Anyway, you may recall that in the last episode, I asked for topic suggestions for Episode 150. While I received a few unrelated emails from listeners, nobody was brave enough to actually sugget a topic. So, being in a glass-half-full kind of mood, I decided to treat the lack of suggestions as a suggestion in itself, and do a topic that has been on my back burner for a while, the history of the number zero.

To start with, we should clarify a few things about zero. In one sense, it is a simple representation of nothing. But perhaps even more importantly, it plays the critical role of a placeholder in our positional number system. For example, how do you know you are listening to episode 150 and not episode 15? It's because of that 0 in the ones place, which pushes the 5 into the tens place and the 1 into the hundreds place. This seems like an obvious idea now, but that wasn't always the case. You may recall that the Roman number system was basically non-positional: with a few minor exceptions, you essentially wrote a bunch of symbols down in any order, added their values, and got a total. Systems like the Roman numerals quickly grow cumbersome in the face of large numbers, and led to the confusing situation of many possible representations for the same number.

One of the earliest positional number systems was used by the Babylonians, well-established by the second millenium B.C. Their system was base-60 instead of base-10, and we still hear echoes of it today when we measure time or angles. Initially, they did not have a symbol for zero, which meant that written numbers were inherently ambiguous: you could not tell whether you had written 61 or 3601, 60 squared plus 1, because there was no symbol marking any non-used powers of 60 in the number. By the first millenium B.C., the system had been enhanced by several authors to use placeholder symbols such as a pair of wedges, but these were only used internally between two digits: trailing zeros in the lower places could only be identified by context. There must have been a lot of arguments with the waiter over the check in ancient Babylonian resturants.

Strangely, even when used by a few mathematicians, this placeholder concept did not take hold very quickly. Despite all their advances in mathematics, the Greeks didn't develop a true positional number system. This may have been partially due to the Euclidean emphasis on geometry, where numbers were respected mainly for their usefulness when talking about drawn figures. A few Greek and Roman astronomers, such as Ptolemy in 130 A.D., used the Babylonian system enhanced by a placeholder zero when recording their observations, but this was still considered an esoteric usage.

We should also mention that the native Olmecs and Mayans of the Americas independently developed a mixed base-20 and base-18 positional system as early as the 1st century B.C., which also included a true zero placeholder symbol. Their Long Count calendar basically counted the days from 3114 B.C, when Raised-up-Sky-Lord caused three stones to be set by associated gods at Lying-Down-Sky, First-Three-Stone-Place. While their numbers were mostly base-20, the second digit from the end rolled over whenever it hit 18 rather than 20. The mixed base is kind of strange, until you think about the fact that 20x18 is 360, very close to an actual year.

Most sources seem to agree that the widespread use of a true zero originated in India between 500-700 A.D. In Brahmagupta's treatise "The Opening Of The Universe", he laid out a number of rules for mathematical operations on numbers including zero. Some of his rules are very familiar to us today, such as adding zero to a negative number gives you a negative, and adding zero to a positive number gives you a positive. But oddly, he tried to define division by zero, claiming that zero over zero equals zero. Today we see that is clearly wrong: if a calculation results in 0/0, you need more context to figure out a reasonable interpretation. For example, look at the function y = x/x. You can see that this is 1 for all nonzero values-- so shouldn't it also be 1 for x=0, thus showing that 0/0 = 1? But on the other hand, look at y=2x/x. With the same reasoning, we find that 0/0 equals two!

The Hindu-Arabic number system, a positional base-10 system with zero, was originally brought into Spain in the 11th century by the Moors. Apparently it had spread into common usage among merchants in that civilization. It was popularized in Christian Europe by Leonardo of Pisa, or Fibonacci, in 1202. Even then it did not exactly take the continent by storm: while mathematicians embraced it, merchants continued using the non-positional Roman numeral system for several more centuries before it slowly died out in the face of a superior competitor. Except, of course, for the critical tasks of expressing motion picture copyright dates, or naming Popes.

As usual, I've barely scratched the surface here. Whole books have been written about the number zero, such as Robert Kaplan's highly regarded "The Nothing That Is". And you can also find some excellent online articles linked in the show notes. But hopefully this podcast has given you a non-zero number of things to think about.

And this has been your math mutation for today.

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