[Verse 1]
Take the real plane, call it R squared
Draw your points as pairs everywhere
But we'll give them rules that are brand new
Addition stays the same, it's true
A plus C, B plus D
Just like vectors, naturally
[Chorus]
Building complex numbers from the ground
A plus B I, that's the form we've found
I squared equals negative one
When we multiply, here's how it's done
A C minus B D, then A D plus B C
That's the construction, can't you see
[Verse 2]
Multiplication gets more complex now
Let me break it down and show you how
Take two pairs, multiply them out
Follow rules without a doubt
Real times real minus imaginary squared
Imaginary parts get cross-shared
[Chorus]
Building complex numbers from the ground
A plus B I, that's the form we've found
I squared equals negative one
When we multiply, here's how it's done
A C minus B D, then A D plus B C
That's the construction, can't you see
[Bridge]
There's another way to see this thing
Polynomials in a ring
Real coefficients, X the variable
Modulo X squared plus one, it's manageable
X plays I, X squared is negative one
Same result when we are done
[Chorus]
Building complex numbers from the ground
A plus B I, that's the form we've found
I squared equals negative one
When we multiply, here's how it's done
A C minus B D, then A D plus B C
That's the construction, can't you see
[Outro]
From R squared to C complete
Mathematical structure, neat
Two approaches, same result
Complex numbers, no occult
Just arithmetic with a twist
Now complex numbers exist