[Verse 1]
Natural numbers can't subtract backwards
When three minus five breaks the rules
So we build a clever workaround
With pairs of numbers as our tools
Take any two and call them (a, b)
The difference that we can't compute
We'll carry both around together
Till equivalence bears fruit
[Chorus]
Pairs that dance the same difference
When a plus d equals b plus c
(5, 3) and (7, 5) are siblings
Both representing +2
Bookkeeping tricks become foundations
When we can't subtract, we create
Integers from natural pieces
Making negatives from fate
[Verse 2]
Check reflexive: every pair equals itself, that's clear
Check symmetric: if first equals second, then reverse is here
Check transitive: chain them together, the relation holds
Equivalence proven, now our construction unfolds
(3, 5) links with (0, 2)
Both carrying negative two
The intended difference lives
In what we cannot do
[Chorus]
Pairs that dance the same difference
When a plus d equals b plus c
(5, 3) and (7, 5) are siblings
Both representing +2
Bookkeeping tricks become foundations
When we can't subtract, we create
Integers from natural pieces
Making negatives from fate
[Bridge]
Addition works on representatives
Add the firsts, add the seconds too
Multiplication gets more twisted
(ac + bd, ad + bc) will do
Well-defined means choice of pair
Never changes the result
Mathematics builds from nothing
When invention fills the fault
[Outro]
From pairs to equivalence classes
We invented what we need
Negative numbers born from nothing
But a bookkeeping deed