' Quantitative Operators Quantitative Operators: Or, Things To Do With Numbers

Addition

What is addition? What does it mean to add two different numbers? One of the nice things about math is that we don't have to worry about that - addition has a strict definition, in terms of other mathematical objects. There's straightforward rules to apply "adding" to two numbers, and get a result. This is why we can build calculators, and why math is called "the universal language." But still, I think this formalist wonder can get in the way of understanding applied math (the kind of math most people care about). If you want, you can rephrase the question as "how do we interpret addition?" There can, of course, be multiple correct interpretations, which we switch between based on the problem at hand.

I view addition as "putting next to" or "mixing." If you put 3 apples next to 2 apples, you now have 5 apples. 3 + 2 = 5. If you pour 4 liters of water into a container, and then pour another 6 liters in, you have 10 liters of water in the container. 4 + 6 = 10. When using quantities with units, note that addition is pretty useless when you have different kinds of things. 3 apples mixed with 2 liters of water gives you... well, 3 apples and 2 liters of water! 3a + 2L = 3a + 2L, and you're not getting anything else out of that.

Images for addition - a square + a triangle = a square joined with a triangle

short line plus long line = even longer line

image of the water thing

You can also view addition as pushing: you can just add up all the pushes on an object to get where it actually moves. In other words, ten people pushing a car left while three push it right will have the same result as if seven people pushed it left.

Image of the pushing thing

Multiplication

Multiplication is "scaling" or, thinking more spatially, "sweeping." For scaling: let's say you sell one lemonade, and make $2. If you were to scale up your operation and sell 50 lemonades, how much would you make? $2 per lemonade * 50 lemonades = $100. Or, let's say that the average person does 5 hours of productive work a day. How many work-hours would be done by a city of 2 million people in a year? 5 work-hours per person per day * 2 million people per city * 365 days per year = 3,650 million work-hours per city per year. Note how units are changing here. You can multiply quantities with different units, and have the result be in sensible units. See https://en.wikipedia.org/wiki/Dimensional_analysis#The_factor-label_method_for_converting_units for examples. The sweeping is best shown, and not told:

line times line equals rectangle

line times triangle equals triangular prism

Multiplication also has its own version of the pushing interpretation - torque. When you're opening a heavy door, the torque applied (the effectiveness of your door-opening) is how hard you're pushing * how far away you're pushing. The same principle is at work when you're on something spinning, like a carousel - how fast you're going is the carousel's speed times your distance from the middle of the carousel.

Complex Multiplication

Multiplying two complex numbers is a scaling, combined with a spinning. When multiplying complex numbers, it's easier to first convert the complex numbers to a "polar" form, where it represents an angle and a magnitude. Then, to multiply two complex numbers, multiply the magnitudes, and add the angles. So (x3, 30 degrees) * (x5, 20 degrees) = (x15, 50 degrees).

Derivation of how this is true?

3Blue1Brown animation or something showing how this is true

Complex Addition

Normally complex addition is just though of as adding the real and imaginary components separately. But if we're committing to viewing complex numbers as representing a "rotation-and-scale," then we need a new view of complex addition. There's no neat way to numerically express the addition of complex numbers in polar form. You have to convert to the a + bi form, add that way, then convert back to polar. However, there does exist a very nice physical model for how complex addition works - the addition of waves, which is familiar to us as sounds.

To interpret a complex number as a wave, we pick whatever frequency we want, and then use the magnitude of the complex number - the scaling part - as the amplitude of wave, and the rotation part as the "phase shift." We can imagine a physical apparatus that traces out a wave. [description here] By default, the arm of the machine has a certain length, and starts horizontal. However, we can scale and rotate the arm before turning the machine on, and when we do, it'll trace out a different wave. This scale-and-rotate is the complex number we care about! So we can see that a complex number can also be interpreted as a wave - the wave the machine makes when it's scale-and-rotated by that complex number before being turned on.

https://www.desmos.com/calculator/cgzouj48jp

When you play the same note on two different instruments, what you hear sounds just like if it was being played on a single instrument - except it might sound quieter or louder. You can see this for yourself by opening https://www.szynalski.com/tone-generator/ on two different devices (for best results, set the audio to be single-channel, left-speaker or right-speaker only) and playing the same tone on both. It'll just sound like the audio is coming from one impossible-to-locate source - freaky!