4.1. COMPLEX NUMBERS

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1 4.1. COMPLEX NUMBERS

2 What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers in standard form. Find complex solutions of quadratic equations.

3 The Imaginary Unit i Some quadratic equations have no real solutions. For instance, the quadratic equation x = 0 has no real solution because there is no real number x that can be squared to produce 1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i = Imaginary unit where i 2 = 1.

4 The Imaginary Unit i By adding real numbers to real multiples of this imaginary unit, the set of complex numbers is obtained. Each complex number can be written in the standard form a + bi. For instance, the standard form of the complex number 5 + is 5 + 3i because

5 The Imaginary Unit i In the standard form a + bi, the real number a is called the real part of the complex number a + bi, and the number bi (where b is a real number) is called the imaginary part of the complex number.

6 For every real number a, you can write a = a + 0i. Thus, the set of real numbers is a subset of the set of complex numbers. In the diagram, non-real complex numbers are called imaginary numbers.

7 Equality of two complex numbers Two complex numbers are equal if their real and imaginary parts are the same.

8 Adding and Subtracting Complex Numbers To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately.

9 Additive identity and Additive inverse The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a + bi is (a + bi ) = a bi. Additive inverse So, you have (a + bi ) + ( a bi ) = 0 + 0i = 0.

10 Example Perform the operation: a. (4 + 7i ) + (1 6i ) b. (1 + 2i ) (4 + 2i ) c. 3i ( 2 + 3i ) (2 + 5i ) d. (3 + 2i ) + (4 i ) (7 + i )

11 Solution a. (4 + 7i ) + (1 6i ) = 4 + 7i + 1 6i = (4 + 1) + (7i 6i ) = 5 + i Remove parentheses. Group like terms. Write in standard form. b. (1 + 2i ) (4 + 2i ) = 1 + 2i 4 2i = (1 4) + (2i 2i ) = = 3 Remove parentheses. Group like terms. Simplify. Write in standard form.

12 Solution c. 3i ( 2 + 3i ) (2 + 5i ) = 3i + 2 3i 2 5i = (2 2) + (3i 3i 5i ) = 0 5i = 5i d. (3 + 2i ) + (4 i ) (7 + i ) = 3 + 2i + 4 i 7 i = ( ) + (2i i i ) = 0 + 0i = 0

13 Remark 1. Note in Examples 1(b) and 1(d) that the sum of two complex numbers can be a real number. 2. Many of the properties of real numbers are valid for complex numbers as well. Here are some examples. Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Property of Multiplication Over Addition

14 Multiplying Complex Numbers (Use Distribution Property and i 2 = 1) Notice below how these properties are used when two complex numbers are multiplied. (a + bi )(c + di ) = a(c + di ) + bi(c + di ) Distributive Property = ac + (ad)i + (bc)i + (bd)i 2 = ac + (ad)i + (bc)i + (bd)( 1) = ac bd + (ad)i + (bc)i = (ac bd) + (ad + bc)i Distributive Property i 2 = 1 Commutative Property Associative Property Rather than trying to memorize this multiplication rule, you should simply remember how the Distributive Property is used to multiply two complex numbers.

15 Complex Conjugates a + bi and a bi are called complex conjugates. When they are multiplied, the product becomes a real number. Here is why: (a + bi )(a bi ) = a 2 abi + abi b 2 i 2 = a 2 b 2 ( 1) = a 2 + b 2 Complex conjugates are useful when you rationalize the quotient of complex numbers.

16 Example Multiply each complex number by its complex conjugate. a. 1 + i b. 4 3i

17 Solution a. The complex conjugate of 1 + i is 1 i. (1 + i )(1 i ) = 1 2 i 2 = 1 ( 1) = 2 b. The complex conjugate of 4 3i is 4 + 3i. (4 3i )(4 + 3i ) = 4 2 (3i ) 2 = 16 9i 2 = 16 9( 1) = 25

18 Rationalizing the quotient of complex numbers To write the quotient of a + bi and c + di in standard form, where c and d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain Standard form

19 Complex Solutions of Quadratic Equations When using the Quadratic Formula to solve a quadratic equation, you often obtain a result such as, which you know is not a real number. By factoring out i =, you can write this number in standard form. The number i is called the principal square root of 3.

20 Example Solve (a) x = 0 and (b) 3x 2 2x + 5 = 0. Hint:(b) Use the quadratic formula: If the quadratic equation is ax 2 + bx + c = 0, then the solution is x = b+ b2 4ac 2a or x = b b2 4ac If b 2 4ac < 0, then the quadratic equation has complex conjugate solutions. 2a

21 Solution a. x = 0 b. 3x 2 2x + 5 = 0 x 2 = 4 x = 2i Remark: 56 = 2 14 i

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