Imaginary & Complex Numbers Mini Unit

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

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, …)

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 ex. ex. ex.

Complex Numbers Find the value of x and y that makes 5𝑥+1+ 3−2𝑦 𝑖=11−13𝑖 true.

REMEMBER: i² = -1 Multiply 1) 2)

Multiply ex)

Complex numbers are defined as 𝑎+𝑏𝑖, where 𝑎 and 𝑏 are real numbers and 𝑖 is the imaginary unit. Given 2−2𝑖 3+5𝑖 , what is 𝑎+𝑏?

Assignment Pg. 250-251 #18−32 even #36−44 even #66

Questions on Assignment Pg. 250-251 #18−32 even #36−44 even #66

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

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

Divide… 12)

You try… 13)