Friday, March 14, 2008

Happy Pi Day

March 14th is Pi Day. Pi is the Greek letter used to name the
ratio of two special lengths: the length around the circle's
edge, the circumference, and the length of the diameter of a
circle. That is, Pi is the ratio you get when you divide the
circumference length by the diameter length.

(Geeks may skip ahead, here.) That length, that ratio, is an
irrational number -- meaning that there is no fraction with an
integer on top and an integer on the bottom that will equal that
ratio. The ancient Greek only liked ratios of integers. (An
integer is, for our purposes, a counting number: 1, 2, 3...)
They liked ratios like 1/2 or 43/295, or even 22/7.

Because they liked these integer ratios so much, the Greeks tried
to come up with an integer ratio that they thought would be equal
to Pi. They used 22/7, which is good to four hundredths of one
percent (0.04%). It's good enough for most construction work.

Later people also tried to use integer fractions, and came up
with 355/113 -- which is 10,000 times more accurate than 22/7
(good to 0.000008%).

These days, we use a decimal point, rather than an integer ratio,
to show Pi. For construction engineers, usually 3.14 is
adequate. It's about as good as 22/7. By the way, this form is
why March 14th is called Pi Day: 3.14 treated as a month and day.

If you compute more digits of Pi you will better come understand
that you never get to a place where a sequence of digits starts
repeating over and over. That is the mark of an irrational
number -- no repetition. The integer fractions you're used to
seeing are different. One half, 1/2, is 0.5000... One third,
1/3, is 0.333... One fourth is 0.25000... One fifth, 1/5, is
0.2000... One sixth, 1/6,is 0.1666... And one seventh, 1/7, is
0.142857142857142857..., where the "142857" pattern repeats forever.

There are a lot of other weird things about Pi. While some are
more advanced than others, let's end with an oddball weirdity.
The first and second decimal places of Pi are the digits "1" and
"4", the digits that make up the "14th" for Pi Day. There is a
spot where Pi has six 9s in a row, in decimal places 762 through
767. Pi doesn't start repeating after this, it never repeats.

The start of this repetition of six 9s is called the Feynman
[Click on the title above, or date stamp below, to see the full post.]

being the practical joker he is, wanted to memorize the decimal
digit sequence Pi up to this point so that he could recite and
say "...2 1 1 3 4 9 9 9 9 9 9 and so on" -- acting as if Pi
really *did* repeat 9s forever.

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