Imaginary & Complex Numbers

Once upon a time…

-In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions. -These numbers were devised using an imaginary unit named i.

-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. -The first four powers of i establish an important pattern and should be memorized. Powers of i

Powers of i Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i.

Powers of i 1.) Find i23 2.) Find i2006 3.) Find i37 4.) Find i828

Complex Number System Reals Rationals (fractions, decimals) Integers Imaginary i, 2i, -3-7i, etc. Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Irrationals (no fractions) pi, e Whole (0, 1, 2, …) Natural (1, 2, …)

Simplify. -Express these numbers in terms of i. 3.) 4.) 5.)

You try… 6. 7. 8.

To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

Multiplying 9. 10. 11.

a + bi imaginary real Complex Numbers The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

Add or Subtract 12. 13. 14.

Multiplying & Dividing Complex Numbers

REMEMBER: i² = -1 Multiply 1) 2)

You try… 3) 4)

Multiply 5)

You try… 6)

You try… 7)

Conjugate -The conjugate of a + bi is a – bi

Find the conjugate of each number… 8) 9) 10) 11)

Divide… 12)

You try… 13)