# You Gotta Know These Mathematicians

These are the ten people that have come up most frequently in NAQT's questions as a result of their accomplishments in pure mathematics.

1. 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 the finding roots. 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.

2. Euclid (c. 300 BC, Alexandrian Greek) is principally known for the Elements, a textbook on geometry and number theory, that was used for over 2,000 years and which grounds essentially all of what is taught in modern high school geometry classes. Euclid is known for his five postulates that define Euclidean (i.e., "normal") space, especially the fifth (the "parallel postulate") which can be broken to create spherical and hyperbolic geometries. He also proved the infinitude of prime numbers.

3. 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. He also proved the fundamental theorem of algebra, the law of quadratic reciprocity, and the prime number theorem. Gauss may be most famous for the (possibly apocryphal) story of intuiting the formula for the summation of an arithmetic series when given the busywork task of adding the first 100 positive integers by his primary school teacher.

4. Archimedes (287-212 BC, Syracusan Greek) is best known for his "Eureka moment" of using 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 presaged the summation of infinite series with his "method of exhaustion."

5. 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 invented binary numbers and did fundamental work in establishing boolean algebra and symbolic logic.

6. Pierre de Fermat (1601-1665, French) is remembered for his contributions to number theory including his "little theorem" that ap - a will be divisible by p if p is prime. He also studied Fermat primes (those of the form 22n+1) and stated his "Last Theorem" that xn + yn = zn has no solutions if x, y, and z are positive integers and n is a positive integer greater than 2. He and Blaise Pascal founded probability theory. In addition, he discovered methods for finding the maxima and minima of functions and the areas under polynomials that anticipated calculus and inspired Isaac Newton.

7. 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 with the Seven Bridges of Königsberg problem and introduced the modern notation for e, the square root of -1 (i), and trigonometric functions. Richard Feynman called his proof that eiπ = -1 "the most beautiful equation in mathematics" because it linked four of math's most important constants.

8. Kurt Gödel (1906-1978, Austrian) was a logician best known for his two incompleteness theorems proving that every formal system that was powerful enough to express ordinary arithmetic must necessarily contain statements that were true, but which could not be proved within the system itself.

9. Andrew Wiles (1953-present, British) is best known for proving the Taniyama-Shimura conjecture that all rational semi-stable elliptic curves are modular. This would normally be too abstruse to occur frequently in quiz bowl, but a corollary of that result established Fermat's Last Theorem.

10. William Rowan Hamilton (1805-1865, Irish) is known for extending the notion of complex numbers to four dimensions by inventing the quaternions, a non-commutative field with six square roots of -1: ±i, ±j, and ±k with the property that ij = k, jk = i, and ki = j.

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