There many types of mathematical proofs like proof by induction, proof by contradiction, etc...
My favorite mathematical proof is proof by contradiction which was used to prove that there are an infinite number of prime numbers. It goes something like this (i'm skipping the math formalism since I don't remember it)
First you need to prove that adding 1 to a number gives a completely different set of prime numbers for the original.
1) Take a number x that is a composite of a set of prime numbers A={a1,a2,..}. e.g. 36 = 2*2*3*3
2) take any prime number a in set A. e.g. a=3
3) the ratio x/a has no remainder because a is one composition of x. e.g 36/3=(2*2*3*3)/3=(2*2*3)=18
4) lets define y=x+1 which is a composite of a set of prime numbers B={b1,b2...}. e.g. 37=36+1
5) the ratio y/a=(x+1)/a=x/a+1/a which gives a remainder of 1/a because the smallest prime number is 2. e.g. 37/3 = (36+1)/3=36/3+1/3=18+1/3
6) set B does not have the prime number a in it.
7) a was a random choice, we can choose any prime number from set A get the same result
8) Therefore, there are no prime numbers that are in set A that are also in set B.
Now we do the prove there is an infinite number of prime numbers by proof by contradiction
1) Assume there is a finite number of prime numbers
2) Make the number x which is a composite of all the prime numbers
3) Let y=x+1
4) By my earlier proof, y doesn't have any of the prime numbers x has.
5) y is then a composite of prime numbers that are not part the set of all prime numbers. Contradiction.
6) Our assumptions is wrong. There is an infinite number of prime numbers.
My favorite mathematical proof is proof by contradiction which was used to prove that there are an infinite number of prime numbers. It goes something like this (i'm skipping the math formalism since I don't remember it)
First you need to prove that adding 1 to a number gives a completely different set of prime numbers for the original.
1) Take a number x that is a composite of a set of prime numbers A={a1,a2,..}. e.g. 36 = 2*2*3*3
2) take any prime number a in set A. e.g. a=3
3) the ratio x/a has no remainder because a is one composition of x. e.g 36/3=(2*2*3*3)/3=(2*2*3)=18
4) lets define y=x+1 which is a composite of a set of prime numbers B={b1,b2...}. e.g. 37=36+1
5) the ratio y/a=(x+1)/a=x/a+1/a which gives a remainder of 1/a because the smallest prime number is 2. e.g. 37/3 = (36+1)/3=36/3+1/3=18+1/3
6) set B does not have the prime number a in it.
7) a was a random choice, we can choose any prime number from set A get the same result
8) Therefore, there are no prime numbers that are in set A that are also in set B.
Now we do the prove there is an infinite number of prime numbers by proof by contradiction
1) Assume there is a finite number of prime numbers
2) Make the number x which is a composite of all the prime numbers
3) Let y=x+1
4) By my earlier proof, y doesn't have any of the prime numbers x has.
5) y is then a composite of prime numbers that are not part the set of all prime numbers. Contradiction.
6) Our assumptions is wrong. There is an infinite number of prime numbers.
