[Verse 1]
Take two integers, make a pair tonight
First one's numerator, second's not zero right
Call it a b, but it's not what it seems
Just a placeholder for fractional dreams
One half could be one two, or two four instead
Three six, four eight, they're equivalent threads
[Chorus]
Equivalence classes, that's the construction way
A d equals b c, that's what we say
When fractions are equal, though they look apart
The rational numbers, now we can start
Add them, multiply them, order and compare
A field that's complete, beyond integers' care
[Verse 2]
Addition's a dance with denominators crossed
A d plus b c on top, nothing's lost
B d underneath, that's your common ground
Multiplication's simpler than what you've found
A c over b d, straight across the line
Operations defined, working every time
[Chorus]
Equivalence classes, that's the construction way
A d equals b c, that's what we say
When fractions are equal, though they look apart
The rational numbers, now we can start
Add them, multiply them, order and compare
A field that's complete, beyond integers' care
[Bridge]
From pairs to quotients, we've built something new
Every fraction living in its own crew
Representative chosen, but the class remains
Ordered field properties, breaking integer chains
Division works here, except for zero's wall
The rationals answer arithmetic's call
[Outro]
So remember the brackets, equivalence bound
Where rational numbers can finally be found
From Z cross Z minus zero we climb
To Q's perfect structure, lasting through time