Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Aim: What are imaginary and complex numbers? Do Now: Solve for x: x = 0 ? What number when multiplied by itself gives us a negative one? No such real number Graph it parabola does not intersect x-axis - NO REAL ROOTS

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Definition: Imaginary Numbers If is not a real number, then is a non-real or imaginary number. i A pure imaginary number is any number that can be expressed in the form bi, where b is a real number such that b ≠ 0, and i is the imaginary unit. In general, for any real number b, where b > 0: b = 5

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Powers of i i –1 i 2 = If i 2 = – 1, then i 3 = ? = i 2 i = –1( ) = –i i 2 i 2 = (–1)(–1) = 1 = i 4 i = 1( ) = i i 4 i 2 = (1)(–1) = –1 = i 6 i = -1( ) = –i i 6 i 2 = (–1)(–1) = 1 i 0 = 1 i 1 = i i 2 = –1 i 3 = –i i 4 = 1 i 5 = i i 6 = –1 i 7 = –i i 8 = 1 i 9 = i i 10 = –1 i 11 = –i i 12 = 1 What is i 82 in simplest form? 82 ÷ 4 = 20 remainder 2 equivalent to i 2 = –1 i 82 i3i3 i 4 = i 6 = i 8 = i5i5 i7i7 –1 i 2 =

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Properties of i i Addition: 4i + 3i = 7i Subtraction: 5i – 4i = i Multiplication: (6i)(2i) = 12i 2 = –12 Division:

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Complex Numbers Definition: A complex number is any number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. a + bi pure imaginary number Any number can be expressed as a complex number: 7 + 0i = i = 2i real numbers a + bi

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Complex Numbers Real Numbers Rational Numbers Integers Whole Numbers Irrational Numbers Counting Numbers The Number System i i i i 75 -i 47 i i -i i3i3 i9i9 i 2 + 3i-6 – 3i 1/2 – 12i

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Graphing Complex Numbers reals pure imaginaries i -i 2i 3i 4i 5i -4i -3i -2i -5i -6i (0 + 0i) (0 + 3i) (0 – 4i) (3 – 2i) (4 + 5i) (–5 – 2i) Complex Number Plane (x, y) a + bi

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Vectors reals pure imaginaries i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i Vector - a directed line segment that represents directed force notation: OP O (3 + 4i) P The length of vectors is found by using the Pythagorean Theorem & is always positive. The length of a vector representing a complex number is

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Model Problems Add: Express in terms of i and simplify: = 10i = 4/5i Write each given power of i in simplest terms: i 49 i 54 i 300 i 2001 = i= -1= i= 1 Multiply: Simplify:

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Model Problems 1) ( i) x i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Which number is included in the shaded region? 2) (1.5 – 3.5i) 3) (3.5) 4) (4.5i) (1) (2) (3) (4)

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Fractals Fractals are self-similar designs. Functions that generate fractals have the form f(z) = z 2 + c, where c is a complex number. The points on the graph are the output values. To find points, use 0 as the first input value, then use each output value as the next input value. Example of FractalSierpinski Triangle

Aim: Complex & Imaginary Numbers Course: Adv. Alg. & Trig. Model Problems Fractals are self-similar designs. Functions that generate fractals has the form f(z) = z 2 + c, where c is a complex number. The points on the graph are the output values. To find points, use 0 as the first input value, then use each output value as the next input value. Find the first three output values for f(z) = z 2 + i, 0 is the first input value. f(z) = z 2 + if(0) = i = i f(i) = i 2 + i = -1 + i first output f(-1 + i) = (-1 + i) 2 + i = [(-1) 2 + (-1)(i) + (-1)(i) -(i) 2 ] + i second output = (1 – 2i – 1) + i = -i third output