# 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:

- Go forward
*a*units of length. - Turn left (by some angle not specified).
- Go forward
*b*units. - Turn towards the starting point.
- 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:

- Go forward
*a*units. - Turn left by
**90 degrees**. - Go forward
*b*units. - Turn towards the starting point.
- 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*):

- Go forward 1 unit.
- Turn left 90 degrees.
- “Go forward
*i*units”, i.e. turn left 90 degrees (again) then go forward 1 unit. - Turn towards the starting point.
- 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.)