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# Complex Numbers Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Introduction to Complex Numbers

Until now, zeros of polynomials have been real numbers. The next topic involves complex zeros. These zeros come from even roots of negative numbers like . Before working with complex zeros of polynomials, we will first learn some complex number arithmetic. Complex numbers are normally written in the form a + bi, where a and b are real numbers and . A number such as would be written as 4 + 3 i because . Real numbers are complex numbers where b = 0.

#### Examples

Write the complex numbers in the form a + bi , where a and b are real numbers.

### Adding and Subtracting Complex Numbers

Adding complex numbers is a matter of adding like terms. Add the real parts, a and c , and the imaginary parts, b and d.

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtract two complex numbers by distributing the minus sign in the parentheses then adding the like terms.

a + bi(c + di) = a + bicdi = (a − c) + (b − d)i

#### Examples

Perform the arithmetic. Write the sum or difference in the form a + bi , where a and b are real numbers.

• (3 − 5 i ) + (4 + 8 i )  =  (3 + 4) + (−5 + 8) i  =  7 + 3 i

• 2 i −6 + 9 i  =  −6 + 11 i

•

• 11 − 3 i − (7 + 6 i )  =  11 − 3 i − 7 − 6 i  =  4 − 9 i

•

## Multiplying and Dividing Complex Numbers

### Multiplying Complex Numbers

Multiplying complex numbers is not as straightforward as adding and subtracting them. First we will take the product of two purely imaginary numbers (numbers whose real parts are 0). Remember that , which makes i 2 = −1. In most complex number multiplication problems, we will have a term with i 2 . Replace i 2 with −1. Multiply two complex numbers in the form a + bi using the FOIL method, substituting −1 for i 2 and combining like terms.

#### Examples

Write the product in the form a + bi , where a and b are real numbers.

• (5 i )(6 i ) = 30 i 2 = 30(−1) = −30
• (2 i )(−9 i ) = −18 i 2 = −18(−1) = 18
• (4 + 2 i )(5 + 3 i ) = 20 + 12 i + 10 i + 6 i 2 = 20 + 22 i + 6(−1) = 14 + 22 i
• (8 − 2 i )(8 + 2 i ) = 64 + 16 i − 16 i − 4 i 2 = 64 − 4(−1) = 68

### Complex Conjugates

The complex numbers a + bi and a − bi are called complex conjugates . The only difference between a complex number and its conjugate is the sign between the real part and the imaginary part. The product of any complex number and its conjugate is a real number.

(a + bi)(a − bi) = a 2abi + abib 2 i 2

= a 2b 2 (−1)

= a 2 + b 2

#### Examples

• The complex conjugate of 3 + 2 i is 3 − 2 i.
• The complex conjugate of −7 − i is −7 + i.
• The complex conjugate of 10 i is −10 i.
• (7 − 2 i )(7 + 2 i ). Here, a = 7 and b = 2, so a 2 = 49 and b 2 = 4, making (7 − 2 i )(7 + 2 i ) = 49 + 4 = 53.
• (1 − i )(1 + i ). Here, a = 1 and b = 1, so a 2 = 1 and b 2 = 1, making (1 − i )(1 + i ) = 1 + 1 = 2.

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