The Mathematical Proof Thread

9th September 2016, 10:09
(This post was last modified: 9th September 2016, 12:08 by Kernel Sohcahtoa.)
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.

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.

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)