Sunday, December 18, 2005
My favorite math proof
Most people think this a given. If you multiply a number (i.e. "something") by nothing, you get nothing. Simple enough. We all know this. But it's not a given, or in math-speak, not an axiom of the real numbers. The reason is that back in the days when the Greek philosophers (early mathematicians) were working out what numbers were and what theories rested with numbers, they didn't believe that 0 was a valid number. How can you have a symbol that represents the concept of nothing? they asked. It seemed counter-intuitive to them (and you can see why when you look at it like that, right?). So, every arithmetical calculation involving 0 that we take for granted had to be proven.
This history is only part of the reason that this is my favorite proof, though. The other reason is that it's such a slick proof. Concise, to the point, with a little trick that I just love! And now, you're curious what this proof looks like (even if you think you're not curious, I promise that you actually are). So, I'll type it out below. It looks better on a chalkboard, and if I could somehow put a chalkboard onto my blog and physically write the proof out on that chalkboard, I would. However, since that's not an option, this will just have to do (although I'm extremely sad that I can't use the shorthand symbols for "there exists," "such that," and "is contained in" because those always make the proof look so much cooler). Read on, and be amazed at the simplicity of this monumentous result of the real numbers...
Prove: x*0=0, where x is any real number.
Since x is a real number, and the set of real numbers is closed under multiplication, then x*0 is also a real number.
We know 0+0=0, thus x*0=x*(0+0)=x*0+x*0.
Every real number has an inverse, so there exists a y in the real numbers such that x*0+y=0.
It follows that x*0+y=x*0+x*0+y. Since x*0+y=0, using substitution, we see that x*0+x*0+y=x*0+0.
Since 0 is defined as the additive identity of the real numbers, then x*0+0=x*0, and thus x*0=0.
Ah, so slick. So sleek. So nice. Every time I go through that proof, it makes me smile. Hope you had fun with it, too!
Today.. I discovered how to prove x*0=0.
Today.. I haven't decided if I'm better off with that knowledge or not :-)