Complex Numbers

Once upon a time…

Reals Rationals (Can be written as fractions) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …)

Since all number belong to the Complex number field, C, all number can be classified as complex. The Real number field, R, and the imaginary numbers, i, are subsets of this field as illustrated below. Real Numbers a + 0i Pure Imaginary Numbers 0 + bi Complex Numbers a + bi

A Little History Math is used to explain our universe. When a recurring phenomenon is seen and can’t be explained by our present mathematics, new systems of mathematics are derived. In the real number system, we can’t take the square root of negatives, therefore the complex number system was created. Complex numbers revolutionized computer graphics

-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.

-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

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.

Powers of i

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

You try…

Multiplying

Complex Numbers Day 2

Recap:

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.

Add or Subtract

Multiplying & Dividing Complex Numbers

REMEMBER: i² = -1 Multiply 1) 2)

You try… 3) 4)

5) Multiply

You try… 6)

You try… 7)

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

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

Divide… 12)

13) You try…