1 C ollege A lgebra Linear and Quadratic Functions (Chapter2) 1

2 Section(2-4) Complex Numbers

3 Objectives After completing this Section, you should be able to: 1.Take the principle square root of a negative number. 2.Write a complex number in standard form. 3.Add and subtract complex numbers. 4.Multiply complex numbers. 5.Divide complex numbers.

4 The equation x 2 = - 1 has no real number solution. To remedy this situation, we define a new number that solves this equation, called the imaginary unit, which is not a real number. The solution to x 2 = - 1 is the imaginary unit I where i 2 = - 1, or Imaginary Unit

5 The imaginary unit i is defined as Example

6 Standard Form of Complex Numbers A complex number has a real part & an imaginary part. Standard form is: Real part Imaginary part Example: 5+4i

7 The set of all numbers in the form a+bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part, of the complex number a+bi. Standard Form of Complex Numbers

8 Complex Number System

9 a + bi = c + di if and only if a = c and b = d In other words, complex numbers are equal if and only if there real and imaginary parts are equal. Equality of Complex Numbers

10 Addition with Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d)i Example: (2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i = i

11 Subtraction with Complex Numbers (a + bi) - (c + di) = (a - c) + (b - d)i Example (3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i = 2 + 3i

12 Multiplying Complex Numbers Step 1: Multiply the complex numbers in the same manner as polynomials.polynomials Step 2: Simplify the expression. Add real numbers together and imaginary numbers together. Whenever you have an, use the definition and replace it with -1. Step 3: Write the final answer in standard form.standard form

13 Example1: Multiply using the distributive property Multiplying Complex Numbers

14 Example2: Multiply using the distributive property Multiplying Complex Numbers

15 Example 1 Perform the indicated operation, writing the result in standard form. (-5 + 7i) - ( i) Combine the real and imaginary parts: (-5-(-11)) + (7-(-6))i = (-5+11) + (7+6)i = 6+13i

16 Example 2: Multiply

17 Example 2 (continued)

18 Example 2 (continued)

19 Example 2 (continued)

20 If z = a + bi is a complex number, then its conjugate, denoted by, is defined as Theorem The product of a complex number and its conjugate is a nonnegative real number. Thus, if z = a + bi, then Conjugate

21 Division with Complex Numbers To divide by a complex number, multiply the dividend (numerator) and divisor (denominator) by the conjugate of the divisor. Example1:

22 Example 2 Divide:

23 Example 2 (continued)

24 Example 2 (continued)

25 Example 2 (continued)

26 Principal Square Root of a Negative Number For any positive real number b, the principal square root of the negative number -b is defined by  (-b) = i  b

27 Example 1: Simplify

28 Example 1 (continued)

29 Example 1 (continued)

30 In the complex number system, the solutions of the quadratic equation where a, b, and c are real numbers and a 0, are given by the formula Since we now have a way of evaluating the square root of a negative number, there are now no restrictions placed on the quadratic formula.

31 Example: Find all solutions to the equation real or complex.

32 Property of the square root of negative numbers If r is a positive real number, then Examples:

33 *For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:

34 Examples

35 The Complex plane Imaginary Axis Real Axis

36 Graphing in the complex plane

37 Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex:

38 Multiplying Treat the i’s like variables, then change any i 2 to -1 Ex: Ex:

39

40 Absolute Value of a Complex Number The distance the complex number is from the origin on the complex plane. If you have a complex number the absolute value can be found using:

41 Examples Which of these 2 complex numbers is closest to the origin? -2+5i

42