Introduction to the question

Both teachers and students tend to give up on ‘having a feel’ for imaginary numbers pretty quickly. Imaginary numbers are chalked up as a book-keeping tool or algebraic gymnastics rather than ‘real-life’ numbers like counting numbers, fractions, or negative numbers. The name, meant to be derogatory, doesn’t help, of course. I personally learned imaginary numbers this way, as a construct of convenience that shouldn’t be thought about too hard, and made it several years into college math with that attitude–I knew the rules, how imaginary and complex numbers behaved and everything. All without ‘having a feel’. However, every once in a while, a teacher would make an offhand comment belying a real intuitive understanding of imaginary numbers, often having something to do with rotation. I would glance around flabbergasted only to see several classmates nodding knowingly. After revisiting imaginary numbers, I’d like to share a more intuitive explanation of what imaginary numbers are and how they add to our system of numbers.

An analogy with negative numbers

Now, I will not make the case the imaginary numbers are anything more than a sophisticated book-keeping device or just a convenient construct. I will make the case that all numbers are that way, including negative numbers and fractions, only most people are more familiar with some of them.¹

Negative numbers were also considered absurd and made-up once. It isn’t so difficult to understand why. How can Jimmy have 4 cookies and give away seven? How can anybody have less than nothing of anything? However, negative numbers fill a gap in the number system and simplify, if not outright facilitate, many mathematical problems. Furthermore, they were more accepted when represented with an elegant geometry that fit cleanly with the geometry of positive numbers.

Consider that . Very simple and straightforward. I might think of this problem as asking “What number x, if added to four, makes seven?” and you would say, “Why, three of course.” Now, anyone could rearrange this question to ask, “What number y, if added to seven, makes four?”. Without negative numbers, even if you know the difference between four and seven, you can’t express this difference in direction. You would need to say, “That doesn’t make any sense, but three added to four is seven.” There is a gap here.

So, we invented a convenient book-keeping device that makes this simpler. Even before negative numbers were considered legitimate, and before mathematicians imagined them on a number line, they were used in algebra. As an example: Suppose you want to find two numbers whose sum is 10 and whose product is 21. You can set that up like so.This problem can be solved quite simply by completing the square, but only with an “intermediate debt”, or with a book-keeping negative number that disappears by the end of the problem.

Voilà! We used some negatives for “keeping track” but they disappeared in the solution. This is how many mathematicians, even Euler, often thought of negative numbers.

Additionally, with negative numbers, it is very easy to say “negative three is the number that added to seven makes four”, and I can simply write . Nowadays, first graders are supposed to understand what centuries of mathematicians were skeptical of–and the intuitive device that has made all the difference is the number line. We can see very clearly here that all numbers exist on a line, with positive numbers extending from the center (0) to the right, and negative numbers to the left. Let’s consider how some operations with each work on this number line. - Magnitude: Magnitude asks “How far is this number from zero?”–and this distance is always positive. Under the hood, it is calculated like so. -
- Addition: Addition translates on the number line. Positive numbers translate in the positive direction (to the right), and negative numbers translate in the negative direction (to the left).
- Multiplication: Multiplication by a number scales by its magnitude. is to say that the magnitude of is . Multiplying by a positive number will scale a number in whatever direction it is already facing. Multiplying by a negative number will scale in the opposite direction the number is facing–or it reflects, then scales.

A helpful way to think about multiplication is to think of an extra 1 in front. For example,

Adding the one in front reminds us to think of all numbers as being scaled from the identity, one. Positive numbers in the positive direction, negative numbers in the negative direction.

This is why Gauss suggested we call negative numbers inverse numbers.² and have the same magnitude, they just “face” different directions. In addition, they add the same amount just in different directions. Likewise, when multiplied, they scale a number by the same amount, just in different directions. Thinking about these numbers and these operations on the number line makes negative numbers intuitive and real.

Enter imaginary numbers

Now let’s consider a different elementary issue of algebra. The equation is simple to solve. Both and squared are equal to , so we have our roots. Now what would happen if I were to tweak this just a little bit? Well now I’m in trouble (without imaginary numbers). Squaring either a negative or a positive number results in a positive number, so there is no way to square something and arrive at . To fill this gap in the number system, a theoretical, convenient number was invented: . The special thing about is that . allows me to solve this problem with and .

This is the same process with negative numbers–an invention to fill a gap and allow a solution to be written. This invention has proved to be as useful and as elegant as negative numbers, and has a beautiful geometric representation. Just as the diagram of the negative numbers extending the number line has helped mathematicians and students understand negative numbers more intuitively, so will a geometric explanation (conceived of after imaginary numbers had proven their mettle) make imaginary numbers more intuitive.

A geometric interpretation

Consider that the equation may be equivalently writtenSo the question posed by this equation can be written, “What transformation, when applied twice, turns into ?”

If we try , then . If , .

The problem with integers, as we saw earlier, is that scaling by any integer twice gets us a positive number. Scaling by a positive number keeps 1 on the same side throughout, and scaling by a negative number twice reflects to the negative side of the number line, then reflects back. So no number on the integer number line can scale 1 twice and make it to the negative side of the number line.

Imaginary numbers offer a whole new kind of transformation–a rotation! Imagine that multiplying by rotates degrees counterclockwise, so it is “halfway” to the negatives on the number line. Then, multiplying by again applies the same transformation and rotates the rest of the way to . We’re there!

Similarly, we can multiply by to rotate degrees clockwise, then multiply by again to get to .

Note that counterclockwise being the positive orientation is purely a human convention, and could just as easily have been reversed.

Since is a degree rotation of the number , is outside of the number line as we used to think of it. Just as we added the arrow pointing off into infinity on the left of zero to make the negative numbers, imaginary numbers add a whole new axis to the number line. There are the positive imaginary numbers off to infinity pointing up from zero, and negative imaginary numbers off to infinity down from zero. is a whole new ‘imaginary’ dimension to measure a number with.

So we asked, essentially, “How can we transform to in two identical steps,” and arrived at “rotate degrees twice.” This proves to be extremely useful.

Note to add somewhere When I learned about complex numbers, I was used to graphing on two real axes. Until fairly recently, I thought of the complex plane as a way to graph complex numbers the same way, without being something inherent to complex numbers. It was mysterious to me because the coordinate system worked by adding. So is the coordinate at . I always thought it was an odd way to run a coordinate system. What I was missing is that the complex plane is just the new way to represent the number-line. When imaginary numbers are used, numbers now have two components: the real part and the complex part. Numbers are two dimensional now.

Cyclical patterns

When multiplying by -1, a pattern emerges:

Multiplying by just flips the sign. In fact, since imaginary multiplication introduced the idea of rotation, we can think of multiplication by as a rotation of degrees (and a multiplication by as a rotation of degrees), which brings us to this pearl of mathematical intuition presented by some online chat.

Now let’s piece out what the sequence of multiplying will be.

• , just where we start.
• 
• , remember that two degree turns is degrees.
• 
• , which is the same as . Alternatively, you could think of this as starting at , rotating degrees, then rotating degrees to end up back at .

Geometrically, we are taking 90 degree steps around the unit circle in the complex plane, repeating every four terms. This makes imaginary numbers great for modeling things that cycle in two dimensions, or circular relationships.

Understanding complex numbers

Up to this point we have only talked about numbers on the real axis (where the imaginary part is zero) and the imaginary axis (where the real part is zero), but the new number system exists at all points in between–and there is a lot of new real estate.

Consider the number . It has a nonzero real part () and a nonzero imaginary part (). Graphed, it is located here on the extended number line.

Rather than rotating a full degrees to the imaginary axis, we are at a degree rotation.

You’ll notice that adding imaginary numbers translates just like adding real numbers, just in new directions perpendicular to the real number line.

For an complex number , we can find the point on this complex number plane (the new name for our new extended number line) and imagine a vector starting at zero and ending at our coordinate to represent the complex number–it’s like a triangle with the hypotenuse representing the number, and the sides representing the real and imaginary parts. This may feel new, but even with the old number line with just positive and negative numbers in one dimension, numbers were represented with a vector starting at zero and going a certain direction with a certain magnitude. We are only expanding the allowed directions to any direction in two dimensions!

“Complex number” is the name usually reserved for numbers with a nonzero real part and a nonzero imaginary part, but remember, imaginary numbers have just expanded the whole playing field for all numbers, so “complex numbers” just refers to “The new system of numbers as they are now constituted, including the innovation of imaginary numbers and numbers with both nonzero real and imaginary parts”. But “Real numbers”, numbers on the real axis with zero imaginary part, are still complex numbers, because they are a part of the new number system too. So complex numbers aren’t a different animal or a different species of numbers, they’re just from a part of the number map that doesn’t always show up.

As for thinking about the size or the magnitude of these complex numbers, they work the same as real numbers–distance from zero. Since we’re talking about two dimensions, we use the pythagorean theorem. For instance, if I want to find the magnitude of the number , or , I can plug it into the formula.

Part of why this is really slick is because it extends very nicely how we used to do magnitude for real numbers, or what we call absolute value. Absolute value measures a number’s distance from zero, so However, the way we compute this under the hood is the square root of the square, since the square of real numbers is always positive. When we added in imaginary numbers and the complex plane, we can say “Actually we were just doing the pythagorean theorem with a zero imaginary part”, because

So not only does this extension of the number line work very cleanly, but the extension of magnitude is also very clean and seamless.

Complex multiplication

The great thing about complex numbers, and the first quality they were imbued with, is that multiplying by a complex number rotates by its angle. Recall that the number is at a degree angle to the real number line (where degrees is the ray pointing in the direction of the positive numbers and positive rotation is counterclockwise), and that multiplying a number by rotates it degrees. The same is true for and degrees.

Just like magnitude, this is a concept that applies cleanly to the plain old real number line. As we discussed earlier, multiplying by a positive real number applies degrees of rotation, and multiplying by a negative real number applies degrees of rotation, which matches their respective angles in the complex number system. So rather than thinking of multiplying by negative numbers as applying a then scaling, we think of all multiplication as applying a then scaling

This rule–that multiplying by a complex number rotates by its angle–is true for all complex numbers. Let’s look at an example.

In conclusion

I hope to have established a jumping off point for complex numbers by establishing a solid, geometrically based intuition for what imaginary numbers are and why they are great and quite reasonable.

Hopefully, coming soon are the deep waters worth diving into, beginning with applying this intuition to complex arithmetic, and working towards understanding Euler’s formula and Fourier transforms!

A new approach to learning math

You may have already made this switch–I just know it took me longer than it should have. I know we have already talked about it too, but I want to hit it again. There was a big shift for me, in between high school and university, away from learning a formulaic approach to math to learning a relational one. Let me explain in terms of being bad at directions.

For much of my life, I haven’t been great at directions. I wasn’t interested in getting a feel for the city I grew up in, I just wanted to get from point A to point B, and 90% of the time A and B where our house, North High School, church, a few friends’ houses, or Culver’s. I was able to learn directions as a linear series of turns: “Turn right on Superior Ave, left at the first intersection, right by the Sport’s Core sign” or “Left onto Superior, left at the big intersection, right on North, left on before the cemetery”. This worked just fine for me, especially when I could use google maps easily when I didn’t know how to get to that Culver’s or to a house I hadn’t visited before.

My series to get to the Sheboygan Youth Sailing Center is “Left on Superior, keep going, keep going, right on third”, even though I knew it wasn’t the best way to get there. When I was in Sheboygan a few months ago, I decided to take Erie. I knew how to get to Erie–“right at the round about, right on after the movie theater, left onto the ramp”–but once I was on it I had no idea when to turn. I didn’t know where I was and I didn’t know where other places were, because I had left one of my linear series of turns. My brother Truman had to guide me step by step. Funny enough, he messed up and had me turn earlier than he usually does, but he was unfazed because he still knew where he was and got me to the sailing center easily.

Truman (and Jack Allen and maybe you) doesn’t think of Sheboygan like a subway line, with discrete tracks in between places of interest. It’s a bunch of places that are all related to each other by direction and distance. If he knows where he is, he knows where everything else is. Rather than thinking about directions like formulas, he thinks about the map.

It is lazier, less useful, and less sticky to learn directions linearly rather than becoming familiar with the city itself. Similarly, learning math by memorizing formulas or the wording of theorems is lazy, hard to apply, and hard to remember. Many people don’t know there is another way, so they unavoidably hate math.

It is drudgery to memorize the eleven properties of a full-rank matrix. It would take a lot of flashcard reps to recall that “full rank square matrix”is equivalent to - Column space spans the whole vector space - Row space spans the whole vector space - Invertible - Diagonalizable - Determinant 0 - Nonzero eigenvalues … And the list goes on! The Jackson of math nonetheless trudges forward and works to drill into his head that these all go together. The Trumans think about it in terms of relationships and ask why a determinant of zero should mean a matrix isn’t invertible. Grant Sanderson does a great job making this clear.

Since a determinant is the factor by which the area of shape changes under the matrix transformation, a determinant of zero would mean the space gets squashed into a lower dimension, which necessarily means lots of points get mapped to the same point under the transformation–so starting with a transformed point, you can’t go backwards. Makes sense.

When that is in place and makes sense, the explanation of eigenvalues as “magical values such that ” won’t be enough. You’ll dig and ask why until you arrive somewhere like,

The transformation denoted by matrix can be decomposed into eigenvectors with associated eigenvalues–and if an eigenvalue is 0, that means there is squashing in the direction of that eigenvector. So it’s the squashing thing again.

You will see that “ is invertible” isn’t just a package deal with “”, they are just two different ways to say the same thing–that doesn’t squash dimensions down, so you can work backwards. Proofs will change from complex recipes to windows into the fundamental nature of things. Good proofs teach you more about the objects that show up in the proof.

Then on homework and exams, you won’t get lost after a few unexpected turns. And when you find yourself in uncharted territory, you’ll have the skills to make a map. You’ll be a mathematician. So be a Truman.

The rest of this is more or less ways to learn math in this way, and common pitfalls to avoid.

I. Think hard

Memory is the residue of thought. ~Daniel Willingham

Learning, in any discipline, is the result of actively engaging with the material to be learned. Thinking hard is what digs the grooves through which thought can flow. An apt analogy is that of lifting weights. Looking at weights or reading about lifting weights or enlisting a friend or robot to lift weights for you can’t make you stronger. You need to do the heavy lifting! Likewise, in order to learn math (or anything), you need to think about it hard.

Some practical examples–which we will go over in more detail–are exercises like working through examples on your own, summarizing or explaining sections and proofs, and giving homework sets real concentrated effort.

This section is relevant always, but especially during test season.

III Limit time spent doing math

There is a universal law of life that every math student must learn.

[Math homework] will fill the amount of time you make available to it.

When you sit down and day, “Well I have until [time] to finish this”, be it an encouraging amount of time or a discouraging amount of time, you will take that amount of time to finish it. Almost always. I’m not certain why this is–maybe because we go slower because we know we can, or put an extra finish on each problem if there’s time, or urgency makes our brain work better.

I just know that even if I hustle, it will take me right to the edge of my available time slot to finish what I am working on.

The solution to this is to 1) give yourself less time and 2) have lower standards for what is good enough. ## 1. Non-negotiable schedule The most important way to constrain the time you spend doing math is by blocking off non-negotiable events.

Set a stop-time later than which you will not do homework. I recommend that it be much earlier than your bedtime. I stop at 6pm. Whether you’re done or not doesn’t matter, no homework past stop-time.

Schedule consistent times to exercise. Exercise at midday is better than exercise in the morning is better than exercise at night, is what I have found. Fun exercise is always the best.

This is hard to do initially but it rewires your brain. You will likely be surprised that you continue to get just as many things done in less time. You will also have more time to do the things you want to do, like spend time with friends/family or on your own hobbies. Not least is the fact that sleeping enough, exercising, and getting good social connection time is another way to buy the nice pair of boots and keep the saw sharp. These three things are part of being a healthy and happy person, and certainly don’t hurt your math education.