Why Beauty is Truth (33 page)

Stevin published an essay describing his new notational system. He was sufficiently alert to marketing issues to include a statement that the ideas had been subjected to “a thorough trial by practical men who found it so useful that they had voluntarily discarded the short cuts of their own invention to adopt this new one.” Further, he claimed that his decimal system “teaches how all computations that are met in business may be performed by integers alone without the aid of fractions.”

Stevin's notation does not use today's decimal point, but it is directly related. Where we would write 3.1416, Stevin wrote 3
1
4
1
6
.
The symbol
indicated a whole number,
indicated one tenth,
one hundredth, and so on. As people got used to the system they dispensed with
,
, etc., and retained only
, which mutated into the decimal point.

We can't actually write the square root of two in decimals—not if we ever plan to stop. But neither can we write the fraction ⅓ in decimals. It is close to 0.33, but 0.333 is closer, and 0.3333 is closer still. An exact representation exists—to use that word in a novel way—only if we contemplate an infinitely long list of 3's. But if that's acceptable, we can in principle write down the square root of two exactly. There's no evident pattern in the digits, but by taking enough of them we can get a number whose square is as close to 2 as we please. Conceptually, if we take
all
of them, we get a number whose square is exactly 2.

With the acceptance of “infinite decimals,” the real number system was complete. It could represent any number required by a businessman or mathematician to any desired accuracy. Every conceivable measurement could be stated as a decimal. If it was useful to write down negative numbers, the decimal system handled them with ease. But no other kind of number could possibly be needed. There were no gaps left to fill.

Except.

That confounded cubic formula of Cardano's seemed to be telling us something, but whatever it was, it was terribly obscure. If you started with
an apparently harmless cubic—one where you
knew
a root—the formula did not give you that answer explicitly. Instead it offered a messy recipe requiring you to take cube roots of things that were even messier, and those things seemed to ask for the impossible, the square root of a negative number. The Pythagoreans had balked at the square root of two, but the square root of minus one was even more baffling.