5.4 Complex Numbers
Goals p Perform operations with complex numbers. p Apply complex numbers to fractal geometry.
Your Notes VOCABULARY
Imaginary unit i The imaginary unit i is defined as i 1.
Complex number A number a bi where a and b are real numbers and i is the imaginary unit
Standard form of a complex number The form a bi where a and b are real numbers and i is the imaginary unit. The number a is the real part of the complex number and bi is the imaginary part of the complex number.
Imaginary number A complex number a bi where b 0
Pure imaginary number A complex number a bi where a 0 and b 0
Complex plane A coordinate plane where each point (a, b) represents a complex number a bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis.
Complex conjugates Two complex numbers of the form a bi and a bi
Absolute value of a complex number If z a bi, then the absolute value of z, denoted z, is a nonnegative real number defined as z a 2 b 2 . Geometrically, the absolute value of a complex number is the number’s distance to the origin.
104 Algebra 2 Notetaking Guide • Chapter 5 Your Notes THE SQUARE ROOT OF A NEGATIVE NUMBER Property Example 1. If r is a positive real number, 5 i 5 then r i r. 2. By Property (1), it follows (i 5)2 i2 p 5 5 that (i r)2 r.
Example 1 Solving a Quadratic Equation
Solve 2x2 3 15.
Solution 2x2 3 15 Write original equation. 2x2 18 Subtract 3 from each side. x2 9 Divide each side by 2 . x 9 Take square roots of each side. x i 9 Write in terms of i. x 3i Simplify the radical. The solutions are 3i and 3i .
Example 2 Plotting Complex Numbers Plot the complex numbers in the complex plane. a. 1 i b. 2 2i c. 3 3i
Solution a. To plot 1 i, start at the origin, imaginary move 1 unit to the right , and 1 i i then 1 unit up . 3 1 1 real i b. To plot 2 2i, start at the origin, 2 2i move 2 units to the left , and 3 3i 3i then 2 units down . c. To plot 3 3i, start at the origin, move 3 units to the right , and then 3 units down .
Lesson 5.4 • Algebra 2 Notetaking Guide 105 Your Notes Example 3 Adding and Subtracting Complex Numbers
Write the expression (5 ؉ i) ؉ (1 ؊ 2i) as a complex number in standard form. (5 i) (1 2i) ( 5 1 ) ( 1 2 )i Complex addition 6 i Standard form
Checkpoint Complete the following exercises.
1. Solve 5x2 2 8. 2. In which quadrant of the complex plane is 1 3i?