# Top 10 mathematical innovations

Of all the mathematical innovations since ancient times, only some are worthy of multicentenary celebrations. Certainly logarithms, celebrating their 400th anniversary this year, are among them. Ranking where logarithms rate among the rest is subjective, of course, but I’d put them 10th (they’d be higher if everybody still used slide rules, though). Here are the rest of my Top 10 mathematical innovations, which you might as well read here because David Letterman isn’t going to get to them before he retires:

#### 10. Logarithms (John Napier, Joost Bürgi, Henry Briggs)

A great aid to anybody who multiplied or messed with powers and roots, logarithms made slide rules possible and clarified all sorts of mathematical relationships in various fields. Napier and Bürgi both had the basic idea in the late 16th century, but both spent a couple of decades calculating log tables before publishing them. Napier’s came first, in 1614. Briggs made them popular, though, by recasting Napier’s version into something closer to the modern base-10 form.

#### 9. Matrix algebra (Arthur Cayley)

An ancient Chinese math text included matrix-like calculations, but their modern form was established in the mid-19th century by Cayley. (Several others, including Jacques Binet, had explored aspects of matrix multiplication before then.) Besides their many other applications, matrices became extremely useful for quantum mechanics. In fact, in 1925 Werner Heisenberg reinvented a system identical to matrix multiplication to do quantum calculations without even knowing that matrix algebra already existed.

#### 8. Complex numbers (Girolamo Cardano, Rafael Bombelli)

Before Cardano, square roots of negative numbers had shown up in various equations, but nobody took them very seriously, regarding them as meaningless. Cardano played around with them, but it was Bombelli in the mid-16th century who worked out the details of calculating with complex numbers, which combine ordinary numbers with roots of negative numbers. A century later John Wallis made the first serious case that the square roots of negative numbers were actually physically meaningful.

#### 7. Non-Euclidean geometry (Carl Gauss, Nikolai Lobachevsky, János Bolyai, Bernhard Riemann)

Gauss, in the early 19th century, was probably the first to figure out an alternative to Euclid’s traditional geometry, but Gauss was a perfectionist, and perfection is the enemy of publication. So Lobachevsky and Bolyai get the credit for originating one non-Euclidean approach to space, while Riemann, much later, produced the non-Euclidean geometry that was most helpful for Einstein in articulating general relativity. The best thing about non-Euclidean geometry was that it demolished the dumb idea that some knowledge is known to be true a priori, without any need to check it out by real-world observations and experiments. Immanuel Kant thought Euclidean space was the exemplar of a priori knowledge. But not only is it not a priori, it’s not even right.

#### 6. Binary logic (George Boole)

Boole was interested in developing a mathematical representation of the “laws of thought,” which led to using symbols (such as x) to stand for concepts (such as Irish mathematicians). He hit a snag when he realized that his system required x times x to be equal to x. That requirement pretty much rules out most of mathematics, but Boole noticed that x squared does equal x for two numbers: 0 and 1. In 1854 he wrote a whole book based on doing logic with 0s and 1s — a book that was well-known to the founders of modern computer languages.

#### 5. Decimal fractions (Simon Stevin, Abu’l Hasan Al-Uqlidisi)

Stevin introduced the idea of decimal fractions to a European audience in a pamphlet published in 1585, promising to teach “how all Computations that are met in Business may be performed by Integers alone without the aid of Fractions.” He thought his decimal fraction approach would be of value not only to merchants but also to astrologers, surveyors and measurers of tapestry. But long before Stevin, the basic idea of decimals had been applied in limited contexts. In the mid-10th century, al-Uqlidisi, in Damascus, wrote a treatise on Arabic (Hindu) numerals in which he dealt with decimal fractions, although historians differ on whether he understood them thoroughly or not.

#### 4. Zero and 3. Negative numbers (Brahmagupta)

Brahmagupta, a seventh-century Hindu astronomer, was not the first to discuss negative numbers, but he was the first to make sense of them. It’s not a coincidence that he also had to figure out the concept of zero to make negative numbers make sense. Zero was not just nothingness, but a meaningful number, the number you get by subtracting a number from itself. “Zero was not just a placeholder,” writes Joseph Mazur in his new book Enlightening Symbols. “For what may have been the first time ever, there was a number to represent nothing.”

#### 2. Calculus (Isaac Newton, Gottfried Leibniz)

You know the story — Newton gets all the credit, even though Leibniz invented calculus at about the same time, and with more convenient notation (still used today). In any event, calculus made all sorts of science possible that couldn’t have happened without its calculational powers. Today everything from architecture and astronomy to neuroscience and thermodynamics depends on calculus.

#### 1. Arabic numerals

Did you ever wonder why the Romans didn’t do much creative quantitative science? Try doing a complicated calculation with their numerals. Great advances in Western European science followed the introduction of Arabic numerals by the Italian mathematician Fibonacci in the early 13th century. He learned them from conducting business in Africa and the Middle East. Of course, they should really be called Hindu numerals because the Arabs got them from the Hindus. In any case, mathematics would be stuck in the dark ages without such versatile numerals. And nobody would want to click on a Top X list. (Wait — maybe they would. But you won’t see any list like that on this blog.)