# 1² + i² = 0²

Observe the above mathematical joke. Really breathe it in - enjoy it. Feel free to click through to the Wikipedia article it came from. I’m about to explain the joke, and explaining a joke is how you ruin it. So enjoy it now while you can.

A key motivation of algebra is the solution of problems in geometry, but sometimes the working, or even the answer itself, is strange. Imaginary numbers are great examples.

I tweeted that there was a way to draw the triangle such that it made sense. Before I get to that though, I will need to provide a small amount of background. I’m going to ignore most of Philosophy of Triangles and, without backing evidence or additional explanation, skip straight to this unusual definition:

Definition 1: A triangle (of lengths a, b, and c) is a shape that can be traced by the following algorithm:

1. Go forward a units of length.
2. Turn left (by some angle not specified).
3. Go forward b units.
4. Turn towards the starting point.
5. Go forward all the way to the starting point, and no further (c units).

A right triangle is usually defined to be a triangle where two of the sides are perpendicular (at right angles to one another). Instead, I will repeat the unusual definition above with a key detail.

Definition 2: A right triangle (of lengths a, b, and c) is a shape that can be traced by the following algorithm:

1. Go forward a units.
2. Turn left by 90 degrees.
3. Go forward b units.
4. Turn towards the starting point.
5. Go forward all the way to the starting point, and no further (c units).

The Pythagorean Theorem applies to right triangles:

Theorem (Pythagorean Theorem): For any right triangle of lengths a, b, and c, $$a^2 + b^2 = c^2.$$ (Proof omitted for brevity.)

Now, you may be wondering why I chose an unusual, yet still understandable, definition of a right triangle above. To understand the joke, we need to deal with an imaginary amount of length. To understand imaginary length, we need the intermediate step of understanding negative length, because the foundational equation of imaginary numbers is that $$i^2 = -1$$. Fortunately, the definitions above make “negative length” easy to deal with: “go forward -x units” can be treated as “go backward x units”. But walking backwards is hard for some people, so for the sake of accessibility I will break it down even further.

Note that every negative number -x is the product of -1 and a positive number x: $$-x = (-1)x$$. Therefore we can factorise the step “go forward -x units” into two sub-steps: the “-1 part”, followed by “go forward x units”, and to avoid “going backwards”, I will make “the -1 part” to mean “turn left 180 degrees”.

For example, “go forward -2 units” really means “turn left 180 degrees then go forward 2 units”, and reality is once again rescued from the jaws of nonsense geometry.

And guess what? The Pythagorean Theorem still holds, because $$x^2 = (-x)^2$$ for any length x! Hooray for the ancient Greeks!

Back to imaginary length. What does it mean to “go forward i units”?

Well, i really means i times 1, so analogously with the above, we need to factorise the step “go forward i units” into two sub-parts: “the i part”, followed by “go forward 1 unit”.

What’s “the i part”? Consider the foundational equation of imaginary numbers: $$i^2 = -1$$, or said another way, we can factorise -1 into i times i. “The -1 part” above was “turn left by 180 degrees”, so, factorising that into two sub-steps, we need something that, when done twice, is the same as turning left 180 degrees. It follows that “the i part” must be “turn left by 90 degrees”! Thus “go forward i units” means “turn left 90 degrees and then go forward 1 unit”.

Armed with this information, I can now provide a more accurate drawing of the right triangle from the joke, as traced by the algorithm (a right triangle of lengths 1, i, and 0):

1. Go forward 1 unit.
2. Turn left 90 degrees.
3. “Go forward i units”, i.e. turn left 90 degrees (again) then go forward 1 unit.
4. Turn towards the starting point.
5. Go forward all the way to the starting point, and no further.

(Proof that step 5 traverses 0 units of length is left as an exercise to the reader.)