Wednesday, October 19, 2011

Circles Are Unreasonable

I was thinking last night about natural numbers, whole number, integers, rationals and irrational numbers. In particular it was rational and irrational numbers that made me think.

According to dictionary.com rational means "agreeable to reason" while irrational means "without the facility of reason; deprived of reason."  So what we are saying is that some numbers are "reasonable" and some numbers aren't.  The more I thought about it, the less I liked it.  What is so unreasonable about pi?

OK, in mathematical terms the meaning must be slightly different.  Rational numbers, according to Wikipedia, are "any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero."  Well, I guess that does sound reasonable.  If you can express it as a fraction, its rational.  In fact, Wiktionary tells me that "ratio" is related to the word "rational."  They both come from the latin word "rat," which means thought.  I get it.  Rational numbers are numbers that make sense, where as irrational numbers dont.


Lets take a look at some "irrational numbers."  The most famous of these is pi.  As we all know, pi is the ratio of a circle's circumference to its diameter.  Hey!  Wait a minute! pi is a ratio!  pi is diameter/circumference.  Isn't that a fraction?  Didn't we just say that rational numbers are numbers that can be expressed as a fraction?  Shouldn't this imply that pi is rational?  


What is so "unreasonable" about pi?  If we look closer, we see that the ratio must be a ratio of integers.  So, its not the ratio part that makes pi irrational.  Instead, this is saying that there is no unit of measurement, meters, feet, or angstroms, in which you can exactly measure both the circumference and the diameter of a circle and have it come out in even integers.  This isn't like a 3, 4, 5 triangle where the ratios of dimensions is exact integers.  No circles are just different.  Hmmm... I have to think about that one.


OK so you are thinking, "circles are weird, but there aren't so many other weird irrational numbers are there?  I mean there's pi, e square root of 2... but that's about it right?"  The answer is a resounding no.  In fact, a fellow named Cantor back in 1891 published something that is now known as Cantor's Diagonal Argument that shows that there are many MORE irrational numbers than rational ones.  They are all over the place.  So here's the thing.  By calling these numbers irrational, it implies that they don't make sense.  In fact, all it really is saying is that our number system isn't able to represent these numbers.  Just the way the ancient Roman's didn't have a way to represent zero, our number system can't represent pi.  Its not the fault of the numbers themselves, its the fault of our system for representing the numbers.

Our system for representing numbers - the one that all of our science and engineering is based on - is not able to represent most of the numbers out there.  I don't know about you, but I think this sucks.  Can you come up with a new way to represent quantities that can represent more than our current system?