Mathematics an example of mathematical proof?

Answer 1

A mathematical proof is a process that combines statements we know to be true to show something else must be true. Basically, it's just a way of saying something is true, along with why each step in the reasoning must be true.

So for example, assume we know "If a<b then a+c<b+c". Makes sense right? If we add the same thing to both sides, the right side still has more. If we treat this as a fact we know, we can use it to show other statements must also be true.

Now suppose we want to show that the average of two different numbers is between them in value. In other words, if x is the lesser number, then x<(x+y)/2<y. You might say "that seems like it would be true", but you might not be sure it works in every case (even strange cases).

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This is how a proof is used. We start with something we know is true (sometimes called "given") and transform it into something else true. For example, we can start with x < y. We said x is the lesser number after all.

We then know "x+x<x+y" because of our earlier reasoning "If a<b then a+c<b+c". To avoid proofs being huge, the reasoning is usually just given a name, but the same idea still holds. In this case, adding the same number to each side doesn't change the inequality. Similarly, we can show "x+y<y+y".

Then we can combine these two to show x+x<x+y<y+y, which we rewrite as 2*x<x+y< 2*y. If we divide everything by 2, we get x<(x+y)/2<y, which is what we wanted to show.