[Verse 1]
Between any two rationals, there's always one more
Infinitely dense, like sand upon the shore
You'd think we have it all, every number we need
But rationals are hiding a deceptive deed
[Pre-Chorus]
Look closer at the line, what do you see?
Invisible gaps where numbers should be
[Chorus]
Q has holes, holes, holes in the rational line
Dense but not complete, missing numbers divine
Square root two is nowhere, though we search and we try
Q looks whole but it's broken, that's the rational lie
Holes, holes, gaps that we cannot fill
Bounded sets with no supremum, missing numbers still
[Verse 2]
Take all rationals where q squared is less than two
They're bounded from above, but here's what's true
No smallest upper bound exists in Q's domain
The least upper bound property breaks the chain
[Pre-Chorus]
Square root two should be there, but it's not
A hole in our number line, an empty spot
[Chorus]
Q has holes, holes, holes in the rational line
Dense but not complete, missing numbers divine
Square root two is nowhere, though we search and we try
Q looks whole but it's broken, that's the rational lie
Holes, holes, gaps that we cannot fill
Bounded sets with no supremum, missing numbers still
[Bridge]
Square root three and square root five
Pi and e are not alive
In the rationals' embrace
Irrational numbers have no place
Get arbitrarily close but never reach
These missing numbers that we seek
[Chorus]
Q has holes, holes, holes in the rational line
Dense but not complete, missing numbers divine
Square root two is nowhere, though we search and we try
Q looks whole but it's broken, that's the rational lie
[Outro]
The continuum's not continuous yet
In Q there are gaps we can't forget
Dense but incomplete, that's the rational way
Holes in the line that are here to stay