[Verse 1]
In the real numbers we thought we knew it all
Every equation had an answer when we'd call
But then we met x squared plus one equals zero
Searching for a solution, couldn't find our hero
The reals are complete, they're ordered and they're fine
But some equations cross a different kind of line
[Chorus]
There's a problem in our paradise
Real numbers can't solve every dice
X squared plus one has no home here
Complex roots that disappear
The reals are missing something more
Polynomials knocking at the door
What we thought was everything
Left us with an empty ring
[Verse 2]
Take a polynomial with real coefficients true
Sometimes the roots are complex, not the ones we knew
They come in pairs conjugate, dancing out of sight
Real mathematics missing half the total light
We built our house on solid ground we thought
But found the foundation had gaps we never caught
[Chorus]
There's a problem in our paradise
Real numbers can't solve every dice
X squared plus one has no home here
Complex roots that disappear
The reals are missing something more
Polynomials knocking at the door
What we thought was everything
Left us with an empty ring
[Bridge]
Complete and ordered, that's what real numbers claim
But completeness has limits in this mathematical game
The fundamental theorem calls for something new
A bigger field where every poly finds its due
[Chorus]
There's a problem in our paradise
Real numbers can't solve every dice
X squared plus one has no home here
Complex roots that disappear
The reals are missing something more
Polynomials knocking at the door
What we thought was everything
Left us with an empty ring
[Outro]
So we learned the reals aren't all
Sometimes we need to break the wall
The problem shows us where to grow
Into the complex world we'll go