Multiplication of squares

On a request from a colleague, I republish a post from my old blog.

My old friend and colleague Dmitry Fon-Der-Flaass left a comment to my post Binary numbers:

Offtopic: an elementary and natural proof that every number dividing X^2+1 for some X, is the sum of two squares. Is it in the spirit of your micromathematics?

Yes, it is very much in spirit of my micromathematics. I was thinking, for some reason, about sums of two integer squares (an archetypal mathematical problem: representation of numbers by forms) and a remarkable fact that the if m and m are sums of two integer squares (let us call them bisquare numbers) then their product mn also has the same property. What is the most natural way to explain this property to a child?

Actually bisquare numbers are exactly areas of squares with vertices at nodes of graphed paper (with unit grid step) — this immediately follows from the Pythagoras Theorem. And the multiplication property of bisquare numbers follows from the following simple rule for multiplication of squares drawn on graphed paper (and with one vertex marked):

Of course, this is multiplication of complex numbers – restricted to the ring of Gaussian integers Z[i], the red square and the blue square being Gaussian integer 3+2i and 2+i, correspondingly, their product being 4+7i (check — it is the new big blue square!). What becomes immediately obvious is the addition rule for arguments:

arg(uv) = arg(u) + arg(v)

as well as the multiplicativity of absolute values:

|uv| = |u||v|

(which was our aim in the first instance, since |u|² is the area of the square representing u).

Teachers, do not underestimate humble graphed paper, a powerful source of mathematical intuition!