[Verse 1]
Pythagoras thought he knew the score
Every number dressed as fractions wore
But square root two came knocking loud
A rebel in the rational crowd
Suppose it's p divided by q
Reduced completely, nothing new
Let's chase this tale where logic leads
And watch assumptions crack and bleed
[Chorus]
Assume, derive, contradict
Follow where the math predicts
Both are even, both divide
But reduced means nothing hides
Square root two breaks all the rules
Gaps exist where ratios cruise
Proof by contradiction shows
What we can't know, now we know
[Verse 2]
If root two equals p over q
Then two q squared equals p squared too
So p squared must be even, friend
Which means that p itself must bend
If p is even, write it double
Some integer without trouble
Substitute and calculate
Watch the contradiction's fate
[Chorus]
Assume, derive, contradict
Follow where the math predicts
Both are even, both divide
But reduced means nothing hides
Square root two breaks all the rules
Gaps exist where ratios cruise
Proof by contradiction shows
What we can't know, now we know
[Bridge]
Now q squared times two appears
Forcing q to join its peers
Both are even, both divisible
But we said irreducible
Give them rope to hang themselves
Logic empties all the shelves
This same pattern echoes through
Gödel's halls and Turing's zoo
[Verse 3]
Ancient Greeks tried to suppress
This numerical wilderness
All is number, they declared
But the number line has tears
Cantor's diagonals dance here
Halting problems interfere
Every time we think we're done
Mathematics has more fun
[Chorus]
Assume, derive, contradict
Follow where the math predicts
Both are even, both divide
But reduced means nothing hides
Square root two breaks all the rules
Gaps exist where ratios cruise
Proof by contradiction shows
What we can't know, now we know
[Outro]
So trust the contradiction's call
When assumptions start to fall
What seems impossible to prove
Shows us how the truth can move