Mar 8, 2010

The meaning of π (3)


Recall the last episode. π is an irrational number? Irrational number are infinitely long? How do you know? How do we get to the end of the rainbow?

Sit back and don’t relax. Rational numbers can be expressed by a fraction m/n, where m and n are integers, with n non-zero. For example, 4 is a rational number (4/1 = 4), 0.25 is rational (1/4 = 0.25), and so on. Irrational numbers, then, by definition, cannot be expressed by such a fraction.
The first famous example for an irrational number was the square root of 2, and when Hippasus, a member of the Pythagorean school of mathematicians (Pythagoras, him, the author of the notorious theorem)…when Hippasus came up with the first proof, he was apparently not lauded for his efforts: according to legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”
How do “we” know π is irrational? Well, “we” did not, for quite some time, although “we” had our suspicions, because all efforts to represent π by a rational number failed. So we had to wait until the 18th century, as the proof requires some serious (infinitesimal) calculus, which had to be invented first (by Newton and Leibniz).
How do we know that irrational numbers have an infinite decimal expansion? Well, because the long division of a rational number either terminates (1/4), or yields a repeating sequence (1/3, for example, or 1/81).

OK, so, π is irrational, and hence enjoys an infinite, non-repetitive decimal expansion. There we go. Equipped with the Infinite Monkey Theorem (last episode) we will be able to find any arbitrary message in the decimal expansion of π. Ellie’s efforts futile, Jodie Foster mislead, Carl Sagan misleading…(first episode)? No conspiracy?

No conspiracy? You mean that's a happy ending?

Stay tuned.

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