Algebra Know-It-ALL

350 Imaginary and Complex Numbers

you’re more likely to go into science or engineering than pure mathematics, so you should get

used to the notation they prefer.

Positive and negative j

The square root of −1 can have either of two values, just as can the square root of any positive

real number. One of these is j. The other is −j, the product of j and −1. These two numbers

are not the same, just as the positive and negative square roots of 1 are not the same!

Are you confused?

Some people have trouble envisioning this unit imaginary number, also called the j operator. Does the idea

escape your “mind’s eye”? If so, don’t worry about it. Recall from Chap. 3 that the natural numbers—the

simplest ones—are built up from a set containing nothing! All numbers are abstract in the literal sense, so

j isn’t any more bizarre than 0, or −1, or any other number.

Here’s a challenge!

All of the laws of real-number arithmetic also apply to the unit imaginary number. Based on that fact,

figure out what happens as j is raised to increasing integer powers starting with the 1st power.

Solution

Keep in mind that j is the positive square root of −1, which is (−1)1/2. The parentheses are important in

this expression. If we leave them out, someone might get the idea that we’re discussing the quantity −(11/2),

which is equal to −1. Because all the laws of the reals also apply to j, we can be sure that j^1 =j. By defini-

tion,j^2 =−1. From this we can calculate

j^3 =j^2 ×j

=− 1 ×j

=−j

Now for the 4th power:

j^4 =j^3 ×j

=−j×j

=− 1 ×j×j

=− 1 ×j^2

=− 1 × (−1)

= 1

And the 5th power:

j^5 =j^4 ×j

= 1 ×j

=j