Games have rules and math has rules, so in that sense, math is like a game. If you follow the rules of math, you win. What do you win? You win a truth, which up till now, you have been calling “a correct answer” – but there is more. . . You already know many math rules, like keeping an equation balanced, or a negative number multiplied by another negative number results in a positive number etc. etc. . . . . There are lots of these kinds of procedural rules for playing with numbers. But there are other types of rules that will become more and more important as you get further into math and they have to do with the properties of numbers and consequences of these properties. The most amazing thing about these rules is that they, combined with other rules can lead you to a truth than no one has discovered yet, in other words, this rule based game of exploration can take you where no one has gone before – and it is that aspect which keeps us pushing the boundaries of mathematics. Many new discoveries are beautiful (fractal geometry), or useful (chaos theory), or weird (quantum mechanics) and some are even a bit disturbing (the n-body problem). Now, the path that leads to a truth in mathematics is called a proof. Guess what? You have been doing proofs all this time, right since you first started to add numbers up until now. For example, if you have a problem like 7x - 10 = 5x + 6, you can prove, using the rules of the game of math, that x can only be equal to 8 in order that 7x - 10 = 5x + 6 becomes a true statement. This is an example of a proof using just the procedural rules. If at each step of the way, you obey the rules, you prove (arrive at the truth) that x = 8.
The proof you are asking about in this video is a proof that uses some properties of numbers and some concepts and their consequences. This takes a bit more considered thinking. The proof that √2 is irrational is the most common introduction to this type of thinking.
So, here we go . . . . .
First, would you agree that any rational number whose numerator and denominator are not co-prime, can be reduced to a co-prime form? (if you don’t agree, look into it, because it is true). Co-prime is just a fancy pants way of saying that the greatest common divisor between two numbers is 1. Lots of times you will hear it said as “reduced form”. So if you have 4/6, you can take out the 2 which is a common term between 4 and 6, to get 2/3 and then the only factor that 2 and 3 have in common is 1, right? This process is true for any rational number, that is, a rational number that is not already in co-prime form can be reduced to co-prime form. So, for example 6/9 = 4/6 = 2/3, the ratio is the same, the result is the same even though the numbers are different. This little detail becomes important in the proof . . . .
A common method of proof is called “proof by contradiction” or formally “reductio ad absurdum” (reduced to absurdity). How this type of proof works is: suppose we want to prove that something is true, let’s call that something S. If we start the proof by assuming that S is false, and then through a series of mathematically sound arguments, we can show that we get a nonsense or contradictory result, well then, that means that the assumption we made that S was false can’t be correct, so S must be true.
In this proof we want to show that √2 is irrational so we assume the opposite, that it is rational, which means we can write √2 = a/b. Now we know from the discussion above that any rational number that is not in co-prime form can be reduced to co-prime form, right? So for the sake of argument, let’s assume that we have done any needed reduction and now a and b are reduced (co-prime), meaning that the only common divisor between the numerator and denominator is 1. Now we square both sides of √2 = a/b to get 2=a²/b² and from there it is a short journey to show that since 2=a²/b², it means that both a and b must be even numbers which means they have a 2 in common. Oops! We said that they only had a 1 in common.
Here is the subtle part:
Even though it is true that having the number 2 in common is in contradiction to the statement that a and b are in reduced form (co-prime), this contradiction is not the big deal. The big deal is that we got to this stage by assuming that √2 was rational in the first place! By assuming that √2 is rational, we were led, by ever so correct logic, to this contradiction. So, it was the assumption that √2 was a rational number that got us into trouble, so that assumption must be incorrect, which means that √2 must be irrational.
Here is a link to some other proofs by contradiction:
https://nrich.maths.org/4717
https://en.wikipedia.org/wiki/Proof_by_contradiction
as an aside, you might want to look up the definition of mathematical soundness:
https://en.wikipedia.org/wiki/Soundness
