Complex Numbers: Trigonometric Form Section 8.3 Complex Numbers: Trigonometric Form

Objectives Graph complex numbers. Given a complex number in standard form, find trigonometric, or polar, notation; and given a complex number in trigonometric form, find standard notation. Use trigonometric notation to multiply and divide complex numbers. Use DeMoivre’s theorem to raise a complex number to powers. Find the nth roots of a complex number. Just as real numbers can be graphed on a line, complex numbers can be graphed on a plane. We graph a complex number a + bi in the same way that we graph an ordered pair of real numbers (a, b). However, in place of an x-axis,we have a real axis, and in place of a y-axis, we have an imaginary axis. Horizontal distances correspond to the real part of a number. Vertical distances correspond to the imaginary part.

What are imaginary and complex numbers? Do Now: Graph it Solve for x: x2 + 1 = 0 ? What number when multiplied by itself gives us a negative one? parabola does not intersect x-axis - NO REAL ROOTS No such real number

i Imaginary Numbers If is not a real number, then is a non-real or Definition: A pure imaginary number is any number that can be expressed in the form bi, where b is a real number such that b ≠ 0, and i is the imaginary unit. i

i2 = i2 = Powers of i i –1 –1 i0 = 1 i1 = i i2 = –1 i3 = –i i4 = 1 If i2 = – 1, then i3 = ? = i2 • i = –1( ) = –i i3 i4 = i6 = i8 = i5 i7 i2 • i2 = (–1)(–1) = 1 = i4 • i = 1( ) = i i4 • i2 = (1)(–1) = –1 = i6 • i = -1( ) = –i i6 • i2 = (–1)(–1) = 1 What is i82 in simplest form? 82 ÷ 4 = 20 remainder 2 equivalent to i2 = –1 i82

a + bi Complex Numbers A complex number is any number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Definition: a + bi real numbers pure imaginary number Any number can be expressed as a complex number: 7 + 0i = 7 a + bi 0 + 2i = 2i

The Number System Complex Numbers i -i i3 i9 Real Numbers Rational Numbers i i75 -i47 Irrational Numbers Integers Whole Numbers Counting Numbers 2 + 3i -6 – 3i 1/2 – 12i

Example Graph each of the following complex numbers. a) 3 + 2i b) –4 – 5i c) –3i d) –1 + 3i e) 2 Solution:

Absolute Value The absolute value of a complex number a + bi is

Example Find the absolute value of each of the following. Solution:

Trigonometric Notation If we let  be an angle in standard position whose terminal side passes through the point (a, b), then

Example Add/Subtract

Example Multiply

Example Divide