Zero: The Biography of a Dangerous Idea

by Charles Seife

“The Achilles,” Zeno proves that swift Achilles can never catch up with a lumbering tortoise that has a head start.

The Greeks were stumped by the problem, but they did find the source of the trouble: infinity.

The infinite must be approached very carefully, but it can be mastered, with the help of zero.

Greeks did not have zero, but we do, and it is the key to solving Zeno’s puzzle.

To them, the numbers 1, 1/2, 1/4, 1/8, 1/16, and so forth aren’t approaching anything;

that the terms have a limit;

They pondered the concept of the void but rejected zero as a number,

The atomists, for example, believed that the universe is made up of little particles called atoms,

Aristotle just wished infinity away by stating that it is simply a construct of the human mind.

there is no infinity; there are a finite number of spheres, and a finite number of things that are moving each other.

Something must be the ultimate cause of motion. Something must be moving the sphere of fixed stars.

God.

When Christianity swept through the West, it became closely tied to the Aristotelian view of the universe an...

Atomism became associated wi...

First, there could be an infinite amount of void—thus infinity exists. Second, there could be a finite amount of void, but since void is simply the lack of matter, there must be an infinite amount of matter to make sure that there is only a finite amount of void—thus infinity exists.

Aristotle’s arguments were accepted, the Greeks were forced to reject zero, void, the infinite, and infinity. There was a problem, though. It is not so easy to reject both infinity and zero. Look back through time. Events have happened throughout history, but if there is no such thing as infinity, there cannot be an infinite number of events. Thus, there must be a first event: creation. But what existed before creation? Void?

(It was a stretch, but many scholars bought the argument; some even chose the creation story as further evidence for God.

Archimedes, the eccentric mathematician of Syracuse. He

Archimedes leapt out of the tub and ran through the streets of Syracuse shouting, “Eureka! Eureka!” Of course Archimedes forgot that he was stark naked.

“silly, childish discussion”—whether the new century begins on the year 00 or the year 01—appears

But count aloud and you always start with “one,” unless

you’re a mathematician or a computer programmer.* It has to do with order. When we are dealing with the counting

was not absolute nothingness. It was a kind of formlessness without any definition. . . . True reasoning convinced me that I should wholly subtract all remnants of every kind of form if I wished to conceive the absolutely formless. I could not achieve this.

infinite and a horror of the void.

Medieval scholars branded void as evil—and evil as void. Satan was quite literally nothing. Boethius made the argument as follows: God is omnipotent. There is nothing God cannot do. But God, the ultimate goodness, cannot do evil. Therefore evil is nothing.

The very act of creation was out of a chaotic void, and theologians like Saint Augustine, who lived in the fourth century, tried to explain it away by referring to the state before creation as “a nothing something” that is empty of form but yet “falls short of utter nothingness.” The fear of the void was so great that Christian scholars tried to fix the Bible to match Aristotle rather than vice versa.

Where there is the Infinite there is joy. There is no joy in the finite. —THE CHANDOGYA UPANISHAD

Love

remote India was insulated from the

Numbers had finally become distinct from geometry;

This was the birth of what we now know as algebra.

Nowadays we recognize that 2 – 3=–1: negative one. However, this was not at all obvious to the ancients.

Many times they solved equations only to get a negative result and concluded that their answer had no meaning.

It had a specific value, a fixed place on the number line. Since zero was equal to 2 – 2, then it had to be placed between one (2 – 1) and negative one (2 – 3).

No longer could zero sit to the right of nine, just as it does on the top of the computer keyboard;

After all, zero multiplied by anything is zero; it sucks everything into itself. And when you divide with it, all hell breaks loose.

Brahmagupta tried to figure out what 0 ÷ 0 and 1 ÷ 0 were, and failed.

In time the Indians realized that 1 ÷ 0 was infinite.

“This fraction of which the denominator is a cipher,

is termed an infinite quantity,” wri...

God was found in infinity—and in zero.

Does man forget that We created him out of the void?

Islam would take zero from India—and the West would eventually take it from Islam.

zero cifra, which became cipher.

the atomists’ ideas; after all, now that zero was around, the void was again a respectable idea.

The Bible told of the creation from the void, while the Greek doctrine rejected the possibility.

Nothingness is being and being nothingness. . . . Our limited mind can not grasp or fathom this, for it joins infinity.

The infinite and the void go hand in hand, and are both part of the divine creator.

fall of Aristotle and the rise of the void and the infinite were clearly beginning.

Worse yet, if the universe were infinite, then there could be no center. How could Earth,

then, be the center of the universe? The answer was found in zero.