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April 23, 2021

A New Look at Complex Numbers

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Textbook Reference: Section 8.5 Polar Form of Complex Numbers

It's time.

"Time for what?" I hear you ask.

Time for the last topic of this class, the culminating set of lessons in which we tie together not just the things we've studied all throughout this class but material from your previous classes as well, especially MAT 120 Introduction to Functions (or whatever equivalent you've taken). Every course deserves a "grand finale" like this.

For us, that topic is complex numbers.

A Visual Imagination

Recall that a complex number is a number of the form \(a+bi\), where \(i\) is the imaginary unit.

Now, honestly, did that sentence mean anything to you?

Most textbooks do a pretty terrible job of motivating the existence of imaginary numbers. They say, "the equation \(x^2=-1\) has no solutions, right? WELL LET'S MAKE ONE UP ANYWAY!" and then proceed to do a whole bunch of calculations that seem like nothing but an exercise in meaningless abstraction.

But there's a much nicer, more visual, more intuitive way to think of \(i\) and complex numbers, one that motivates everything else we're going to talk about. And it all comes down to two key ideas:

Key Idea #1:
Numbers can be two-dimensional.
(There's more than just the number line.)

Key Idea #2:
Multiplying by \(i\) means rotating by \(90^\circ\).

Drag the point to move it around and see the corresponding complex number.

Once you think of \(i\) in this way, everything else about complex numbers is a consequence of these fact.

For instance, you probably once learned that the powers of \(i\) follow a cyclic pattern that you had to memorize: \[ \begin{align*} i^0&=1\\ i^1&=i\\ i^2&=-1\\ i^3&=-i\\ i^4&=1\\ i^5&=i\\ i^6&=-1\\ i^7&=-i\\ i^8&=1\\ &\vdots \end{align*} \] But this just comes from the fact that repeated multiplication by \(i\) means repeated rotation by \(90^\circ\).

No need to memorize anything.

Once we know how \(i\) works, we can understand how to treat points in the 2D plane like numbers. That means we can do numbery things to them — add them, subtract them, multiply them, divide them, even raise them to powers and take roots. And it's all just a matter of following your gut (that is, your basic rules of arithmetic and algebra you've learned) and occasionally remembering that \(i^2=-1\).

But this only scratches the surface. Complex numbers are one of the most beautiful and powerful objects in the entirety of mathematics.

And we've just spent a whole semester studying the tools that will really allow us to understand them.

Over the next couple of weeks, you'll see exactly what I mean.


Preview Activity 29

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

  1. Calculate the following powers of \(i\).

    1. \(i^{12}\)
    2. \(i^{31}\)
    3. \(i^{613}\)
      (Hint: How many times does \(4\) go into \(613\)? How many full rotations have been made? How much more do you have to go after that?)
  2. Observe that \(3i\cdot 3i=9i^2=9(-1)=-9\). So, we can reasonably say that \(3i\) is a square root of \(-9\), and we write \(\sqrt{-9}=3i\).

    1. Calculate \(\sqrt{-100}\) and \(\sqrt{-13}\).
    2. Notice we said that \(3i\) is a square root of \(-9\), not the square root of \(-9\). This suggest that there's another square root. How can we find it?
  3. Consider two complex numbers \(z=2+3i\) and \(w=7-2i\).

    1. Plot \(z\) and \(w\) in the complex plane.
    2. Calculate \(z+w\), \(z-w\), \(zw\), and \(\dfrac zw\).
      Remember, just follow the basic rules of arithmetic/algebra and when necessary use the fact that \(i^2=-1\).
      (Hint: When calculating \(\dfrac zw\), multiply the numerator and denominator by \(7+2i\).)
    3. Plot \(z+w\), \(z-w\), \(zw\), and \(\dfrac zw\) in the complex plane.
  4. How do you think we're going to connect the concepts we've learned this semester to complex numbers?
  5. Answer AT LEAST one of the following questions:
    1. What was something you found interesting about this activity?
    2. What was an "a-ha" moment you had while doing this activity?
    3. What was the muddiest point of this activity for you?
    4. What question(s) do you have about anything you've done?

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