Math is fascinating to me, though I struggled with math in high school and only took the minimum I needed. (Age changes things, man.) I’m reading a book on Wiles’ proof for Fermat’s Last Theorem and got curious about proofs.
At what point does something move from an assumption with examples (well yeah. Look at this) to a full proof?
Simple example that came to mind:
For any number n, where n is a prime >2, the sum of the factors of n cannot be odd.
In: Mathematics
> For any number n, where n is a prime >2
That’s a bit of a weird example, as the “proof” is the definition of a prime. All primes n > 2 have two factors, 1, and the odd number n. if it didn’t it wouldn’t be a prime.
But the main goal of proofs is to take some assumptions and logical steps, and use them to show the truthfulness of the thing you want to prove. If you take a true assumption and apply true steps to those assumptions, everything you derive is also true. If you end in a logical contradiction either the steps you applied are false, or the assumption.
Let’s try a different proof, the proof that sqrt(2) is not rational.
1. Assume that sqrt(2) can be expressed with the integer ratio a/b. (the definition of a rational number).
2. Assume a/b is in it’s lowest form (a and b share no factors).
3. If sqrt(2) = a/b, then 2 = a^2 / b^2 (square both sides)
4. rearrange, a^2 = 2b^2
5. a^2 thus is even. If a^2 is even, then it must be that a itself is also even (odd numbers squared are always odd).
6. a is even, and thus can be represented as 2k, where k is a positive integer.
7. 2 = (2k)^2 / b^2
8. rearrange, 2b^2 = 4k^2 => b^2 = 2k^2
9. By the same logic as steps 4 to 6, b is even and can be represented as 2j where j is a positive integer.
10. therefore, sqrt(2) = 2k / 2j
11. This is a contradiction, as 2k and 2j share a factor of 2: however our assumption in 2 stated they had no common factors.
12. Thus, our assumptions about the rationality of sqrt(2) are false, and therefore sqrt(2) is irrational.
QED (end of proof).
Here we started with an assumption ( The square root of 2 is rational), and from there just took steps to show what the result of that assumption is. Eventually we arrived at something that we know is false, and so we can see the original assumption was false, since that’s the only thing we used that wasn’t automatically know to be true.
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