Sometimes you work off memory, and it is usually just fine.

I am teaching one group of kids for the second year. We did some algebra, a full year of geometry, and now on to Algebra II, but with lots of proof (sort of a hybrdized precalc). And I want to work on the Fundamental Theorem of Algebra, and specifically I want to talk about roots showing up in conjugate and complex conjugate pairs. So I do a one day review of radicals. And the next day a long homework review (since some of the radical equations got messy), and then a complex number review with the remaining half hour.

1) Complex numbers are easy and fun
2) With strong algebraic preparation, there is not really much new in complex topics

I was concerned that the review was going too fast. Did they really retain everything, with no questions? So after about 15 minutes of review, I added a parenthetical, “You remember this well from last year?” “We didn’t do this last year….”

(read on beneath the fold —>)So what do I do? They got everything from $\sqrt{-1} = i$ to $\frac{1}{a + bi} = \frac{a-bi}{a^2 + b^2}$ in 30 minutes, with no prior background. Getting complex out of the denominator was the hardest I reviewed.

I left them with a challenge: Find $\sqrt{i}$ . I guided them to discover that the answer could not be imaginary ( $(ni)^2$ gives a real answer). So they are hunting for a + bi so that the real part of its square = 0 and the imaginary part = 1. We’ll see Friday what they found.

Anyway, they never had full lessons on this stuff. Do I:
1) accept that they “got it” and move on?
2) go back and do a proper job?

And what do I conclude?
1) Complex numbers are easy and fun (I think there is an element of truth to this, up to a point)
2) With strong algebraic preparation, there is not really much new in complex topics

How do you advise, and what do you think?

1. March 8, 2007 pm31 3:18 pm 3:18 pm

I don’t recall being taught complex numbers “properly” either. It took me a long time to be comfortable using them because I thought I must be missing things, but I was fine…

I think you’ll get your answer on Friday when you see what they come up with. They might be just fine and ready to go on, or a bit more review might give them the needed confidence to move forward!

2. March 9, 2007 am31 7:36 am 7:36 am

I would agree with your conclusions. As for what do you do… You’ll see what they come back with, but I might make a short problem set that goes over the stuff you think you “reviewed” last time and see how well they do. That should tell you.

Btw, you prove things for them? Or with them. Did you do that in geometry as well? Dan over at dy/dan said he wouldn’t want to spoil the fun in order to prove that say area of a traingle is what it is. He said they have a section on it, but he doesn’t dwell on it too much (at least that’s what I understood). This is quite contrary to what I heard some more experienced geometry teachers say. What say you?

3. March 9, 2007 am31 7:45 am 7:45 am

In this course (alg 2/precalc) I set up a proof, and run through it, eliciting assistance from the class, giving them opportunities to prove ahead of me, etc. I also ask them to complete shorter proofs on their own.

In geometry (which I avoid teaching), I prove theorems in the same way. The kids do lots of particular proofs.

So, the remainder theorem? I prove. The factor theorem? I outline the theorem and ask them to propose a proof (it is a fairly natural extension).

1. Nothing new is introduced without proof.
2. They might a) watch, b) help, c) do it themselves, depending on the proof.

4. March 9, 2007 pm31 5:42 pm 5:42 pm

Great! Btw, you said earlier that you avoid teaching geometry? Why? I love geometry and can’t imagine why one wouldn’t like it? Is it the subject itself, the way it’s taught or what is taught that you dislike?

5. March 9, 2007 pm31 6:05 pm 6:05 pm

I agree with your point 1, but point 2 depends on how far you delve into complex analysis. If you include such topics as Euler’s formula (e^it = cos t + i sin t) and its ramifications, or complex numbers as transformations and their application to geometry, then there’s potentially a lot of new material!

I have fond memories of complex analysis as a fun subject, with both theoretical interest and practical applications, at A-level maths and further maths (age 16-18), in the UK in the late 70s. Sadly, complex numbers have now been banished from the A-level syllabus; students must take further maths (usually taken only by those who intend to read maths or physics at university) to see some of the splendour of this area of mathematics. This is consistent with the relegation of related ideas, such as matrices, to the further maths syllabus.

March 9, 2007 pm31 9:35 pm 9:35 pm

What proof of FTA do you use? The one I like uses rather a lot of the geometry of complex numbers, and you need a feel for the re^(i theta) version of complex numbers to appreciate it. If you have a more elementary proof, I’d love to see it.

7. March 9, 2007 pm31 11:42 pm 11:42 pm

There are purely algebraic proofs that only assume very elementary analysis (i.e. the intermediate value theorem), but require you to know various things about Galois theory.

8. March 10, 2007 am31 4:45 am 4:45 am

Alon, LSquared,

there are things I cannot prove at this level – the Fundamental Theorem of Algebra being one. Further, while some of my proofs look like detailed high school proofs, I drill that proof is “that which convinces,” lowering or raising the bar, appropriate to the kids’ experience with mathematics. For the FTA there will be no proof. There will be some observations that this seems to make sense, a number of examples, etc. Oh, and when I don’t prove, I do so explicitly. They will know that they have gotten something without proof.

I wish this class was more like my precalc from a few years ago, though. They weren’t as strong overall, but one time I tried to introduce a formula (tangent of a sum, I seem to recall), and they absolutely rebelled. The claim was that they could look up the formula, and apply it by copying the models from the text. Class was for me to prove and for them to ask question. I can still see the girl scolding me. That’s a proud, memorable, moment.

Nick, these are high school kids, so they are only getting a taste of complex numbers here. My question about $\sqrt{i}$ was way out of curriculum. For them complex is a bit novel, and will give meaning to the FTA (which they will learn about, but not prove!)

Finally, for all: I gave a practice lesson today, to make sure that all of this was solid. They had good control of basic operations, though finding $\sqrt{i}$ was not possible without significant help. I gave them but one problem this weekend: try to find [wanted cube root of i, but couldn’t latex it. Tried latex \sqrt[\3]{i} .]

9. March 10, 2007 am31 5:22 am 5:22 am

\sqrt[3]{i}?

10. March 10, 2007 am31 5:35 am 5:35 am

Finally, for all: I gave a practice lesson today, to make sure that all of this was solid. They had good control of basic operations, though finding $\sqrt{i}$ was not possible without significant help. I gave them but one problem this weekend: try to find $\sqrt[3]{i}$.

Thanks e.

March 10, 2007 am31 9:35 am 9:35 am

Haven’t taught math for 30 years. But, my approach was – before introducing complex numbers, take them back to natural numbers, show that integers are artificial, created for the purpose of solving certain equations, give an example of such an equation, give a real life example of why they are useful, that the symbolism is arbitrary, and what these new numbers do to the number line Then do the same with rationals. when they are comfortable with the concept of inventing numbers to solve certain equations to arise normally, and that symbols are arbitrary, and that the number line can be expanded, give them an equation that can’t be solved with a rational number (x^2=-1). INVENT a new number, propose some possible symbols for it, explain who first thought of it and why the particular synbol was used. Where does this fit on the number line? What uses does it have ? Then go to complex. If they UNDERSTAND, they will have a lot less trouble, will not be just manipulating symbols.

March 10, 2007 pm31 5:59 pm 5:59 pm

I came up with something else that might be useful for introducing complex numbers. Start by introducing the evil numbers. The basic evil number is e, non-evil numbers are called good numbers, and mixed numbers are called moral. They use all the same rules as complex numbers. The only difference? e^2 = 1. Once they’ve gotten all of this, reveal that they were just learning negative numbers, and imaginary numbers work in the exact same fashion except that i^2 = -1. I imagine being able to just draw a single line on the board to magically transform the evil into the imaginary.

This should help overcome the mystique of complexes as something unreal or crazy. Definitely integrate what Karl is saying as well, where you emphasize that numbers are used to model things; they aren’t real in and of themselves. Focus on the non-reality of negatives if you go this route to link up with the above. This should be easy enough, since major mathematicians didn’t accept negatives as real until the 19th century!

13. March 10, 2007 pm31 6:07 pm 6:07 pm

Karl,

I love this approach. I use something similar in the introduction (which I abbreviated with this group, not knowing they hadn’t seen it before)
$2 \times \square = 7$ I write on the board. Before anyone can answer seven halves or three-point-five, I ask what a first grader would say. The idea that fractions extend the number system makes sense.

Then I write $4 + \square = 1$. We have the same discussion. For a young enough student, this equation has no solution in their number system. So we extend the number system again.

The game continues. $x^2 = 11$ (which reminds me I owe the class a proof of the irrationality of $\sqrt{2}$) By this point a student can jump in without prompting that we used to think there was no solution, but we extended our number system to include square roots (side discussion of other irrational numbers).

And then when I write $x^2 = -1$ I can almost hear the little voice now: “Are we going to extend the number system again?”

We forget, each one of these extensions is a minor trauma for the most thoughtful kids. I guess part of my overall concern here is that this class accepted the algebra of complex numbers without challenging their existence. That still worries me, even after seeing that most of the students easily manipulate a + bi.

14. March 11, 2007 am31 3:13 am 3:13 am

jd, have you considered mentioning that, historically, imaginary numbers first came to mathematicians’ attention through study not of quadratic, but of cubic equations? This arises when using Cardano’s method to solve a cubic with three real roots: even though the roots are real, the formula expresses them in terms of imaginary numbers! This meant that mathematicians were forced to work with “imaginary” numbers, in contrast to an equation such as x^2 + 1 = 0, where they could instead declare that there was no solution.

This situation is known as Casus Irreducibilis; see e.g. http://www.sosmath.com/algebra/factor/fac111/fac111.html

15. March 11, 2007 am31 3:32 am 3:32 am

Nick,

no, I had not considered mentioning the history of imaginary numbers, because I neither knew it, nor Cardano’s method.

I have printed your link, will study it a bit, and share appropriate parts with the kidlets.

I will also, btw, share how I learned it. It is important for them to understand the collaborative nature of mathematical learning, and to keep in mind that I can’t really be a teacher of mathematics unless I continue to be a student.

Thank you.

16. March 14, 2007 am31 2:18 am 2:18 am

please don’t use the square root symbol so freely.
\sqrt{-1} is slang; if it’s to be made at all formal
it should always have \pm (“plus or minus”)–
for example, as in the quadratic formula.

there simply isn’t a function on {\Bbb C}
that does what “\sqrt” does for the postitive reals.
–yes, i really do think the difference matters).

17. March 14, 2007 am31 3:19 am 3:19 am

Vlorbik,

your comments are (to me) a bit cryptic. Can you explain the problem with the fourth root of i ?

March 17, 2007 am31 5:52 am 5:52 am

The problem he’s pointing out is that there are four fourth roots of 1, and no nice convention to distinguish between them. In fact, though, I think the complaint about the square root (or any rational power) on the complexes is silly: each number has two square roots, but it is absolutely possible to define a square-root function; the only “problem” is that it can’t be continuous everywhere.

19. May 5, 2010 am31 4:49 am 4:49 am

I think you don’t get any questions for two reasons… 1) the material is so easy that anyone (i mean anyone) can understand or 2) they just don’t get it.

I can’t believe that imaginary numbers are easy enough for anyone to understand. I speak from experience!