No letter was decided upon for "five thousand." In an cient times there was little need in ordinary life for num bers that high. And if scholars and tax collectors had oc casion for larger numbers, their systems did not percolate down to the common man.
One method of penetrating to "five thousand" and be yond is to use a bar to represent thousands. Thus, V would represent not "five" but "five thousand." And sixty-seven thousand four hundred eighty-two would be LX-VIICD LXXXII.
But another method of writing large numbers harks back to the primitive symbol (1) for "thousand." By adding to the curved lines we can increase the number by ratios of ten. Thus "ten thousand" would be (1)), and "one hundred thousand" would be (1) Then just as "five hundred" was 1) or D, "five thousand" would be 1)) and "fifty thousand" would be I))).
Just as the Romans made special marks to indicate tbou sands, so did the Greeks. What's more, the Greeks made special marks for ten thousands and for millions (or at least some of the Greek writers did). That the Romans didn't carry this to the logical extreme is no surprise. The Romans prided themselves on being non-intellectual. That the Greeks missed it also, however, will never cease to astonish me.
Suppose that instead of making special marks for large numbers only, one were to make special marks for every type of group from the units on. If we stick to the system I introduced at the start of the chapter-that is, the one in which ' stands for units, - for tens, + for hundreds, and = for thousands-then we could get by with but one set of nine syrrbols. We could write every number with a little heading, marking off the type of groups -+-'. Then for "two thousand five hundred eighty-one" we could get by with only the letters from A to I and write it GEHA. What's more, for "five thousand five hundred fifty-five" we could write EEEE. There would be no confusion with all the E's, since the symbol above each E would indicate that one was a "five," another a "fifty," another a "five hundred," and another a "five thousand." By using additional symbols for ten thousands, hundred thousands, millions, and so on, any number, however large, could be written in this same fashion.
Yet it is not surprising that this would not be popular.
Even if a Greek had thought of it he would have been re peucd by the necessity of writing those tiny symbols. In an age of band-copying, additional symbols meant additional labor and scribes would resent that furiously.
Of course, one might easily decide that the symbols weren't necessary. The Groups, one could agree, could al ways be written right to left in increasing values. The units would be at the right end, the tens next on the left, the hun dreds next, and so on. In that case, BEHA would be "two thousand five hundred eighty-one" and EEEE would be "five thousand five hundred fifty-five" even without the little symbols on top.
Here, though, a difficulty would creep in. What if there were no groups of ten, or perhaps no units, in a particular number? Consider the number "ten" or the number "one hundred and one." The former is made up of one group of ten and no units, while the latter is made up of one group of hundreds, no groups of tens, and ont unit. Using sym bols over the columns, the numbers could be written A and A A, but now you would not dare leave out the little sym bols. If you did, how could you differentiate A meaning "ten" from A meaning "one" or AA meaning "one hun dred and one" from AA meaning "eleven" or AA meaning "one hundred and ten"?
You might try to leave a gap so as to indicate "one hun dred and one" by A A. But then, in an age of hand-copy ing, how quickly would that become AA, or, for that mat ter, how quickly might AA become A A? Then, too, how would you indicate a gap at the end of a symbol? No, even if the Greeks thought of this system, they must obviously have come to the conclusion that the existence of gaps in numbers made this attempted simplification impractical.
They decided it was safer to let J stand for "ten" and SA for "one hundred and one" and to Hades with little sym bols.
What no Greek ever thought of-not even Archimedes himself-was that it wasn't absolutely necessary to work with gaps. One could fill the gap with a symbol by letting one stand for nothing-for "no groups." Suppose we use $ as such a symbol. Then, if "one hundred and one",is made up of one group of hundreds, no groups of tens, an one + - I unit, it can be written A$A. If we do that sort of thing, all gaps are eliminated and we don't need the little symbols on top. "One" becomes A, "ten" becomes A$, "one hun dred" becomes A$$, "one hundred and one" becomes A$A, "one hundred and ten" becomes AA$, and so on.
Any number, however large, can be written with the use of exactly nine letters plus a symbol for nothinc, Surely this is the simplest thing in the world-after you think of it.
Yet it took men about five thousand years, counting from the beginning of number symbols, to think of a sym bol for nothing. The man who succeeded (one of the most creative and original thinkers in history) is unknown. We know only that he was some Hindu who lived no later than the ninth century.
The Hindus called the symbol sunyo, meaning "empty."
This symbol for nothing was picked up by the Arabs, who termed it sifr, which in their language meant "empty." This has been distorted into our own words "cipher" and, by way of zefirum, into "zero."
Very slowly, the new svstem of numerals (called "Ara bic numerals" because the Europeans learned of them from the Arabs) reached the West and replaced the Roman sys tem.
Because the Arabic numerals came from lands which did not use the Roman alphabet, the shape of the numerals was nothing like the letters of the Roman alphabet and this was good, too. It rerroved word-number confusion and reduced gematria from the everyday occupation of anyone who could read, to a burdensome folly that only a few would wish to bother with.
The Arabic numerals as now used by us are, of course, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the all-important 0. Such is our reliance on these numerals (which are internationally accepted) that we are not even aware of the extent to which we rely on them. For instance, if this chapter has seemed vaauely queer to you, perhaps it was because I had delib eratclv refrained from using Arz.bic numerals all through.
We ail know the great simplicity Arabic numerals have lent 'Lo arithmetical computation. The unnecessary load they took off the human mind, all because of the presence of t' e zero, is simply incalculable. Nor has this fact gone unnot.ccd in the Engl'sh language. Tle importance of the zero is reflected in the fact that when we work out an arithmetical computation we are (to use a term now slightly old-fashioned) "ciphering." And when we work out some code, we are "deciphering" it.
So if you look once more at the title of this chapter, you will see that I am not being cynical. I mean it literally.
Nothing counts! The symbol for nothing makes all the dif ference in the world.
13. C For Celeritas
If ever an equation has come into its own it is Ein stein's e = mc 2. Everyone can rattle it off now, from the highest to the lowest; from the rarefied intellectual height of the science-fiction reader, through nuclear physicists, college students, newspapers reporters, housewives, busboys, all the way down to congressmen.
Rattling it off is not, of course, the same as understand ing it; any more than a quick paternoster (from which, in cidentally, the word "patter" is derived) is necessarily evi dence of deep religious devotion.
So let's take a look at the equation. Each letter is the initial of a word representing the concept it stands for.