WEEKLY WHINE

Whose zero was it?

One of my many errors has come, gone, and come again. At one time I claimed in Numbers that the Mayans invented zero. Then, on the advice of my pal Parag Pathak, I amended that to say that the Indians had beaten the Mayans out. But, as you might expect, the story is much more complex than that.

Like the other nine digits, zero exists as both a digit and a number. There is the number zero, and there is the 0 that we use to represent it. We also use 0 in numbers that aren't zero if necessary, like 507. But for some time nobody used anything to do that. In Babylon, cuneiform tablets that contained numbers would express them without zeroes. That is equivalent to writing 507 as 57 - the context shows whether 57 actually means 57, or 507, or 570, or 50,007, or whatever. Of course, the Babylonians' sexagesimal system means that far fewer numbers would require zeroes. In sexagesimal, 507 would be expressed as 8 27; that is, your digit that means eight followed by your digit that means 27.

But by 400 BC, they started using marks as placeholders, like 5"7 for 507. But they would never write 57" for 570. Hence these marks don't qualify as a placeholding zero. The first confirmed use of zero as a placeholder was by the Mayans in the seventh century AD, followed by the Hindus in the ninth century. Some items from India show evidence of a placeholding zero as early as AD 200, but the authenticity of these is in question.

So much for that. What about the number zero? The Indian mathematician Brahmagupta was apparently the first to identify zero as a number, defining it as what you get when you subtract a number from itself. He described what happens when you add or subtract zero, namely nothing. He also made an attempt to define division by zero, but only got so far as to say that a number divided by zero can only be expressed as x/0. Arab mathematicians in the 1100s and later cited the Hindus' use of the zero.

Who really invented the zero, then? The first known use of zero as a placeholder goes to the Mayans, but the first known use of zero as a number goes to Brahmagupta and the Hindus.

It's also interesting to note how long it took for zero to catch on. Quite a few mathematicians, especially pre-1600 Europeans, didn't use the number zero. For instance, Girolamo Cardano worked on solutions of cubic and quartic equations without the use of zero, a number that undoubtedly would have made his work much easier. Oddly enough, Cardano inadvertently stumbled upon complex numbers in his work.

Many of these facts come to you courtesy of the MacTutor History of Mathematics Archive at the University of St Andrews in Scotland.