## Imaginary Numbers – Powers of i

Yesterday in my intermediate algebra class we were finishing up the section on complex numbers. The last topic was simplifying powers of i. Instead of having my students memorize a table that shows the pattern I took a different approach.

Starting point: I showed my students that $i^4=1$. There are a couple of ways to do this.

$(i^2)(i^2)=(-1)(-1)=1$

or

$i^3=i(i^2)=i(-1)=-i$
$i^4=i(i^3)=i(-i)=-i^2=-(-1)=1$

Now, for any power of i, we can rewrite the expression in terms of $i^4$ and some other power of i. For example, $i^{73}=(i^4)^{18} \times i$. Since $i^4=1$, we are left with i.

For $i^n$, divide the exponent n by 4 and note the remainder r. We can then rewrite $i^n$ as $i^r$. There are only four possible cases for r: 0, 1, 2, or 3.

• $i^0=1$
This is easy for students to remember, because they already know that any real number raised to the 0 power is equal to 1.
• $i^1=i$
Again, an easy case for students to remember, because they know that we do not have to write an exponent of 1.
• $i^2=-1$
This is one of the two definitions we used to introduce imaginary and complex numbers. (The other is $\sqrt{-1}=i$.)
• $i^3=-i$
For this one, I recommend rewriting $i^3$ as $i \times i \times i$ and then simplify as shown.
$i \times i \times i = ( i \times i) \times i = -1 \times i=-1$
(By the way, my students refer to this as the “i i i” case – pronounced like “ay ay ay”.)

I think that understanding these 4 cases is much better than trying to memorize some table, because we know that while it is easy to recall that information incorrectly, when we understand those errors don’t occur.

Do you have any techniques for teaching imaginary numbers? As a student, do you have any questions about imaginary numbers? You can reach me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

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• 1. ZeroSum Ruler  |  November 28, 2010 at 10:04 pm

It’s important to start the year with i- I learned this the hard way at the end of last year, my students facing a final exam with i all over it. Parabolas with no roots, simplifying, “foiling” (I hate using that acronym). The number had been compartmentalized into one chapter section in the book, how was I to know?

I posted an Excel worksheet in i on my WordPress blog. It’s something I came up with THIS year to battle the i end-year invasion!