Zero - Charles Seife [50]
This logic may seem like splitting hairs, like an argument as mystical as Newton’s “ghosts,” but in reality it’s not. It satisfies the mathematician’s strict requirement of logical rigor. There is a very firm, consistent basis for the concept of limits. Indeed, you can even dispense with the “I challenge you” argument entirely, as there are other ways of defining a limit, such as calling it the convergence of two numbers, the lim sup and lim inf. (I have a wonderful proof of this, but alas, this book is too small to contain it.) Since limits are logically airtight, by defining a derivative in terms of limits, it becomes airtight as well—and puts calculus on a solid foundation.
No longer was it necessary to divide by zeros. Mysticism vanished from the realm of mathematics and logic ruled once more. The peace lasted until the Reign of Terror.
Chapter 6
Infinity’s Twin
[THE INFINITE NATURE OF ZERO]
God made integers; all else is the work of man.
—LEOPOLD KRONECKER
Zero and infinity always looked suspiciously alike. Multiply zero by anything and you get zero. Multiply infinity by anything and you get infinity. Dividing a number by zero yields infinity; dividing a number by infinity yields zero. Adding zero to a number leaves the number unchanged. Adding a number to infinity leaves infinity unchanged.
These similarities were obvious since the Renaissance, but mathematicians had to wait until the end of the French Revolution before they finally unraveled zero’s big secret.
Zero and infinity are two sides of the same coin—equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers. The troublesome nature of zero lies with the strange powers of the infinite, and it is possible to understand the infinite by studying zero. To learn this, mathematicians had to venture into the world of the imaginary, a bizarre world where circles are lines, lines are circles, and infinity and zero sit on opposite poles.
The Imaginary
…a fine and wonderful refuge of the divine spirit—almost an amphibian between being and non-being.
—GOTTFRIED WILHELM LEIBNIZ
Zero is not the only number that was rejected by mathematicians for centuries. Just as zero suffered from Greek prejudice, other numbers were ignored as well, numbers that made no geometric sense. One of these numbers, i, held the key to zero’s strange properties.
Algebra presented another way of looking at numbers, entirely divorced from the Greek geometric ideas. Instead of trying to measure the area inside a parabola as the Greeks did, early algebraists sought to find the solutions to equations that encode relationships between different numbers. For instance, the simple equation 4x - 12 = 0 describes how an unknown number x is related to 4, 12, and 0. The task of the algebra student is to figure out what number x is. In this case x is 3. Substitute 3 for x in the above equation and you will quickly see that the equation is satisfied; 3 is a solution for the equation 4x - 12 = 0. In other words, 3 is a zero or a root of the expression 4x - 12.
When you start stringing symbols together to get equations, you can wind up with something unexpected. For instance, take the above equation and change the - sign into a + sign. This leaves us with a very innocent-looking equation, 4x + 12 = 0, but the solution to that equation is now - 3, a negative number.
Just as Indian mathematicians accepted zero while Europeans rejected it for centuries, the East embraced negative numbers while the West tried to ignore them. As late as the seventeenth century, Descartes refused to accept negative numbers as roots of equations. He called them “false roots,” which explains why he never extended his coordinate system to the negative numbers. Descartes was a late holdover, a victim of his success in marrying algebra to geometry. Negative numbers had long been useful to algebraists—even Western algebraists. Negative numbers came up all the time in solving equations,