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Rationals are Nonobvious

Piper (my second grader) is learning fractions while we cook in the kitchen, and it is fascinating to see the ways in which fractions are not intuitive. For example, if you remove one egg from a four-egg recipe, it seems intuitive to her to remove one whole cup from the one-and-a-half cups of flour to match.

On the other hand, if you reduce a four-egg recipe to one egg, she gets the intuition right.

That matches the ancient history of fraction notation where Egyptians had a way to write "1/n" without generalizing to "m/n". They formed other fractions using sums (using only distinct n). That system is actually fully general but very complicated.

The ancient Babylonians had a more practical, but less general fractional decimal notation using base 60. (That is still used in minutes and seconds today.) And the Romans had a similarly practical but imprecise notation for "m/12".

The idea of allowing both numerator and denominator to be written was a Hindu innovation. Thanks to the Hindus, it is now possible to successfully pull off a cookie recipe no matter how many eggs are in the refrigerator.

Posted by David at January 2, 2009 02:01 PM