You’re wandering through a forest, and spot a binary operator \star.

(a, b) \star (c, d) = (a + c, b + d)

"Cute," you say, "it’s just vector addition."

It retreats into a bush.

"Gone but not forgotten," you murmer as you scribble notes in the sand.

• \vec{x} = (a, b)
• \vec{y} = (c, d)

Maybe \star is the simpliest binary operator you know. After all, it obeys all the pleasing properties you could possibly imagine.

• The lovely symmetry, \vec{x} \star \vec{y} = \vec{y} \star \vec{x}
• The forgiving order of operations, \big(\vec{x} \star \vec{y}) \star \vec{z} = \vec{x} \star \big(\vec{y} \star \vec{z}\big)
• The unimposing zero, \vec{x} \star \vec{0} = \vec{x}
• The inverted twin, \vec{x} \star -\vec{x} = \vec{0}

The sound of something else familiar catches your eye.

(a, b) \star_2 (c, d) = (ad + bc, bd)

"Funny little thing, aren’t you?"

You find comfort in what \star_2 does to the second terms. It simply multiplies them, bd.

"I know you."

You realize it’s nothing more than adding fractions \frac{a}{b} and \frac{c}{d}.

As the sky darkens, mysterious creatures wander closer.

(a, b) \star_3 (c, d) = (ac - bd, ad + bc)

There’s no way you recognize it.

This is the process of multiplying complex numbers a + bi and c + di.

"How peculiar, " you say, "I would have never guessed."

As you look at the next beast ahead, you shout, "this has got to be nonsense."

(a, b) \star_4 (c, d) = (ac + 2bd, ad + bc)

What you see is the multiplication of numbers of the form a + b \sqrt{2} and c+ d \sqrt{2}.