19.1. Complex Numbers in Rectangular Form

As usual with any new topic, we start with some definitions.

19.1.1. Imaginary Numbers

Recall that in the real number system, the equation

x2 = −1

had no solution because there was no real number such that its square was −1. Now we extend our number system to allow the quantity to have a meaning. We define the imaginary unit as the square root of −1 and represent it by the symbol i.

As complex numbers find wide use in electric circuits, the letter j is sometimes used for the imaginary unit, reserving i for current.

An imaginary number is the imaginary unit multiplied by a real number. An example of an imaginary number is 9i. In general, we usually denote the real number by b, so the imaginary number would be written as bi.

19.1.2. Complex Numbers

A complex number is any number, real or imaginary, that can be written in the form

a + bi

where a and b are real numbers, and is the imaginary unit. A complex number written this way is said to be in rectangular form.

Example 1:

The following numbers are complex numbers in rectangular form.

4 + 2i−7 + 8i5.92 − 2.93i8327i

When b = 0 in a complex number a + bi, we have a real number. When a = 0, ...

Get Technical Mathematics, Sixth Edition now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.