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8.1 Complex Numbers Definition: Imaginary Unit The number 푖, called the imaginary unit, is such that 푖2 = −1. (That is, 푖 is the number whose square is -1.
Example 1 Write each expression in terms of i.
a. √−9 b. √−12 c. √−17
To simplify expressions that contain square roots of negative numbers, it is necessary to write each square root in terms of 푖 before applying the properties of radicals.
For example: √−4√−9 =
Definition: Complex Number A complex number is any number that can written in the form 푎 + 푏푖 where a and b are real numbers and 푖2 = −1. The form 푎 + 푏푖 is called standard form for complex numbers. The number a is called the real part of the complex number. The number b is called the imaginary part of the complex number. If b = 0, then 푎 + 푏푖 = 푎, which is a real number. If 푎 = 0 and 푏 ≠ 0, then 푎 + 푏푖 = 푏푖, which is called an imaginary number. Example 2 a) 3 + 2푖
b) −7푖
c) 4
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Example 3 Find x and y if (−3푥 − 9) + 4푖 = 6 + (3푦 − 2)푖.
Definition: Sum and Difference
If 푧1 = 푎1 + 푏1푖 and 푧2 = 푎2 + 푏2푖 are complex numbers, then the sum and difference of 푧1 and 푧2 are defined as follows:
푧1 + 푧2 = (푎1 + 푏1푖) + (푎2 + 푏2푖)
=
푧1 − 푧2 = (푎1 + 푏1푖) − (푎2 + 푏2푖)
=
Example 4 If 푧1 = 3 − 5푖 and 푧2 = −6 − 2푖, find 푧1 + 푧2 and 푧1 − 푧2.
Powers of 풊.
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Example 5 Simplify each power if 푖. a. 푖20 = b. 푖23 = c. 푖30 =
Definition: Multiplication
If 푧1 = 푎1 + 푏1푖 and 푧2 = 푎2 + 푏2푖 are complex numbers, then their product is defined as follows:
푧1푧2 = (푎1 + 푏1푖)(푎2 + 푏2푖)
=
You can also FOIL, simplify and combine like terms to get this.
Example 6 Multiply (3 − 4푖)(2 − 5푖).
Example 7 Multiply (4 − 5푖)(4 + 5푖).
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Definition: Complex Conjugates
The complex numbers 푎 + 푏푖 and 푎 − 푏푖 are called complex conjugates. Their product is the real number 푎2 + 푏2.
Try multiplying (푎 + 푏푖)(푎 − 푏푖) to confirm this.
We use complex conjugates to divide two complex numbers.
5푖 Example 8 Divide . 2−3푖
7+2푖 Example 9 Divide . 5푖
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