Proof

Question 1

How to prove, by contradiction, that the square root of something (e.g. 2) is irrational?

Answer 1

State: Root 2 is a rational number
Prove:
set root 2 equal to a/b
square both sides
times both sides by b^2 - therefore a is even (anything squared is even)
therefore a = 2k
sub into 2b^2 = a^2
solve
so b is even

If a and b are both even then cannot be coprime, which is a contradiction

Conclusion: root 2 is irrational

Question 2

How to prove, by contradiction, that there are infinitely many prime numbers?

Answer 2

State: There is a definite amount of prime numbers
Prove:
Assume there is a finite number of prime numbers such that they can be listed: p1, p2, p3,…..,pn
- where all prime numbers are listed and pn is the largest prime number

consider a new number - N = p1 x p2 x ….. x pn+1

therefore N is not divisible by any of the prime numbers in the list as it will have a remainder of 1
or N is a new prime number greater than pn

this contradicts the list as it is complete

Conclusion: There are infinitely many prime numbers