Inconceivable! The Origins of Imaginary Numbers

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SWBAT to explain the reasoning that lead to the invention of the imaginary and complex numbers. SWBAT represent complex numbers in the complex number plane.

Big Idea

Imaginary numbers aren't any more imaginary than negative numbers!

Discussion: What's an Imaginary Number?

15 minutes

Using our recent work solving quadratic equations as a springboard, I spend the bulk of this class period directing a Socratic discussion of real and imaginary numbers.  My overall aim is to move students from a clear understanding of the real number system to the recognition that there is a need for still more numbers.  To generate enthusiasm for the conversation, I emphasize the strangeness of numbers and the concept of infinity. (MP 2)

If I've done my job well, my students will walk out of the class understanding clearly what imaginary numbers are, but feeling a little dazed.  If I've done my job really well, I'll get word from some parents that their kids came home still talking about how math class blew their mind!

To get feel for what this really looks and feels like, please see the sample dialogue of a Socratic discussion of imaginary numbers from my own class and the short video, Imaginary What?!.

Discussion: What's a Complex Number?

20 minutes

Once we've established what the imaginary unit is, why it's "necessary", and how different imaginary numbers arise out of square roots, it's time to talk about complex numbers.  Typically, I find that once students have accepted imaginary numbers, it isn't much of a jump for them to combine them with real numbers.

I like to kick of the conversation with this question:  "Which number is greater, 5 or (5 + 3i)?"

This usually generates a wide variety of responses and allows us to reiterate the point that 3i is neither positive nor negative.  This means that (5 + 3i) is somehow neither greater than, less than, not equal to 5.  Bizarre!  Using the strangeness of these numbers to generate excitement, I then introduce the complex number plane so that we can "see" how these numbers are all related and make some sense of them. (MP 2)  Please see my short video, Complex Numbers, for some thoughts on the important points of this conversation.

Once I'm satisfied that the students understand what a complex number is and how to place it in the complex number plane, it's time to move on to the next section of the lesson.

Student Practice

10 minutes

The classwork, Complex Numbers, includes problems requiring students to express roots of negative numbers in terms of i, problems asking them to plot complex numbers in the complex number plane, and a final problem asking them to graph the first four powers of i in the complex number plane and then describe "what seems to be happening to the graph each time the power of i is increased by 1."

While I do not plan to make much of it at this point, the final problem sows the seeds for understanding the pattern inherent in the powers of i, as well as the rotational property of multiplication by i.  Advanced students may be encouraged to dig more deeply into these concepts. 

Students can watch the excellent video below that explains this rotational property in more detail and in "real world" terms.


If students do not complete all of the problems during the 10 minutes of classwork time, they should complete the remainder for homework tonight.  I've included a sample of one typical student's work.