Use Color!

I have a new resolution: I will use color in equations from now on. My inspiration was a post on the Fourier transform [thanks to Kalid for referring me to it]. The author, Stuart Riffle, begins with a familiar experience… a mathematical formula that looks like a complicated hodgepodge of symbols.

What I found was the Discrete Fourier Transform (DFT), which looks like this:

The Discrete Fourier Transform

This formula, as anyone can see, makes no sense at all. I decided that Fourier must have been speaking to aliens, because if you gave me all the time and paper in the world, I would not have been able to come up with that.

What follows in the article is Stuart’s aha moment, and a careful picking apart of what each piece of the formula means:

Don’t worry if this doesn’t make sense: there’s a lot of context and background math here. The point is that each piece of the formula means something very specific. For example, one thing that’s happening is we’re taking an average of a bunch of points. Without the color, it would be really hard to parse that particular meaning inside the equation. (Notice that the symbols that indicate to us that we’re taking an average are spread out over the whole equation. Nothing is harder or slower to read than math.)

So from now on, when I write the Pythagorean Theorem, on a chalkboard or in a post, I’ll try to write

Clearer than the “a squared plus b square equals c squared” so many of us committed to memory? I think so.

Comments 7

  1. Kalid

    Hi Dan, I love colorized diagrams! Forcing yourself to write a sentence really makes you think about what each part of the equation is doing. The sentence Pythagorean Theorem might even be:

    “The total area swept by two sides of a right triangle equals the area swept by the third”

    I like this because it breaks down what each component is describing (area).

    1. Post

      Hi Kalid,

      Nice addition. Having words, equations, and diagrams working together gives you the most clarity, I agree. Interestingly, that’s what they’re trying to teach kids to do these days in their explanations in school. I’m curious how well the effort is working; most kids (and adults) I talk to have never seen clearly why taking the second power is known as “squaring,” though there it is above: the area of the square with side a is a^2.

  2. Michael Paul Goldenberg

    It seems a little magical that in the second version of the DFT, 1/N makes an appearance, which tells us to divide by the number of measurements we’ve summed. Now THAT makes it clear that there’s averaging going on, of course (well, maybe not “of course,” but that’s certainly one reasonable conclusion), whereas it’s not at all obvious to me that the original version suggests averaging. Reading the original post from which you took the two formulas doesn’t explain how the author knew to pull out 1/N nor from whence he plucked it, so I’m still at a loss as to how I’d have seen it in the original formula for the DFT.

    Also interesting was his comment about the “magic” nature of rotation coming out of raising e to a complex power according to Euler’s formula. As he says, the author’s not a expert mathematician (in fact, he’s a graphics designer who appears to be working in video games), so he may have never tried going through an explanation of Euler’s formula to find out that it’s not magic. I only did so myself with the help of a video lecture by Edward Burger that I found on one of his DVD courses on mathematics for The Learning Company: Burger is generally extremely clear and a good resource if you can find his stuff in your library as I did. Not sure what’s available on YouTube, etc. to unpack that particular bit of “mathematical magic.” However, it shouldn’t come as TOO much of a shock to anyone who knows at least enough about complex numbers to know that multiplying a vector by i gives a 90 counterclockwise rotation on the complex plane.

    All that said, the use of color in formulas seems like a great way to organize one’s own thinking about what’s going on as well as to do presentations to others. Might be a very nice thing to have students do for themselves/each other when they explain their ideas and problem solutions. Kids do like color and coloring, I can say that for sure based on one of the most successful units I’ve ever taught to “alternative education” students back in 1999-2000: graph coloring. Students who’d never done squat in my classes suddenly were writing “A” exams and enjoying themselves. Too bad I didn’t figure that out a lot sooner.

    1. Post

      It’s so often true that there seem to be “magic” steps in a derivation of a formula or a proof. Each idea is usually inspired by some analogous idea in a slightly different field or topic, or is a trick from another result. And of course, as you say, there’s nothing magic once you think your way through everything and get back to ground. Getting to ground can sometimes take a while, or require a really thorough, clear explanation, or better, someone to talk through things with.

      I think that the 1/N you refer to isn’t actually in the original formula, either. He’s using a slightly different formulation, and moving some terms around as well. As he says:
      “So this equation is a little different from what we started with. I’ve added a normalization factor of 1/N, and changed the sign of the exponent. I also rearranged the terms slightly for clarity. This form is normally called the inverse DFT, which is confusing, but apparently the difference between the DFT and IDFT is a matter of convention, and can depend on the application. So, let’s call that close enough.”

      Anyway, all details that are beside the main point: color is good. I absolutely agree that kids like color and coloring. Graph coloring, map coloring, coloring boards to see how to tile them–there are great activities out there which reach kids who might not respond well to the “normal” math curriculum.

  3. Sue VanHattum

    In my intermediate algebra course last week, after we’d been multiplying and dividing rational expressions (ugh!), one of the students suggested using 4 colors, one for top or bottom of each factor, so they could more easily see the pieces. I loved it. I’ve taught for over 20 years using chalk, and just recently started using marker boards. I’m slowly learning how to use color helpfully.

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