Via Lara Alcock (How to Study as a Math Major, 2013: oxford university press), math proofs are like an internal combustion engine: we drive just fine without knowing how a car actually works, but if we are ever curious why a car operates the way it does, then we need to understand the internal combustion engine, which is the broader system that makes driving a reality. 

I've created this thread for people to post, share, discuss, or inquire about any proof they want.  It would be really cool if the proof was something beautiful to you. Perhaps we could even form a mini proof library.  

Here's a basic example to get things rolling.

Inquiry: suppose we have the even numbers 2,4,6.  We know that these numbers are even.  We also know that if we square these numbers then the squares are also even.  But, although we intuitively know this, how could we show that this result is true for 2,4,6, or more importantly, for all even integers x?

Definition of an Even Number: An integer n is even if n=2a for some integer a. (Hammack, Book of Proof, 2013, pg 89)


Proposition: If x is an even integer, then x^2 is even. 

Proof (direct). Suppose x is an even integer.  Then x=2a for some integer a via the definition of an even number.  Now substitute x=2a into x^2, which gives x^2=(2a)^2=4a^2=2(2a^2).  Consequently, x^2=2b for some integer b=2a^2.  Thus, x^2 is even by the definition of an even number.

Don't know much math, but that sounds like WIFOM to me.

My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.

(September 14, 2016 at 12:04 am)Nymphadora Wrote: My kid is studying some stuff in her 8th grade advanced honors algebra class. I'll have to run this by her to see if she understands. Once you start throwing letters into a math equation, I get lost. Letters don't belong with numbers for us dumb folk.

if she's doing advanced honors algebra, then this is just basic stuff for her.

Whatevers, cool, after a first reading, it looks watertight to me. As far as I am concerned, you could shorten your second step to
"
n^3-n=(n-1) * n * (n+1)
If n is odd, n-1 and n+1 are even. In fact, one of them is divisible by four because that os always the case for two consec. even numbers.
"
without introducing a. That made it harder to read and I don't see why you needed it.

I'll try to write an induction proof for it later, the problem seems perfectly suited for that

(September 14, 2016 at 1:59 am)Alex K Wrote: Whatevers, cool, after a first reading, it looks watertight to me. As far as I am concerned, you could shorten your second step to

n^3-n=(n-1) * n * (n+1)
If n is odd, n-1 and n+1 are even

without introducing a. That made it harder to read and I don't see why you needed it.

Glad to hear you say so.  Felt like sticking legs on a snake to me too.  I guess I was trying to make it more math-y.

The more I think about it, the more it becomes apparent that an induction proof could only be more complicated than yours because it would basically contain it.