Electronics makes extensive use of imaginary numbers. They’re really not “imaginary” – they do exist in the real world, they’re just phase-shifted by 90 degrees. And this is where language colors perception again. Our ancestors were thinking linearly in one dimension, and didn’t account for situations where two things could happen at the same time but in different directions. Additionally, when I was in high school, I was taught that proper functions drawn on graph paper couldn’t have 2 simultaneous values for y. Or rather, curves couldn’t loop on themselves. Meaning you weren’t supposed to draw circles on the graph because it wasn’t a “proper” function. I’m pretty sure those days are long in the past now, but we still use “real” and “imaginary”, and we shouldn’t. Especially since in 3D space we have 3 sets of numbers – real (x-axis), imaginary (y-axis) and up/down (z-axis)…

As I mentioned in my earlier blog series, “i” is how we change directions in 2D space, switching from the x- to the y-axis and back. Or, rather from the “real” to the “imaginary” axes, when we have an equation in the form of y = a + i*b.

Say we have a boat trying to cross a river. The boat is going from east to west at 10 miles per hour, while the river is flowing north to south at three miles per hour. If the river is 200 feet wide, how far downstream will the boat go before hitting the opposite shore? It’s easier to work in meters. 10 miles/hour = 16093 meters/hour, or 4.47 meters per second. 3 miles/hour = 1.49 meters/second. 200 feet = 60.96 meters. In the first second, the boat will go diagonally 4.47 meters, placing us 1.49 meters downstream. Straight east-west, we’ve gone sqrt(4.47^2 – 1.49^2) = 4.21 meters. At this rate, it will take us 60.96/4.21 = 14.46 seconds. We hit the opposite shore 14.46 * 1.49 = 21.5 meters farther downstream than when we started. Which part of this scenario is “real” and which is “imaginary”? (To double check the numbers, 64.65^2 = 4180.63. And 21.55^2 + 60.96^2 = 464.51 + 3716.12 = 4180.63.) If “y” is our position in the river at any given time, “t”, and downstream is “minus y”, then y = 4.21*t – i*1.49*t. Or, y = (a + i*b) * t, where a = 4.21 and b = 1.49. Both “a” and “b” are real in that we can measure them in 3-D space, it’s just that we need to use “i” to change directions along the way. That is, “i” gives us a phase shift (and is often simply called a placeholder).

In 2D space, 1 * i = i

i * i = -1

-1 * i = -i

-i * i = 1

What if we step into 3D space? We need a second placeholder. Instead of being in a boat, let’s use a glider. We have a tailwind for a, a crosswind for b, and gravity for c. Starting with the glider on the side of a 50-meter tall cliff, where will we land if we glide 10 meters forward for every meter we drop, and there’s a 2 meter/s crosswind? Same math. How do we change axes?

In 3D space, we could do something like y = a + i*b + j*c

The rule would be:

1 * i = i

i * i = -j

-j * i = -ji

-ji * i = j*j

j * j = -1

We’re now moving around in a unit sphere. And you know what? We can do this in 4D space, too. j*j = -k. k*k = -1. As long as we keep enough spare placeholders in our alphabet, we can extend the math into as many dimensions as we want (which brings us perilously close to string theory…)

Somehow, as the math implies, it should be possible to keep taking right angle turns (a point to a line, a line to a plane, a plane to a solid, a solid to a hyper-solid). Is there an upper limit to the number of dimensions that do exist? (Note that I use “do”, not “can”.) And, what is a “dimension”, really?

When we graph an equation, such as y = a*x + b, or y = 2 * x^2, we’re making some assumptions. Mainly, that either the units don’t matter, or that they work out right. A simple line, y = 2 * x + 4, has an implied unit of “1”. For every “1” of whatever it is that x is, y will increase by “2” of the same thing. If “x” is apples, then if x increases by 1 apple, y increases by 2 apples.

Let’s look back at the boat example. The boat had a velocity of 4.47 meters/second. The wind had a velocity of 1.49 meters/second. The river had a width of 60.96 meters. And “t” had a time in seconds. y = 4.21*t – i*1.49*t. Technically, this should have been y = (4.21m/s – i*1.49m/s) * t. Then, if t = 5 seconds, y = 20.15ms/s – i*7.35ms/s. The seconds cancel and y is the positional distance from the shore, as 20.15m – i*7.35m. In this case, if we plot the boat’s position on a map, the x-axis and y-axis (latitude and longitude on the map) will both be marked off in 1 or 10 meter increments, and we’ll have multiple data points, one each for each second the boat is in the water. However, if we want to plot “y” as a function of time, then maybe it’s more convenient to make the x-axis “t” in seconds, the y-axis as meters/second for boat speed, and z-axis as meters/second for wind speed (actually, it’d be easier to use vector math, or just draw points on the map of the river using different representational curves).

But, regardless, in one case we have meter – meter axes, and another we have seconds – meter/s – meter/s. We could have dollars/second, or pounds of colored dye/batch size (if we’re making hair dye). What’s the difference between an axis plotting factory production costs, and 3D spaces?