Math Blog | About Proof by Contradiction
My first post will be about proof by contradiction! (It's not a proof itself, haha)
I simply decided on this because I want to be able to explain proof by contradiction to others!
To put it simply,
when you want to prove something,
it is a method where you intentionally assume that 'it is wrong'..
If you follow the logic from that assumption,
you will encounter something 'strange' or a 'contradiction'.
The fact that a contradiction arises means that the original assumption was wrong, which means you can conclude that
**'What I wanted to prove at the beginning is correct!'**
Want to prove proposition A
↓For now, assume that proposition A does not hold ← This is important!
↓Calculating based on the assumption leads to a contradiction
↓The assumption is wrong
↓The proposition is correct
1. A simple example
Let's think about the proposition 'odd number + odd number = even number' using proof by contradiction.
↓Assume 'odd number + odd number ≠ even number (it becomes an odd number)'.
↓Odd numbers can be written in the form 2n+1, 2m+1.
(2n+1) + (2m+1) = 2n + 2m + 2 = 2(n+m+1)
This is a multiple of 2, which means it's an even number!
↓Contradicts the assumption!
↓Therefore, 'odd number + odd number = even number' is correct.
2. Example for high school students
√2 is an irrational number.
↓Assume '√2 can be expressed as a fraction'. ← (because fraction = rational number)
√2 = a/b (a, b are integers, in simplest form)
↓Squaring both sides gives
2 = a² / b²
→ a² = 2b²
Since a² is even, a is also even.
→ If a = 2k, then a² = 4k²
Substituting this gives 4k² = 2b²
→ b² = 2k², so b is also even.
↓This means both a and b are even.
But we assumed it was in 'simplest form', so both being even is a contradiction!
➡ Therefore, the assumption that '√2 can be expressed as a fraction' is wrong.
↓√2 is an irrational number
🔑 The key point is to "deliberately assume the opposite, find a contradiction, and conclude that the original statement must be true."