Imaginary Numbers

Have you been wondering what j actually means in expressions of impedance? Well, j is nothing but a number: the positive square root of −1. There’s a negative square root of −1, too, and it is equal to −j. When either j or −j is multiplied by itself, the result is −1. (Pure mathematicians often denote these same numbers as i or −i.)

The positive square root of −1 is known as the unit imaginary number. The set of imaginary numbers is composed of real-number multiples of j or −j. Some examples are j 4, j35.79, −j25.76, and −j25,000.

The square of an imaginary number is always negative. Some people have trouble grasping this, but when you think long and hard about it, all numbers are abstractions. Imaginary numbers are no more imaginary (and no less real) than so-called real numbers such as 4, 35.79, −25.76, or −25,000.

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The imaginary number line.
The unit imaginary number j can be multiplied by any real number on a conventional real number line. If you do this for all the real numbers on the real number line, you get an imaginary number line (above figure). The imaginary number line should be oriented at a right angle to the real number line when you want to graphically portray real and imaginary numbers at the same time.

In electronics, real numbers represent resistances. Imaginary numbers represent reactances.