[Verse 1]
When you want to prove something's not true
And direct proof just won't break through
There's a clever trick that mathematicians do
Assume the opposite, then see what's new
[Pre-Chorus]
Give that assumption some rope to hang
Let it dance until it breaks its own legs
[Chorus]
Proof by contradiction, set the trap
Assume the negative, then watch it snap
When logic leads to something impossible
R and not-R dancing together
That's your signal the assumption's wrong forever
Contradiction tells the truth at last
[Verse 2]
Take root two, is it rational?
Let's assume it is, seems plausible
Say it's p over q in lowest terms
Greatest common divisor equals one, we confirm
[Chorus]
Proof by contradiction, set the trap
Assume the negative, then watch it snap
When logic leads to something impossible
R and not-R dancing together
That's your signal the assumption's wrong forever
Contradiction tells the truth at last
[Verse 3]
Square both sides, get p squared equals two q squared
So p squared is even, that much is clear
If p squared is even, then p is even too
Write p as two k, see what that will do
[Bridge]
Four k squared equals two q squared
Divide by two, q squared equals two k squared
Now q is even, but wait a minute there
Both p and q are even, that's not fair
We said their gcd was one, but even numbers share
A common factor two, contradiction's in the air
[Chorus]
Proof by contradiction, set the trap
Assume the negative, then watch it snap
When logic leads to something impossible
R and not-R dancing together
That's your signal the assumption's wrong forever
Root two's irrational at last
[Outro]
When something can't exist or isn't true
Contradiction's got the proof for you
Assume, derive, and watch it fall apart
That's the contradictory art