# You Gotta Know These Mathematicians

- The work of Isaac Newton (1643–1727, English) in pure math includes generalizing the binomial theorem to non-integer exponents, doing the first rigorous manipulation with power series, and creating Newton’s method for finding roots of differentiable functions. He is best known, however, for a lengthy feud between British and Continental mathematicians over whether he or Gottfried Leibniz invented calculus (whose differential aspect Newton called the method of fluxions). It is now generally accepted that they both did, independently.
- Euclid (c. 300 BC, Alexandrian Greek) is principally known for the
*Elements*, a textbook on geometry and number theory, that has been used for over 2,000 years and which grounds essentially all of what is taught in modern high school geometry classes. The*Elements*includes five postulates that describe what is now called Euclidean space (the usual geometric space we work in); the fifth postulate — also called the parallel postulate — can be broken to create spherical and hyperbolic geometries, which are collectively called non-Euclidean geometries. The*Elements*also includes a proof that there are infinitely many prime numbers. - Carl Friedrich Gauss (1777–1855, German) is considered the “Prince of Mathematicians” for his extraordinary contributions to every major branch of mathematics. His
*Disquisitiones Arithmeticae*systematized number theory and stated the fundamental theorem of arithmetic (every integer greater than 1 has a prime factorization that is unique notwithstanding the order of the factors). In his doctoral dissertation, he proved the fundamental theorem of algebra (every non-constant polynomial has at least one root in the complex numbers), though that proof is not considered rigorous enough for modern standards. He later proved the law of quadratic reciprocity, and the prime number theorem (that the number of primes less than*n*is is approximately*n*divided by the natural logarithm of*n*). Gauss may be most famous for the (possibly apocryphal) story of intuiting the formula for the summation of an arithmetic sequence when his primary-school teacher gave him the task — designed to waste his time — of adding the first 100 positive integers. - Archimedes (287–212 BC, Syracusan Greek) is best known for his “eureka” moment, in which he realized he could use density considerations to determine the purity of a gold crown; nonetheless, he was the preeminent mathematician of ancient Greece. He found the ratios between the surface areas and volumes of a sphere and a circumscribed cylinder, accurately estimated pi, and developed a calculus-like technique to find the area of a circle, his method of exhaustion.
- Gottfried Leibniz (1646–1716, German) is known for his independent invention of calculus and the ensuing priority dispute with Isaac Newton. Most modern calculus notation, including the integral sign and the use of
*d*to indicate a differential, originated with Leibniz. He also did work with the binary number system and did fundamental work in establishing boolean algebra and symbolic logic. - Pierre de Fermat (1601–1665, French) is remembered for his contributions to number theory including his little theorem, which states that if
*p*is a prime number and*a*is any number at all, then*a*^{p}–*a*will be divisible by*p*. He studied Fermat primes, which are prime numbers that can be written as 2^{2n}+ 1 for some integer*n*, but is probably most famous for his “last theorem,” which he wrote in the margin of*Arithmetica*by the ancient Greek mathematician Diophantus with a note that “I have discovered a marvelous proof of this theorem that this margin is too small to contain.” The theorem states that there is no combination of positive integers*x*,*y*,*z*, and*n*, with*n > 2*, such that*x*^{n}+*y*^{n}=*z*^{n}, and mathematicians struggled for over 300 years to find a proof until Andrew Wiles completed one in 1995. (It is generally believed that Fermat did not actually have a valid proof.) Fermat and Blaise Pascal corresponded about probability theory. - Leonhard Euler (1707–1783, Swiss) is known for his prolific output and the fact that he continued to produce seminal results even after going blind. He invented graph theory by solving the Seven Bridges of Königsberg problem, which asked whether there was a way to travel a particular arrangement of bridges so that you would cross each bridge exactly once. (He proved that it was impsosible to do so.) Euler introduced the modern notation for
*e*, an irrational number about equal to 2.718, which is now called Euler’s number in his honor (but don’t confuse it for Euler’s constant, which is different); he also introduced modern notation for*i*, a square root of –1, and for trigonometric functions. He proved Euler’s formula, which relates complex numbers and trigonometric functions:*e*^{i x}= cos*x*+*i*sin*x*, of which a special case is the fact that*e*^{i π}= –1, which Richard Feynman called “the most beautiful equation in mathematics” because it links four of math’s most important constants. - Kurt Gödel (1906–1978, Austrian) was a logician best known for his two incompleteness theorems, which state that if a formal logical system is powerful enough to express ordinary arithmetic, it must contain statements that are true yet unprovable. Gödel developed paranoia late in life and eventually refused to eat because he feared his food had been poisoned; he died of starvation.
- Andrew Wiles (1953–present, British) is best known for proving the Taniyama-Shimura conjecture that all rational semi-stable elliptic curves are modular forms. When combined with work already done by other mathematicians, this immediately implied Fermat’s last theorem (see above).
- William Rowan Hamilton (1805–1865, Irish) is known for a four-dimensional extension of complex numbers, with six square roots of –1 (±
*i*, ±*j*, and ±*k*), called the quaternions.