### 19.1. Complex Numbers in Rectangular Form

#### 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 + 8i 5.92 − 2.93i 83 27i

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

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