Mathematics

C. F. Gauss (1777-1855)

The prime numbers have been an object of fascination for a long time. These are the counting numbers having no divisors other than one and themselves:

\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \ldots. \]

The key fact about the primes is that every natural number can be written as a product of primes, and the product is unique up to the order of the factors. Euclid proved that there are infinitely many prime numbers in 300 BC in Book IX, Proposition 20 of the Elements. Like all of Euclid, the proof is geometrical, with line segments representing numbers, but it's valid and recognizable. The modern proof goes like this:

Chebyshev's Mémoire sur les nombres premiers (1852)

§1.^[1] All questions which depend on the law of distribution of prime numbers in the series

\[ | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \text {etc. } \]

present in general great difficulties. What we manage to conclude with a very high probability from the tables of prime numbers remains most often without rigorous proof. — For example, the tables of prime numbers lead us to believe that for \( a > 3, \) there is always a prime number greater than \( a \) and less than \( 2a-2 \) (which is known as the postulate of M. Bertrand *); but so far the proof of this proposition has failed for values ​​of \( a \) which exceed the limit of our tables. The difficulty becomes even greater when we give ourselves narrower limits, or when we ask to assign the limit of \( a \) above which the series

Emmy Noether Google Doodle for March 23, 2015

There are people great and small who sense the future with a clairvoyance not given to the rest of us, seers with a preternatural instinct for the arc of what is to come. Abraham Lincoln was one, Emmy Noether another in the world of mathematics. She was the daughter of a German mathematics professor renowned in his day,^[1] a lucky fluke in a society with little use for women as thinkers and academics. Emmy Noether revolutionized mathematics in the early twentieth century by identifying and explaining the underlying abstract principles behind the mathematics bequeathed to her. Like all geniuses, indeed all human beings, she existed in a social context. Asked to explain her deep insights, she would say "Es steht alles schon bei Dedekind" (Everything is already there in Dedekind)^[2], and that is not false modesty. Not false modesty, but modesty all the same, because she was second to none in triggering a new direction in mathematics.

David Hilbert, 1862-1943 (photo 1912)

Hilbert first proved a form of the basis theorem in 1890.^[1] It was so revolutionary at the time that Paul Gordan reportedly exclaimed, “This is not mathematics, it is theology!”.^[2] van der Waerden gave an updated and generalized proof in Moderne Algebra in 1931, crediting Hilbert for the basic idea and Emil Artin for the specifics.^[3] The proof here is updated still more, though still retaining van der Waerden's degree reduction strategy.

Hilbert Basis Theorem. Let \( R \) be a commutative ring with \( 1 \). If \( R \) is Noetherian, then \( R[x] \) is Noetherian as well.

Proof. Recall that a ring is Noetherian if its ideals satisfy the Ascending Chain Condition, or equivalently, if every ideal is finitely generated.

B. L. van der Waerden in 1930 (fl. 1903-1996)

A truism in mathematics, and perhaps most subjects, is that its disseminators are as important as its creators. Niccolò Tartaglia (1499-1557) comes to mind, a great and prolific encyclopedist whose works were consulted in his own day and for generations to come. Perhaps the premiere example is Frans van Schooten and his associates, who did so much to organize and present Descartes' deep but scattered and somewhat opaque geometric approach in La Geometrie. van Schooten's "appendices" amplified La Geometrie massively, the 1659-61 edition becoming a textbook for Leibnitz and Newton and the rock that calculus was built on. Of course the disseminators are mathematicians too, among the select few who expand the new path as well as trod it ("Cartesian" coordinates appear nowhere in Descartes, for example, but were introduced by his acolytes). They cast a great net, subsuming existing mathematics into the new framework as they organize and simplify, all the while spreading the word and making it available to a broad audience of practitioners, many having already worked obscurely to clear patches of brambles and more than ready for the big breakthrough.

Tartaglia on the cover of his Quesiti (1546).

Niccolò Tartaglia (1499-1557) was badly used by Cardano and his disciples and has been deprecated ever since, as if to prove the old adage about hating those the most whom you've unjustly injured the most. His partial solution of the cubic equation, expropriated by Cardano, should have propelled him into the first rank of mathematicians of his day^[1]. Instead, it's been used to run him down, the supposedly small-minded foil to the great Cardano. In fact, the impoverished, self-trained Tartaglia was highly productive and exemplified the forward-looking mathematizing approach to the new science leading in direct line to Galileo and the modern approach generally.^[2] First and foremost, Tartaglia was a maestro d'abaco, that is, a teacher of the new art of calculation with pen and paper. This tradition centered in vernacular commercial schools outside the universities, sidestepping the barren Boethian preoccupations of "mathematicians" debating endlessly in Latin quaestios about such matters as the true nature of even and odd numbers and what that might mean for the cosmos.

Gerbert's abacus with representative numbers

Gerbert of Aurillac was a Benedictine monk from this small village in south-central France who was born about 950 AD and died in 1003. He was a great teacher and scholar, politician and church leader. He became Pope Sylvester II in 999 and served in that role till his death. He was the first French pope and the only mathematician to become pope. And mathematician he was, sent as a young man by his abbot to study in Christian Spain, bordering Al-Andalus, then a shining light of Islamic civilization. The first documented appearance of the Hindu-Arabic numerals in Christian Europe was in 976 in a manuscript by Vigila, a monk in the vicinity of Barcelona, then a Christian outpost in northern Spain near where Gerbert resided from 967 to 970. Gerbert brought the Hindu-Arabic numerals he encountered in Spain back to Christian Europe for an abacus / counting board he invented which was subsequently named after him.

Blaise Pascal (1623–1662)

Pascal's celebrated paper Traité du triangle arithmétique (Treatise on the Arithmetic Triangle)^[1] was written in 1654 and published in 1665. Though since named for him because of this work, "Pascal's" Triangle was long known in Europe before 1650 and in other civilizations too^[2]. The Traité contains a short appendix, Potestatum Numericarum Summa (The Sum of Powers of Numbers)^[3], showing how to derive polynomial expressions for sums of powers of integers like:
\begin{equation}{\sum_{k=1}^n k^2 = {1^2 + 2^2 + 3^2 + \cdots + n^2} = {{1 \over 6} n (n + 1)(2n+1)} = {{1 \over 3}n^3 + {1 \over 2}n^2 + {1 \over 6}n}.}\tag{1} \end{equation}

How CORDIC trig works, courtesy of Wikipedia.

CORDIC is is a complex of fast algorithms to calculate transcendental functions using only table lookup, addition and bit shifting. Here I take up Volder's original scheme from 1959 to calculate sines and cosines quickly (CORDIC stands for COordinate Rotation DIgital Computer). My original article from 1992 holds up reasonably well, The CORDIC Method for Faster sin and cos Calculations, except for the Borland graphics — nice pre-Windows, but long-gone. Converting to Visual C requires only one change to the core routines related to moving from 16-bit to 32-bit integers (see comment in code). I've bundled up the source files for anyone who wants to take them into a Visual C Win 32 application, and here is a minimal C implementation. Wikipedia has a first-class article on this subject with lucid explanations and many pertinent references. Volder himself published an illuminating retrospective in 2000, explaining that the initial motivation was digitizing the navigation system of Convair's B-58 jet bomber (Volder worked for Convair).

David Hilbert, 1862-1943 (photo 1912)

It's easy to prove that \( e \) is irrational, but takes more work to prove that it's transcendental, meaning not the root of any equation:

\[ \begin{equation}{a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + a_n x^n = 0, \hspace{16pt} a_i \in \mathbb{Z}.}\tag{1} \end{equation} \]

Put otherwise, it is never the case that \( a_n e^n + a_{n-1} e^{n-1} + \cdots + a_1 e + a_0 = 0 \; \) for any choice of integers \( a_0, a_1, \cdots, a_{n-1}, a_n \). We already have the result for \( n = 0 \) since \( e \) is not an integer, and for \( n = 1 \) since \( e \) is not the root of any equation \( bx - a = 0 \) for integers \( a \) and \( b \) (simply restating that \( e \) is irrational). In 1840, Liouville proved that \( e \) is not the root of a quadratic equation with integer coefficients, but the final object is to prove that \( e \) never satisfies \( (1) \) for any choice of integers \( a_i \) and any \( n = 0, 1, 2, 3, 4, \ldots \).