And the 6th:
j^6 =j^5 ×j
=j×j
=j^2
=− 1
Can you see what will happen if we keep going like this, increasing the integer power by 1 over and over?
We’ll go in a four-way cycle. If you grind things out, you’ll see for yourself that j^7 =−j, j^8 = 1 ,j^9 =j,
j^10 =− 1 , and so on. In general, if n is a positive integer,
jn=jn+^4
The Imaginary Number Line
The unit imaginary number j can be multiplied by any real number to get the positive square
root of some negative real number. Conversely, the positive square root of any negative real
number is equal to some positive-real multiple of j. If we want to multiply j by a positive real
numberb, we write jb. If we want to multiply j by a negative real number −b, we write −jb,
putting the minus sign in front of j rather than between j and b.For example,
j^5 = (−1)1/2× 25 1/2
= (− 1 × 5)1/2
= (−5)1/2
and
(−4)1/2= (− 1 × 4)1/2
= (−1)1/2× 4 1/2
=j 2
If we take the real number line and multiply the value of every point by j, the result is the
imaginary number line (Fig. 21-1).
Are you confused?
“Why,” you might ask, “do we write j before the real-number numeral and not after it?” It’s a matter of
preference. Engineers usually write the j before the real number. If you see other notations for imaginary
numbers such as 2j, 2 i (the way most pure mathematicians write it), or even i 2 , keep in mind that they all
refer to the same quantity, which we would call j 2.
Be careful!
In the “challenge” calculation at the end of the previous section, j was raised to integer powers. If we’re not
careful, we can confuse expressions like these with integer multiples of j. We must pay close attention to
whether that real number is meant to be a multiple of j (as in j4), or a power of j (as in j^4 ).
The Imaginary Number Line 351
