as one of the greatest mathematical minds of his era.

Key Discoveries

Fermat’s most famous contribution is Fermat’s Last Theorem. He proposed that there are no integer (whole number) solutions to the equation xⁿ + yⁿ = zⁿ when n is greater than 2. He claimed to have a proof, but it was never found. The problem remained unsolved for over 350 years. It was finally proved by Andrew Wiles in 1994.

His other major contributions include:

• Co-founding probability theory — alongside Blaise Pascal, he developed the mathematical principles of chance, a field now used across statistics, economics, and science
• Analytical geometry — his work, developed in parallel with René Descartes, bridged algebra and geometry
• Early calculus — his contributions predated and influenced the formal work of Newton and Leibniz

Fermat rarely published his findings. He shared most of his ideas through letters to fellow mathematicians. His intuitive insights, though sometimes lacking formal proofs, were ahead of their time.

Legacy

Fermat inspired future generations of mathematicians, including Euler and Gauss. His work continues to influence fields from pure mathematics to physics. He is a reminder that remarkable mathematics can come from outside the professional world of academia.

6. Blaise Pascal (1623 – 1662)

• Born: 1623, France
• Died: 1662
• Nationality: French
• Known for: Co-founding probability theory (with Fermat), Pascal’s Triangle, Pascal’s law in hydraulics, work on conic sections, Pensées

Background

Blaise Pascal was a French mathematician, physicist, and inventor. He made significant contributions to mathematics, physics, and philosophy. He showed remarkable intellectual ability from childhood.

Key Discoveries

Pascal’s most important mathematical contribution was co-founding probability theory alongside Pierre de Fermat. Together, they developed the mathematical rules of chance. This work has since become fundamental to statistics, economics, and the sciences.

His other major contributions include:

• Geometry — his work on conic sections (curves formed when a cone is cut at different angles, such as circles, ellipses, and parabolas)
• Pascal’s Triangle — a triangular arrangement of numbers in which each number is the sum of the two numbers directly above it. It has surprising connections to probability, combinations, and many other areas of mathematics
• Pascal’s law — his work in fluid mechanics and pressure showed that pressure applied to a fluid is transmitted equally in all directions. This is the principle behind hydraulics (technology that uses liquid pressure to move heavy loads)

Pascal’s philosophical work, Pensées (meaning "Thoughts"), reflects his religious and existential thinking. It is a reminder that great mathematicians often have wide intellectual lives beyond numbers and equations.

Legacy

Pascal’s contributions span mathematics, physics, and philosophy — a remarkable range. The SI unit of pressure, the pascal, is named in his honour.

7. Isaac Newton (1643 – 1727)

• Born: 1643, England
• Died: 1727
• Nationality: English
• Known for: Laws of motion, universal gravitation, calculus (called "fluxions"), optics, the reflecting telescope

Early Life

Sir Isaac Newton was born in England in 1643. He is regarded as one of the most influential scientists in history. His work transformed physics, mathematics, and astronomy.

Key Discoveries

Newton’s most famous work is Philosophiæ Naturalis Principia Mathematica (known as the Principia), published in 1687. It laid the foundation for classical mechanics (the science of how objects move and respond to forces). The Principia introduced Newton’s three laws of motion. These laws describe how objects interact with forces and respond to changes in momentum.

Newton also discovered the law of universal gravitation. It explains that every object in the universe attracts every other object. The force is proportional to the product of their masses and inversely proportional to the square of the distance between them. Put simply: bigger objects and closer objects pull on each other more strongly.

Newton developed calculus independently from Leibniz. He called it "fluxions." Calculus became a powerful tool to describe motion and predict planetary orbits. It is now fundamental in mathematics, physics, engineering, and economics.

Newton also made major advances in optics (the study of light). His experiments showed that white light is composed of a spectrum of colours. By passing sunlight through a prism, he demonstrated that white light can be separated into its component colours, laying the groundwork for the modern study of optics. He also invented the reflecting telescope, which uses mirrors instead of lenses. This improved telescope design and enhanced astronomical observation.

Legacy

Newton’s emphasis on empirical evidence (evidence gathered through observation and experiment) and the scientific method changed how humans pursue knowledge. Though he was private and often eccentric, his legacy forms the foundation of modern physics. His work remains relevant centuries after his death.

8. Leonhard Euler (1707 – 1783)

• Born: 1707
• Died: 1783
• Nationality: Swiss
• Known for: Euler’s identity, graph theory and topology, introducing the symbols e, i, and π, producing over 70 volumes of mathematical work

Background

Leonhard Euler was a Swiss mathematician who made prolific contributions across a range of mathematical fields, including calculus, number theory, topology (the study of shapes and spaces, focusing on properties that don’t change when objects are stretched or bent), and graph theory (the mathematical study of networks and connections).

Euler also introduced much of the notation mathematicians use today:

• e — for the base of natural logarithms (approximately 2.718)
• i — for the imaginary unit (the square root of −1)
• π — for the ratio of a circle’s circumference to its diameter

Key Discoveries

One of Euler’s most celebrated achievements was solving the Seven Bridges of Königsberg problem. The city of Königsberg had seven bridges connecting its different parts. The puzzle: could you walk through the city crossing each bridge exactly once? Euler proved it was impossible. In doing so, he invented graph theory and topology — entirely new branches of mathematics for studying networks and structures.

Euler’s work in number theory linked prime numbers to natural logarithms. His exploration of infinite series (sums of infinitely many numbers) led to his most celebrated result, Euler’s identity:

𝑒^𝑖𝜋 + 1 = 0

This single equation combines five of the most fundamental constants in mathematics — 𝑒, 𝑖, 𝜋, 1, and 0. It is often described as the most beautiful equation in mathematics.

Euler also made important contributions to:

• Mathematical physics
• Fluid dynamics (the study of how liquids and gases move)
• Astronomy
• Engineering

Legacy

Euler wrote extensively, producing over 70 volumes of work — more than any other mathematician in history. His legacy spans mathematics and science. His contributions continue to inspire discoveries across countless fields.

9. Carl Friedrich Gauss (1777 – 1855)

• Born: 1777, Brunswick, Germany
• Died: 1855
• Nationality: German
• Known for: Disquisitiones Arithmeticae, modular arithmetic, the Gaussian (normal) distribution, the fundamental theorem of algebra, Gauss’s law

Early Life

Carl Friedrich Gauss is often called the "Prince of Mathematicians." He was born in Brunswick, Germany in 1777. He showed extraordinary mathematical ability from a very young age. One of his early accomplishments was discovering a formula to add up all the numbers in an arithmetic sequence (a list of numbers with a fixed gap between each one) — which he reportedly did as a young schoolboy.

Key Discoveries

Gauss made groundbreaking contributions across many areas of mathematics and science.

In number theory (the study of whole numbers and their properties):

• His book Disquisitiones Arithmeticae (1801) became a foundational text for the field
• He introduced modular arithmetic (a way of doing maths that "wraps around" after reaching a certain number, like hours on a clock) — now essential in cryptography (the science of secret codes)
• He proved the fundamental theorem of algebra, showing that every non-constant polynomial equation (an equation involving a variable raised to one or more powers) has at least one complex root

In statistics and probability:

• He developed the Gaussian distribution — also called the normal distribution or "bell curve" — which is fundamental to statistics and probability, and is used across science, economics, and medicine

In science and engineering:

• His work in geodesy (the science of measuring the Earth’s shape and size) led to the development of the Gaussian error curve
• He derived equations describing electric and magnetic fields, known today as Gauss’s law in electrostatics and magnetism

Legacy

Gauss’s deep insights across different fields continue to influence mathematics and science. His commitment to mathematical rigour and elegance earned him a lasting place among history’s greatest mathematicians.

10. David Hilbert (1862 – 1943)

• Born: 1862, Königsberg, Prussia
• Died: 1943
• Nationality: German
• Known for: Foundations of Geometry, the 23 Hilbert Problems, Hilbert spaces, formalism in mathematics

Early Life

David Hilbert was a pioneering German mathematician whose work profoundly shaped modern mathematics. He was born in Königsberg, Prussia in 1862. He studied at the University of Königsberg. He later established himself at the University of Göttingen, turning it into a world-renowned centre for mathematics.

Key Discoveries

One of Hilbert’s earliest and most influential works was Foundations of Geometry (1899). In it, he introduced a complete and consistent set of axioms (basic rules assumed to be true without needing proof) for Euclidean geometry. His goal was to remove ambiguities and build geometry on a perfectly solid logical foundation. This set new standards for clarity and rigour across all of mathematics — not just geometry.

In 1900, Hilbert presented his famous 23 unsolved problems at the International Congress of Mathematicians. These problems spanned many fields, from number theory to mathematical physics. They set the agenda for much of 20th-century mathematical research. Some remain unsolved today — including the Riemann Hypothesis. Others led to major breakthroughs. For example, attempts to solve his tenth problem on Diophantine equations (equations where only whole-number solutions are considered) produced powerful new tools in mathematical logic.

Hilbert also developed the concept of Hilbert spaces — infinite-dimensional mathematical spaces (spaces with unlimited dimensions, beyond the usual two or three of everyday experience). These became foundational in quantum mechanics and modern physics.

As a proponent of formalism (the belief that all of mathematics can be built from a fixed, complete set of rules and axioms), Hilbert sought to place all of mathematics on an unshakeable logical foundation. This vision was later challenged by Kurt Gödel’s incompleteness theorems, which showed that no fixed set of axioms can ever fully capture all mathematical truth. Even so, Hilbert’s ideas revolutionised mathematical logic and the philosophy of mathematics, influencing generations of thinkers.

Legacy

Hilbert influenced generations of mathematicians. His 23 problems continue to shape mathematical research more than a century later. He remains one of the most visionary figures in the entire history of mathematics.

11. Srinivasa Ramanujan (1887 – 1920)

• Born: 1887, Erode, India
• Died: 1920
• Nationality: Indian
• Known for: The partition function, modular forms, mock theta functions, thousands of original theorems

Early Life

Srinivasa Ramanujan was a self-taught Indian mathematician. He was born in Erode, India in 1887. Despite limited formal education and resources, he showed an extraordinary gift for mathematics from a young age. Working largely on his own, he produced results that were completely original.

In 1913, Ramanujan sent a letter filled with his mathematical findings to G.H. Hardy, a prominent mathematician at Cambridge. Hardy recognised his potential and brought him to England, where the two worked together closely.

Key Discoveries

Ramanujan’s discoveries in number theory (the mathematical study of whole numbers and their properties), analysis, and continued fractions (a way of expressing numbers as a series of nested fractions) are particularly famous.

His work on the partition function is especially well known. The partition function represents the number of ways a number can be expressed as a sum of other numbers. For example, the number 4 can be partitioned as: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1 — giving it 5 partitions. Ramanujan introduced new perspectives in combinatorics (the study of counting and arrangements) and integer partitions.

His other key contributions include:

• Modular forms — complex mathematical structures with far-reaching applications in physics, computer science, and cryptography
• Mock theta functions — a set of mathematical functions he discovered that mathematicians are still actively studying today
• Thousands of original theorems and conjectures recorded in his notebooks — many still being explored by researchers around the world

Legacy

Despite facing cultural and health challenges, Ramanujan’s productivity was astonishing. His notebooks contain results that were decades ahead of their time. Today, mathematicians continue to study his work. Ramanujan’s life and achievements inspire mathematicians worldwide, demonstrating the power of natural talent and perseverance.

Glossary of Mathematical Terms

This page introduces several technical terms. Here is a quick reference to help you understand them whenever you encounter them.

• Acoustics — the science of sound: how it travels, behaves, and is perceived.
• Analytical geometry — a branch of mathematics that uses algebra to describe and study geometric shapes and positions.
• Axiom / Postulate — a basic rule assumed to be true without proof; the starting point of a mathematical system.
• Buoyancy — the upward force a fluid (liquid or gas) exerts on an object placed in it.
• Calculus — a branch of mathematics dealing with change, motion, and areas. It has two main branches: differentiation (rates of change) and integration (totals and areas).
• Classical mechanics — the science of how physical objects move and respond to forces.
• Combinatorics — the mathematical study of counting, arrangements, and combinations.
• Complex root — a solution to an equation that involves imaginary numbers (numbers involving the square root of −1).
• Conic sections — curves formed by cutting a cone at different angles: circles, ellipses, parabolas, and hyperbolas.
• Continued fractions — a way of expressing numbers as a sequence of nested fractions, one inside another.
• Cryptography — the science of creating and breaking secret codes.
• Deductive reasoning — a logical process where conclusions follow necessarily from given rules or premises.
• Diophantine equations — equations where only whole-number (integer) solutions are considered valid.
• Empirical evidence — evidence gathered through observation and experiment rather than pure reasoning alone.
• Fluid mechanics — the study of how liquids and gases behave and move.
• Formalism — the belief that all of mathematics can be derived from a fixed, complete set of rules and axioms.
• Geodesy — the science of measuring the Earth’s shape, size, and gravitational field.
• Graph theory — the mathematical study of networks and connections made up of points (nodes) and lines (edges).
• Hydraulics — technology that uses the pressure of liquids to move heavy loads or perform mechanical work.
• Imaginary unit (i) — a mathematical symbol representing the square root of −1; the basis of complex numbers.
• Integral calculus — the branch of calculus used to calculate areas, volumes, and accumulated totals.
• Modular arithmetic — a system of arithmetic that "wraps around" after reaching a certain number, the way clock hours reset at 12.
• Modular forms — complex mathematical structures with deep connections to number theory, physics, and cryptography.
• Normal distribution (Gaussian distribution) — a bell-shaped statistical curve describing how measurements tend to cluster around an average.
• Number theory — the branch of mathematics that studies whole numbers and their properties.
• Optics — the study of light: how it travels, bends, reflects, and interacts with objects.
• Partition function — a function that counts how many ways a whole number can be written as a sum of positive integers.
• Polynomial equation — an equation involving a variable raised to one or more powers (for example, x² + 3x + 2 = 0).
• Probability theory — the mathematical study of chance and likelihood.
• Topology — the study of shapes and spaces, focusing on properties that remain the same even when objects are stretched, bent, or deformed.