Since it's March 14th today, I am going to celebrate it somehow. So I will do a proof on why the heck √2 is irrational( like literally a serious proof).
(As if it's related...)
Assume that √2 is rational, which would imply that it could be expressed as a/b(you have to believe me that this holds true for all rational numbers), where a and b are relatively coprime integers( that means they are pretty close, like a = 1.4 something b).
Okay, now we have an equation: √2 = a / b
It would be more interesting if we square both side. So we get: 2 = a**2 / b**2 (excuse me for my manner using ** for the square operation. It doesn't mean bad words, just a standard way of expressing in a high level machine programming language like Python, which you guys will probably see in Uni).
Great! What does this mean? It means that the square of a is like twice the square of b.
Before we say anything, remind me that a and b are relative prime numbers, which means they are coprime, which means that they can not be all even numbers(Hint: coprime numbers can not have common factors besides 1, two even numbers mean that they all have a common factor of 2).
Alright, what the heck does it mean that the square of a is twice the square of b?
WELL, a**2 = 2(b**2)
- The term 2(b**2) is even.
- The square of a is gotta be even. Only even number's square is even. Therefore a is gotta be an even number.
Let a = 2c. We get 4(c**2) = 2(b**2). Therefore, b**2 = 2(c**2).
- Same logic we used previously, since 2(c**2) is even, b**2 is even, therefore b is an even number.
Heck, did we not just mention that "a and b can not be two even numbers" before we say anything?
So here we go, since there is no rational number a and b that would make √2 = a/b, √2 is irrational. Quod erat demonstrandum
What we proved could be expressed in the following first-order-logic expression:
¬ (∃ a ∈ N) (∃ b ∈ N) [ 2 = p**2 / q**2 ]
What's cool about it is that it allows us to see what given conditions, properties, and target equation very clearly. Which is very helpful to see how logical mathematics can be.
Which is why pi is not a rational number. Could someone prove it in a similar way?
Some reference + follow-up reading: