A Few Mathematical Concepts that Blow My Mind (Part One)

As I said in my first post, the first real thing I discuss will have something to do with math. I was thinking about what specifically I wanted to do, so I decided to search the web as well as some of my old math notebooks for inspiration. While flipping around, I discovered the remnants of me showing a friend of mine the concept that 0.99999.... is equal to 1. Every time I see that proof, I still can't help but think "whoa!" and that's when I decided to make this post about an assortment of mathematical things that just blow me away. First, Let's start with the proof that 0.9999.... = 1:

• lets start with x = 0.99999....
• multiply by ten to get 10x = 9.999999......
• subtract by x on both sides 10x - x = 9.99999999.... - 0.99999....
• this gives you 9x = 9
• divide both sides by 9 to give you x = 1, which means 0.9999....=1

One of the things that gets me about this proof explained this way is that the technique used to perform this proof is vastly simple, yet we get a pretty profound conclusion. I find it to be quite beautiful, actually. to borrow from mathematician Steve Strogatz, "Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity." I find this symbolic interpretation to be beautiful as well. I think this concept is really cool if you apply it generally to all things as well. It's like the simplest of things that we thought we understood fully and thoroughly can still surprise us with hidden characteristics of vast complexity and deep profoundness. to illustrate, I'll pull up a quote of Gandalf from Lord of The Rings (isn't it only fitting that a wizard would be a good example to explain math?) "My dear Frodo. Hobbits really are amazing creatures. You can learn all there is to know about their ways in a month, and yet after a hundred years they can still surprise you." If I've lost you, All I'm trying to say is that Gandalf's quick understanding of  "all there is to know about Hobbits" is like our quick understanding of the concept of 1. both are simple and don't require much thought to wrap your head around. But, both of these things have another side more profound; in Hobbits, this is explained through Gandalf still being surprised about some of their actions, and in math, this is explained through .9999.... stretching on to infinity and giving us a much harder time understanding. This proof is great, but it doesn't really tell us much about how in the world .999.... and 1 represent the same real number. Let's take a different approach and prove this concept with the help of infinite series and sequences (and Wikipedia!).

Alright, so first let's think of the number 0.9999..... as an infinite series; that is to say that:

• 0.9999..... = b0.b1b2b3b4..... = b0 + b1(1/10 ) + b2(1/10)^2 + b3(1/10)^3 + b4(1/10)^4 + ....
•  now, with the convergence theorem, we can say that if  |r| < 1 then
  ar + ar^2 + ar^3 + ... = (ar)/(1-r)
•  now simply applying this theorem to our series, we get:
  0.999... = 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + .... = (9(1/10))/(1-(1/10)) = 1
  (I know this isn't the most visually appealing way to display this equation, but the blogger word processor doesn't seem to have a function that lets me display fractions in a more readable way. Next time, I'll try writing this all in a real google doc and see what happens when i try to paste it in here)

Pretty much what's happening here is that if we keep adding up all those 9's infinitely we're eventually going to get 1. This points to another rule in math that says the sum of an infinite series is defined as the limit of what that series approaches and this concept can be applied to any kind of infinitely repeating number like 0.999... They are two equally valid ways to express the same real number (but for all our sakes, just refer to it as 1 in conversation with someone else). Well, This post got longer than I had imagined it to be, so I've decided to give all the mathematical concepts I came up with their own section, and I'll post about them little by little along the way. I won't post them one right after the other so that I can have a little more diversity and so that I don't sound like a text book :P.  If anyone wants to discuss this or if I've made some sort of error somewhere, feel free to say something about it in the comments. thank you all for being awesome and taking the time to listen to me ramble on about math! For those of you who found this post interesting, as a sneak peak, I'll be discussing Euler's identity in part 2, although I don't know yet when I'll begin writing about it.
-Carlos