Zero - Charles Seife [59]
Figure 44: Covering the rationals
Zero.
How big are the rational numbers? They take up no space at all. It’s a tough concept to swallow, but it’s true.
Even though there are rational numbers everywhere on the number line, they take up no space at all. If we were to throw a dart at the number line, it would never hit a rational number. Never. And though the rationals are tiny, the irrationals aren’t, since we can’t make a seating chart and cover them one by one; there will always be uncovered irrationals left over. Kronecker hated the irrationals, but they take up all the space in the number line.
The infinity of the rationals is nothing more than a zero.
Chapter 7
Absolute Zeros
[THE PHYSICS OF ZERO]
Sensible mathematics involves neglecting a quantity when it is small—not neglecting it because it is infinitely great and you do not want it!
—P. A. M. DIRAC
It was finally unmistakable: infinity and zero are inseparable and are essential to mathematics. Mathematicians had no choice but to learn to live with them. For physicists, however, zero and infinity seemed utterly irrelevant to the workings of the universe. Adding infinities and dividing by zeros might be a part of mathematics, but it is not the way of nature.
Or so scientists had hoped. As mathematicians were uncovering the connection between zero and infinity, physicists began to encounter zeros in the natural world; zero crossed over from mathematics to physics. In thermodynamics a zero became an uncrossable barrier: the coldest temperature possible. In Einstein’s theory of general relativity, a zero became a black hole, a monstrous star that swallows entire suns. In quantum mechanics, a zero is responsible for a bizarre source of energy—infinite and ubiquitous, present even in the deepest vacuum—and a phantom force exerted by nothing at all.
Zero Heat
When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science.
—WILLIAM THOMSON, LORD KELVIN
The first inescapable zero in physics comes from a law that had been in use for half a century. This law was discovered in 1787 by Jacques-Alexandre Charles, a French physicist already famous for being the first to fly aboard a hydrogen balloon. Charles isn’t remembered for his aeronautic stunts, but for the law of nature that bears his name.
Charles, like many physicists of his time, was fascinated with the very different properties of gases. Oxygen makes embers burst into flame, while carbon dioxide snuffs them out. Chlorine is green and is deadly; nitrous oxide is colorless and makes people giggle. Yet all these gases have very basic properties in common: heat them up and they expand; cool them down and they contract.
Charles discovered that this behavior is extremely regular and predictable. Take an equal volume of any two different gases and put them in identical balloons. Heat them up by the same amount and they expand by the same amount; cool them down together and they contract in unison. Furthermore, for each degree up or down you go, you gain or lose a certain percentage of the volume. Charles’ law describes the relationship of the volume of a gas to its temperature.
In the 1850s, however, William Thomson, a British physicist, noticed something odd about Charles’ law: the specter of zero. Lower the temperature and the volume of the balloons gets smaller and smaller. Keep lowering at a steady pace and the balloons keep shrinking at a constant rate, but they cannot go on shrinking forever. There is a point at which gas, in theory, takes up no space at all; Charles’ law says that a balloon of gas must shrink to zero