kw: book reviews, nonfiction, mathematics, history

Did Pythagoras re-invent the "Pythagorean Theorem" on his own, or did he get it from the Babylonians (whom he visited), who'd already been using it for ten centuries? Do the Mayans use a "defective" base-20 arithmetic (20 alternating with 18), as most textbooks have it, or do their day counts mean "X years, Y months (of 20 days), Z days"? Did writing begin when the shapes of divers clay tokens, used for counting various commodities, were reproduced in wet clay?

You'll get probable answers to the first two questions in *How Mathematics Happened: The First 50,000 Years* by Peter S. Rudman. The answer to the third is in a book my brother is writing, but the issue is tangentially skimmed in Rudman's book.

Rudman is a lucid and engaging writer, and he has a lot to say. He makes it clear that his survey stops just prior to the Christian Era, to avoid getting into higher math than most people are comfortable with. So a more accurate subtitle, though less interesting, might be *From 50,000 to 2,000 years BP*.

Actually, considering that the Babylonians and others could solve quadratic equations using algebraic geometry, their scribes were well ahead of many modern high school graduates. When did mathematics proceed beyond simple "4-banger" arithmetic (add, subtract, multiply, divide) needed by animal herders and landowners? It isn't clear, though the author makes a good case that it occurred almost coincident with writing, which goes back six thousand years (and perhaps longer on perishable media).

Of particular interest is the evolution of numbering systems, that began with counting, then added replacement (a bigger pebble meaning 5 or 10 or whatever, and bigger or differently-shaped pebbles for bigger replacements—what we call "carries" in arithmetic). Some number systems developed a zero place-holder and a point (our "decimal point" separates digits that refer to whole things from those that refer to less-than-whole, like half an apple or a quarter-acre). This allowed units to be re-used once positional notation was invented. The Egyptians went partway to positional notation, but never developed either a zero or a point. Thus a number such as 24 needed a different "2", because the leftmost symbol still had to carry the meaning of "20".

Suppose we used **1-9** for singles, **A-I** for tens, and **a-i** for hundreds. Then 365 would be written **cF5**, and could be written **5Fc** or even **c5F**, with no loss of meaning, but a little longer thinking process to recognize and understand the quantity. Roman numerals have this property (365 = CCCLXV), and though they are conventionally written large-left-small-right...they don't have to be. This property plus the lack of a point makes hieroglyphic numbers tricky to decode, because if the text elsewhere says the scribe is interested in tens of sheep, **cF5** might refer to 3,650 sheep!

Mathematics before the Greeks got pretty sophisticated. What the Greeks added was the rigorous proof. In about 400 BC Hippasus proved that the square root of two was not a rational number (you couldn't write it as a/b where a and b are integers). Legend has it the Pythagoreans drowned him, as their philosophy required all numbers to be rational. When they were confronted with an irrational number, they became irrational philosophers!

Prior to proof, the Babylonians and Egyptians (and others) could construct or derive many powerful results, but such derivations did not always suffice for proof. With the Greek development, Mathematics got the final foundation stone for the great range of math techniques in use today.

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