Monday, March 14, 2011

Cow Pi Day

Happy Pi Day!  Woo hoo!  March 14!  3.14 (btw, MIT releases admission decisions today at 1:59, the next three digits of pi).

Ok, enough happiness and joy and thoughtless love of mathematical constants that sound like yummy desserts.  I'm going to risk putting a big cow pie in your Pi Day and talk about the absurdities of the whole idea.

Base number systems

Pi (the ratio of the circumference to the diameter of ANY circle) is approximately 3.14159265358.  Well, that's what it is in base-10.  By base-10, I mean using a number system with 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9).  In this system we use these ten symbols to count up and then when we run out we use a combination of these symbols and the power of place value to describe larger values (372 is really shorthand for 3 hundreds, 7 tens, and 2 ones).  Humans haven't always used based-10 though and there isn't anything particularly special about it.  The Mayans used a base-20 system and the Babylonians used a base-60 system.  Yes, that means you'd have to learn 60 different symbols, but the number 3599 is only 2 digits.  Some believe that when computers take over the world we'll be forced to use a base-2 number system.  Computers like this system because they can really only understand two things: "electricity is flowing" and "electricity is not flowing."  You might have seen binary numbers before (0, 1, 10, 11, 100, etc).

Some mathematical concepts remain regardless of what base you're in.  A prime number in base 10 will be a prime number in base 2, base 20, base 60, and base 243112609-1 (which is currently the largest known prime number).  Pi, on the other hand, will not always be 3.14159265358.....

Pi in base-2?

First of all, what in the world does that even mean.  Remember how someone (that would be me) mentioned earlier that 372 is really shorthand for 3 hundreds, 7 tens, and 2 ones.  We can also think about this as $3\times10^2+7\times10^1+2\times10^0$.  Remember that $10^0$=1 (which can be explained using, among other methods, pattern sniffing with decreasing powers).  Well this isn't a coincidence and if we wanted to describe the number of days in a non-leap-year year in base-5, we'd write it as 2430 or $2\times5^3+4\times5^2+3\times5^1+0\times5^0$.  Well, we might use symbols other than 2, 4, 3, and 0 but you hopefully get the idea.  Anyway, this idea can be continued into the decimal realm.  The first few digits of pi in base-10 are 3.14159 which, in its expanded form, can be written as $3\times10^0+1\times\frac{1}{10^1}+4\times\frac{1}{10^2}+1\times\frac{1}{10^3}+5\times\frac{1}{10^4}+9\times\frac{1}{10^5}$ or $3\times10^0+1\times10^{-1}+4\times10^{-2}+1\times10^{-3}+5\times10^{-4}+9\times10^{-5}$. This just means that there are 3 ones, 1 tenth, 4 hundredths, 1 thousandth, 5 ten thousandths, and 9 hundred thousandths.  So what would this look like in base 2?  Well we'd just need to figure out how many eights, fours, twos, ones, halves, fourths, eighths, sixteenths, etc there are in pi.  Well that sounds like a fun project to figure out.

Pi Day in other bases
One nice thing about figuring out pi in other bases is that I can give you the answers without really giving anything away (the method is the hard...err fun...part).  So...

Base 2: 11.00100100001111 (1 two, 1 one, no halves and fourths, 1 eighth, etc)
Base 3: 10.01021101222201
Base 4: 3.02100333122220
Base 5: 3.03232214303343
Base 6: 3.05033005141512
Base 7: 3.06636514320361
Base 8: 3.11037552421026
Base 9: 3.12418812407442

Anyway, you can make the argument that base-10 is our normal base and that makes today special (although our calendar system really isn't a base 10 system.  It's more of a mixture of base-7, base-10, base-12, and base 28/29/30/31).  Sort of...

My point is not to rain on your Pi Day parade.  I'm all for having days where people actually want to talk about and think about math. I'm just saying that October 1st could also be Pi Day (in base 3).  Or March 2nd, 3rd, 5th, 6th, 11th, and 12th.

Maybe we should just call March Pi Month?

1. Dissidence Day: