Pi Day, and all that.

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What I'm going to say --my usual diatribe against Pi Day-- won't make sense unless you know a little of the background.

Our present way of representing numbers is based on the number 10, which is why it is called the Decimal System.  The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 all have absolute meanings, but we can put them together to make compound symbols, which represent a variety of numbers far greater than you would expect from the compact symbols themselves.  For instance, a person unfamiliar with the number system, such as an ancient Roman, for instance, would assume that three -digit numbers, such as 999, were roughly three times as big as a 1-digit number, such as 9.  We of course know that 999 is a much more enormous number than three times 9.

Why 10?  Because, anthropologists believe, we have ten fingers.  In fact some primitive societies are said to use base 11 (a society in which, I should explain, they use for counting the ten fingers and another body part that remains a mystery).

As I have described in earlier posts, there are interesting alternative bases for place-value number representation, including binary (base 2), octal (base 8) and hexadecimal (base 16), all used in computer science.  The number 273, for instance looks different in each of these.

In base 10, we don't quite think of it this way, but 273 really means
"2 times 100 + 7 times 10 + 3."

In base 8, instead of 100, and 10, which are powers of 10, we use 64 and 8 instead.  So to get 273, we have to first figure the largest multiple of 8x8x8 that can be subtracted out of 273.  Let's see ... that's 512, which is too big.  So let's go down to: what is the largest multiple of 64 that can be subtracted from 273?  I think 4 times 64, which would be 256.  Let's see: 273 take away 256 is ... 15.  Now, what is the largest multiple of 8 that can be taken from 15?  Just one, which leaves 7.  So, the number 273 would be represented, in base 8, as 417_[8].  Notice that all the digits we used, 4, 1, and 7, all lie between 0 and 7.  (Notice that in Decimal representation, all our digits lie between 0 and 9.)

In base 2, we use the digits 0 and 1 only.  We must use multiples of 1, 2, 4, 8, 16, 32, 64, 128, 256, and so on.  The number 1 is written the usual way.  The number 2 is written as 10.  The number 4 is written as 100.  The number 8 is written as 1000.  The number 16 is written as 10000.  You get the idea.  (Putting an extra 0 at the end of one number gives you double that number.)

To represent 273, we write it as 256 + 8 + 4 + 2 + 1, which would be written as  100001111_[2] in binary representation.

Other bases are just as usable; the few above have gained importance because of how computers are implemented electronically.  Using chips, we can build a vast array of memory units, each of which can be "up" (representing a 1) or ("down" representing a 0), so binary representation is particularly useful.

Fractions can be represented with only a little more work.  For instance, 3.14159 simply means 3 + (1/10) + (4/100) + (1/1000) + (5/10000) + (9/100000), right?  Let's see:

3.14159
= 2 + 1 + .14159
= 2 + 1 + 1/2 (.28318)
= 2 + 1 + 1/4 (.56636)
= 2 + 1 + 1/8 (1.13272)
= 2 + 1 + 1/8 + 1/8 (.13272)
= 2 + 1 + 1/8 + 1/16 (.26544)
= 2 + 1 + 1/8 + 1/32 (.53088)
= 2 + 1 + 1/8 + 1/64 (1.06176)
= 2 + 1 + 1/8 + 1/64 + 1/128 (.12352), and proceeding similarly,
= 2 + 1 + 1/8 + 1/64 + 1/2048 + 1/4096 + 1/8192 + 1/16384 + 1/32768 (1.62112)

So, 3.14159 would be represented by 11.0010010000111111011010101..._[2] in binary.

Will it ever stop?  No.  The pattern will repeat however, because 3.14159 is a simple fraction, 314159/100000.  The place-value representation (think: representation as a "decimal") of any fraction will either stop (or, as we say, terminate) in any base, just like it does in decimal, or will repeat, like 2/7 = 0.2857142857142857142857....

Numbers like Pi are not simple fractions, so their representations in any base will continue forever.  The example we used above is a rough approximation of Pi, so it behaves like a simple fraction.  The real thing behaves differently.

So when people connect the value 3.14 with Pi, they are doing a number of crazy things.

First of all, they're ignoring the remaining digits.  So, okay, it is an approximation.  But a lot of non-mathematicians are not sharp enough to appreciate the difference between two numbers in the third decimal place.  These are your basic "dollars and cents" people.  In Canada, for instance, they've passed a law to get rid of their pennies, so that the smallest denomination will be their equivalent of a nickel.  Okay; I guess if you live in Canada, after a few years, people will be celebrating Pi day on  3/15, since the closest thing to 3.14 using only nickels and higher will be 3.15.

Secondly, 3.14 stands for 3 + 1/10 + 4/100.

But in the date 3/14, the 3 stands for the third month out of twelve months, not ten, and 14 stands for the 14th day out of 31, not out of 100.  It's crazy.

So Pi day arises because non-math people love to obsess over mathematical things that they can barely understand.  And they tend to celebrate these things in irrelevant ways.  They tend to make tunes based on the sequence 3-1-4-1-5-9- and so on, though that sequence only makes sense if you keep going forever.  This brings us to:

Thirdly, why celebrate the decimal representation of Pi?  Why not use Octal, or Binary, or some other base?  In fact, it just occurred to me: why not use base pi itself?

This is a crazy idea --probably not original-- because even to represent the number 4 would be a challenge.  Here would be the numbers that you could represent easily:

1 (which means 1),
2 (which means 2),
3 (which means 3),
10 (which means pi),
11 (which means  pi + 1, which, in turn, is a little more than four),
12 (which means pi + 2, which is a little more than five),

and so on.  1.1 would represent 1 + 1/Pi, and so on.

[I want to make clear that this is not a practical base for numbers, and is of only theoretical importance!]

So celebrating March 14th as Pi Day makes no sense, because  it celebrates the sequence of symbols ['3', '.', '1', '4'], which has only a tenuous relationship to the fabulous number Pi.  So, while the non-mathematical world (and the slightly-mathematical world, I suppose,) celebrates March 14 with great delight, as far as real mathematicians are concerned, it is of no importance, except for the sake of publicity.  So, carry on, by all means, but leave me out of it.

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