Prentice Hall Algebra 2

Chapter 4 Quadratic Functions and Equations > 4-8 Complex Numbers

Problem 1 Simplifying a Number Using i

How do you write square root of negative 18 end root by using the imaginary unit i?

− 18 = − 1 ⋅ 18 = − 1 ⋅ 18 Multiplication Property of Square Roots = i ⋅ 18 Definition of i = i ⋅ 3 2 Simplify . = 3 i 2 table with 5 rows and 3 columns , row1 column 1 , square root of negative 18 end root , column 2 equals . square root of negative 1 dot 18 end root , column 3 , row2 column 1 , , column 2 equals , square root of negative 1 end root , dot square root of 18 , column 3 cap multiplicationcap propertyofcap squarecap roots , row3 column 1 , , column 2 equals i dot square root of 18 , column 3 cap definitionof . i , row4 column 1 , , column 2 equals i dot 3 square root of 2 , column 3 cap simplify , . , row5 column 1 , , column 2 equals 3 i square root of 2 , column 3 , end table

Think

Is − 18 square root of negative 18 end root a real number?

No. There is no real number that when multiplied by itself gives negative 18 . You must use the imaginary unit i to write − 18 . square root of negative 18 end root , .

✓ Got It?

How do you write each number in parts (a)–(c) by using the imaginary unit i?
1. − 12 square root of negative 12 end root
2. − 25 square root of negative 25 end root
3. − 7 square root of negative 7 end root
4. Reasoning Explain why − 64 ≠ − 64 . square root of negative 64 end root , not equal to negative square root of 64 .

An imaginary number is any number of the form a + bi, where a and b are real numbers and b not equal to 0 . Imaginary numbers and real numbers together make up the set of complex numbers.

Take Note Key Concept: Complex Numbers

You can write a complex number in the form a + bi, where a and b are real numbers.

If b equals 0 comma the number a + bi is a real number.

If eh equals 0 and b not equal to 0 comma the number a + bi is a pure imaginary number.

a + b i ↑ ↑ Real part Imaginary part table with 3 rows and 3 columns , row1 column 1 , eh , column 2 plus , column 3 b i , row2 column 1 , upwards arrow , column 2 , column 3 upwards arrow , row3 column 1 , cap realpart , column 2 , column 3 cap imaginarypart , end table

**Complex Numbers (a + bi)**
Real Numbers (a + 0 i)	Imaginary Numbers ( a + bi , b ≠ 0 ) open eh plus bi comma b not equal to 0 close Pure Imaginary Numbers ( 0 + bi , b ≠ 0 ) open 0 plus , bi , comma b not equal to 0 close

In the complex number plane, the point (a, b) represents the complex number a + bi. To graph a complex number, locate the real part on the horizontal axis and the imaginary part on the vertical axis.

The absolute value of a complex number is its distance from the origin in the complex plane.

| a + b i | = a 2 + b 2 absolute value of , eh plus b i , end absolute value , . equals . square root of eh squared , plus , b squared end root

A complex graph.
Image Long Description

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Problem 1 Simplifying a Number Using i

Think

✓ Got It?

Take Note Key Concept: Complex Numbers

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