Brian Clegg - Math: A Crash Course

For many uses, the math developed over 2,000 years ago largely serves today. No one needs topology to deal with the basics of everyday life. But from the sixteenth century onward, the possibilities for applying math blossomed, becoming applicable not just to reality, but to what might beprobability, the mathematics of chance, for instance, would transform our ability to think about the future.

Early mathematicians had dealt only with futures where there were distinct rules. Given, say, a known rate of interest, it was possible to calculate exactly how much money you might have in a bank account in a years time (if nothing was added or removed). These were very artificial conditions. The real world was full of unknowns. But, as US defense secretary Donald Rumsfeld would famously observe, there are both known unknowns and unknown unknowns. Probability would prove an ideal mechanism for dealing with known unknowns. We dont know what will come up when we toss a coin, but we do know that, with a fair coin, theres a 50 percent chance of a head and a 50 percent chance of a tail.

Sometimes a new field of math opens up when there is a clear application. In the seventeenth century, Isaac Newton and Gottfried Leibniz devised the techniques now collectively known as calculus in order to deal with specific problemshow to work out, for example, the distance traveled when a moving body is accelerating. However, in many cases, mathematicians were simply happy exploring their mathematical universe without any thought of an application. Yet surprisingly often, such apparently pointless work would later have major practical applications.

All manner of mathematical oddities have eventually proved useful. Imaginary numbersthe square root of negative numberswere devised long before applying them; the sixteenth-century Italian mathematician Girolamo Cardano commented that an imaginary number was as subtle as it is useless. Yet by the nineteenth century they were widely employed by both physicists and down-to-earth electrical engineers, and continue to be used today.

Another example is the importance of math in the history of computing. Originally, computers were people. Up to the 1940s, the term largely applied to individuals who performed manual calculations with pencil and paper. The potential advantages of mechanizing the work of a computer were clear for centuries, and there were early attempts with mechanical calculators, but electronic computersfirst developed in the 1940stook the mathematical content to a whole new level.

Math and the digital era

Three huge mathematical leaps were required to develop the electronic infrastructure at the heart of the modern world. The first was using binary digits: no longer working with everything from 0 to 9, but simply with 0s and 1s. Humans could, of course, have always done their mathematics this waybut it would have been tedious in the extreme. However, for a computer there is no disadvantage, and a huge positive. The values 0 and 1 can be represented very easily by physical things. A switch that is on or off. A device that either has an electrical charge or no charge. Either/or situations are easily represented by electronic components.

Secondly, computers are designed around mathematical logic. Those same 0 or 1 values can also be considered as false or true. And in the nineteenth century, a mathematical approach to logic had already been developed. This would underpin the architecture of the innards of computers: they are, in essence, huge logic engines, connecting together vast numbers of very simple logic gatesmechanisms for making logical combinations or transformations of those 0s and 1s.

The final mathematical component came when electronics moved from vacuum tubes to solid-state devices such as transistors. The design of such devices required an understanding of quantum mechanics, the physics of the very small particles that, for example, make up matter. And quantum mechanics, like most modern physics, is extremely mathematical. Not only did solid-state design require an understanding of the tools of higher mathematics, quantum physics has probability at its heart. Without a profound understanding of mathematics, the modern computer could never have been constructed.