What are imaginary and complex numbers? 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 b = 5 In general, for any real number b, where b > 0:

i2 = i2 = i Powers of 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 little saying to help you remember Once I Lost one Missing eye  

i Properties of i Addition: 4i + 3i = 7i Subtraction: 5i – 4i = i Multiplication: (6i)(2i) = 12i2 = –12 Division:

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

Model Problems Express in terms of i and simplify: = 10i = 4/5i Write each given power of i in simplest terms: i49 = i i54 = -1 i300 = 1 i2001 = i Add: Multiply: Simplify: