√2 is irrational
It is really fun proving something by contradiction. Here is an interesting example I want to share.
We are going to prove that square root of 2 is irrational. By contradiction, we start with the assumption that √2 is rational. And later we’ll prove that our assumption is wrong and hence √2 is irrational.
Proof:
1. √2 is rational.
2. We know for a rational number, there exist two integers a and b with common factors no other than 1. Hence,
√2 = a/b.
3. Squaring both sides (since x=y implies x2=y2), we get
2 = a2/b2
4. Multiplying both side by b2, we get
2b2 = a2
5. Here, b is an integer and hence b2 is also an integer. Any integer multiplied by 2 is even. Hence 2b2 is even which implies that a2 is even.
6. Since a2 is even, a is also even. (If square of an integer is even, the integer is also even).
7. Since a2 is even, there exists some integer c such that,
a = 2c
8. Now, we can write
2b2= 4c2
Hence, b2= 2c29. Since c is an integer, c2 is an integer and hence b2 is even.
10. Hence this proves that b is also even.
11. Now we have proved that both a and b are even . This means a and b have a common factor 2, other than 1.
12. Hence, the assumption √2 is rational is false and hence √2 is irrational.
This is it. Isn’t it interesting?
