Zero: The Biography of a Dangerous Idea

by Charles Seife

Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity.

But as natural as zero seems to us today, for ancient peoples zero was a foreign—and frightening—idea.

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.

Nobody knows whether Gog the caveman had used the bone to count the deer he killed, the paintings he drew, or the days he had gone without a bath, but it is pretty clear that early humans were counting something.

In the very beginning of mathematics, it seems that people could only distinguish between one and many.

Over time, primitive languages evolved to distinguish between one, two, and many, and eventually one, two, three, many, but didn’t have terms for higher numbers.

These people count by twos. Mathematicians call this a binary system.

Modern mathematicians would say that Gog, the wolf carver, used a five-based or quinary counting system.

You don’t have to have a number to express the lack of something, and it didn’t occur to anybody to assign a symbol to the absence of objects.

the invention of the art of geometry.

Egyptians

never progressed beyond measuring volumes and counting days and hours.

they did not take their system of mathematics and turn it into an abstract system of logic.

not inclined to put math into their philosophy.

The Greeks were different; they embraced the abstract and the philosophical, and brought mathematics to its ...

Eastern invention: the Babylonian style of counting. And thanks to this system, zero finally appeared in the East, in the Fertile Crescent of present-day Iraq.

each grouping had a different value, depending on its position. In this way the Babylonian system was not so different from the system we use today. Each 1 in the number 111 stands for a different value; from right to left, they stand for “one,” “ten,” and “one hundred,” respectively. Similarly, the symbol in stood for “one,” “sixty,” or “thirty-six hundred” in the three different positions.

How would a Babylonian write the number 60? The number 1 was easy to write: . Unfortunately, 60 was also written as ; the only difference was that was in the second position rather than the first.

could represent 1, 60, or 3,600. It got worse when they mixed numbers. The symbol could mean 61; 3,601; 3,660; or even greater values.

Zero was born out of the need to give any given sequence of Babylonian digits a unique, permanent meaning.

nowadays everybody knows that zero can’t really sit anywhere on the number line, because it has a definite numerical value of its own. It is the number that separates the positive numbers from the negative numbers. It is an even number, and it is the integer that precedes one. Zero must sit in its rightful place on the number line, before one and after negative one.

zero was dangerous.

zero was inexorably linked with the void—with nothing.

void and chaos.

The Greeks claimed that at first Darkness was the mother of all things, and from Darkness sprang Chaos. Darkness and Chaos then spawned the rest of creation.

The Hebrew creation myths say that the earth was chaotic and void before God showered it with light and formed its features.

To the ancients, zero’s mathematical properties were inexplicable, as shrouded in mystery as the birth of the universe.

Add a number to itself and it changes. One and one is not one—it’s two. Two and two is four. But zero and zero is zero. This violates a basic principle of numbers called the axiom of Archimedes,

Zero refuses to get bigger. It also refuses to make any other number bigger.

multiplication is a stretch—literally.

The rubber band has broken. The whole number line has collapsed.

For everyday numbers to make sense, they have to have something called the distributive property,

Thus, no matter what you do, multiplying a number by zero gives you zero. This troublesome number crushes the number line into a point.

multiply a number by zero: the number line is destroyed. Division by zero should be the opposite of multiplying by zero. It should undo the destruction of the number line. Unfortunately, this isn’t quite what happens.

that (2 × 0)/0 will get us back to 2. Likewise, (3 × 0)/0 should get us back to 3, and (4 × 0)/0 should equal 4.

Multiplying by zero collapses the number line. But dividing by zero destroys the entire framework of mathematics.

The whole Greek universe rested upon this pillar: there is no void.

in Greek mathematics there was no significant distinction between shapes and numbers. To the Greek philosopher-mathematicians they were pretty much the same thing. (Even today, we have square numbers and triangular numbers thanks to their influence [Figure 5].)

And to Pythagoras the connection between shapes and numbers was deep and mystical. Every number-shape had a hidden meaning, and the most beautiful number-shapes were sacred.

the most important property of the pentagram was not in this self-replication but was hidden within the lines of the star. They contained a number-shape that was the ultimate symbol of the Pythagorean view of the universe: the golden ratio.

In ancient Greece, Pythagoras was remembered for a different invention: the musical scale.

To Pythagoras, playing music was a mathematical act.

Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions.

This philosophy—the interchangeability of music, math, and nature—led to the earliest Pythagorean model of the planets.

In words, it doesn’t seem particularly special, but figures imbued with this golden ratio seem to be the most beautiful objects.

Zero was a number that didn’t seem to make any geometric sense,

a ratio of two numbers was nothing more than the comparison of two lines of different lengths.

rational.

these ratios had to be written in the form a/b, where a and b were nice, neat counting numbers like 1, 2, or 47.

Some numbers cannot be expressed as a simple ratio of a/b. These irrational numbers were an unavoidable consequence of Greek mathematics.