Imaginary & Complex Numbers Section 4-8 Page 248

Today…. Identify, graph, and perform operations with 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

Generalizing Powers of i Evaluate the examples by first finding a pattern for raising i to a power: Pattern Divide the exponent by 4 and find the remainder. The remainder is where it falls in the pattern: Examples Divide by 4 Remainder After 4, it starts to repeat itself!

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, …)

Evaluating a Negative Square Root Calculate the value of the expression below: “Factor” out a -1 Rewrite Simplify

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

You try… 6. 7. 8.

Warning Consider It is tempting to combine them The multiplicative property of radicals only works for positive values under the radical sign Instead use imaginary numbers

Try It Out    

Try It Out Use the correct principles to simplify the following:   16 + 81 = 97 9 -24i – 144 -135 – 24i

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

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.

REMEMBER: i² = -1 Multiply 1) 2)

You try… 3) 4)

Multiply 5)

You try… 6)

You try… 7)

The sum and product of complex conjugates are always real numbers For any complex number: The Complex Conjugate is: The sum and product of complex conjugates are always real numbers Example: Find the sum and product of 5 – 3i and its complex conjugate.

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

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

Operations on Complex Numbers Division technique Multiply numerator and denominator by the conjugate of the denominator

Divide… 12)

You try… 13)

Graphical representation of a complex number A complex number has a representation in a plane. Simply take the x-axis as the real numbers and an y-axis as the imaginary numbers. Thus, giving the complex number a + bi the representation as point P with coordinates (a,b).

Graphing a Complex Number Therefore, complex numbers can be represented by a two dimensional graph. Here we see the graph of the complex number 3 – 2i.

Graphing in the Complex Plane Plot: 6 – 4i 6 – 4 What is the distance from 0+0i? Imaginary Axis Real Axis 6 4

Complex Absolute Value The distance from the complex number to 0 in the complex plane: Example: Evaluate a = b = -2 11

Assignment Textbook page 253: 8-12, 13-15 (on the same graph), 18-30 even, 48-60 mo3, 67-69