Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

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.

Definition of imaginary numbers: It's any number you can imagine

a + bi Complex Numbers real imaginary 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.

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

Simplify complex numbers Remember 28

Answer: -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. 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.

When adding or subtracting complex numbers, combine like terms.

Multiplying complex numbers. To multiply complex numbers, you use the same procedure as multiplying polynomials.

Examples-6: FOIL

Answer: 21-i

Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

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

Find the conjugate of each number… a) b) c) d)

Example-7:

Example-8: Realize the denominator of:

Discriminant of a Quadratic Equation

Fundamental Theorem of Algebra

Complex Conjugate Zeros

Quick survey I feel I understand “complex Number” a) Very well b) With some review, I’ll be good c) Not really d) Not at all

Pair-work