Sunday, January 25, 2015

Thinking About Complex Numbers Geometrically

There are two different ways to think about complex numbers -- algebraically or geometrically. Most math students only learn how to think algebraically about complex numbers. This is a shame. It’s like playing piano with one hand tied behind your back. Without the geometric perspective, many complex numbers problems become harder.

(I also think that without the geometric perspective complex numbers hardly make any sense at all.)

Here's a fairly typical complex numbers problem (from a PARCC Algebra II sample test):

The vast majority of our students are going to attempt to solve this problem algebraically, which means lots of multiplying.

A fundamentally different approach involves understanding how multiplication by a complex numbers encodes rotations and scalings.

Most multiple choice problems expect your students to take an algebraic approach, and to make algebraic mistakes. Here’s a great example of that tendency (from a past NY Regents exam).

From an algebraic perspective, yes, 8-6i and 8+6i are awfully similar options. But if you see things geometrically, 8 +6i is a fairly silly option. It makes no sense! How could we have ended up back in the first quadrant by performing a modest rotation or two in the clockwise direction?

The vast majority of mistakes that students currently make with complex numbers are algebraic. This speaks volumes about how our kids are generally taught and the limitations of doing so. To really understand complex numbers with any depth is to understand them both algebraically and geometrically. 


  1. Hear, hear ! (British parliamentary sound of approval).
    I hunted through the CCSS doc and found NOTHING on polar coordinates. There seems to be a blindness at work.

    1. Another member of Team Geometry! Glad to have you aboard, Howard.

      I found one mention of the polar coordinates in CCSS, and I think it's the real deal:

      "Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°."

      Here for more details.

    2. About polar coordinates I was thinking about equations in r and theta.
      The real deal, yes, but it is flagged with (+), not needed for the "rest", only for the specialists, which is a big shame, since otherwise 2 + 3i is almost meaningless, and only taught for test purposes. Better to have put all the complex number stuff for the specialists.
      When I was at school I had lots of fun with projective geometry, a topic which seems to have been lost in history.
      I have a really nice geometrical problem, which arose from my design for magnetic blocks - real world geometry. - I will post it soon. Do look out for it.

  2. This is screaming out for explicitly teaching polar form right? As easy as addition and subtraction are in the rectangular world, multiplication, division and exponentiation are in the polar world.

    1. Absolutely. I think comfort in moving between rectangular and polar coordinates is so key to understanding what's going on with complex numbers.

      The start of my complex numbers unit pushes on this point: say that you rotate (6,8) around the origin by 45 degrees. Where does your point end up?

      This is a tough problem, but it's exactly the tough problem that complex numbers are built for.

    2. I love how the topic of complex numbers can bridge together so many important fields of (HS) mathematics: trig, coordinate plane, algebra, vectors, binomial expansion ((a+bi)^8 without polar), transformations, etc...

  3. As I am not a teacher by profession and my kid hasn't reached complex numbers in school, I had no clue that complex numbers were being taught without geometry. Why wouldn't we start with the complex plane? Imaginary numbers don't belong on the real number line, so we need to go orthogonal. Rotating vector, e^(iθ), ties it up with trigonometry. Real multiplication scales, and imaginary rotates. Vector algebra. Complex numbers are such a beautiful confluence of multiple concepts, and geometry enables us to see it.

    Once we get all this and are fluent with complex numbers and their connections, we can just work with complex algebra. However, bypassing the fertile medium of geometry makes no sense at all.

    Michael, thanks for sharing this. Now I know what to expect, and what I need to ensure for my kids.

  4. You're certainly right that the geometry of the complex plane is hugely important. But I'm okay with postponing lessons on this multiplicative structure, for two reasons:

    (1) Often, kids encounter the complex plane before they know real trig. So they don't necessarily have the tools to handle polar form of complex numbers.

    (2) Often, you're meeting the complex plane specifically because you want to solve quadratics. You don't really need the trigonometric structure to do that.

    Strikes me as really important to teach, but okay to teach later.

    -Ben Orlin (your favorite anonymous commenter, I hope)

    1. (1) Is that true? EngageNY has right triangle trig in their geometry materials. Are people teaching complex numbers in their first algebra classes?

      I think that even without right triangle trig we could get pretty far with the fact that multiplication by i encodes 90 degree rotations.

      (2) I would love for kids to meet the complex plane to solve some geometry/rotation problems instead of to solve quadratics.

      (You are my favorite anonymous commenter! Apologies to Anonymous, Anonymous and (especially) Anonymous.)

  5. (1) Often, kids encounter the complex plane before they know real trig. So they don't necessarily have the tools to handle polar form of complex numbers.

    I don't understand this. All polar form uses right angle geometry, simple arctan will find the argument. Using the cis form requires only cosine and sine.

    How do you teach finding roots/de Moivre without polar form?

    My students don't evn know the sine and cosine rules at all, and yet I have never found trig an issue with complex numbers. NZ has a major stress on polar form, and not just for the top students.

  6. As I mentioned on Twitter, I do teach complex numbers geometrically, but not on the polar coordinates, unless that's the imaginary plane. And while I can see the trig connection, I didn't cover this in my trig class. I cover it in my algebra II class, specifically to solve quadratics. While I agree it's not needed, I think that just saying "here's this thing called i" without giving it a presence is less than optimal.

    This is educationrealist, but it doesn't always acknowledge me as such in the commenting software.