Aim: What are imaginary and complex numbers?

1. Aim: What are imaginary and complex numbers? Graph it Solve for x: x2 + 1 = 0 Do Now: ? What number when multiplied by itself gives us a negative one? parabola does not intersect x-axis - NO REAL ROOTS No such real number

2. then is a non-real or imaginary number. Definition: i In general, for any real number b, where b > 0: Imaginary Numbers If is not a real number, 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. b = 5

3. i If i2 = – 1, then i3 = ? = i2 • i = –1( ) = –i i4 = i5 = i4 • i = 1( ) = i i6 = i7 = i6 • i = -1( ) = –i i8 = What is i82 in simplest form? equivalent to i2 = –1 i82 Powers of i i2 = i2 = –1 –1 i0 = 1 i1 = i i2 = –1 i3 = –i i4 = 1 i5 = i i6 = –1 i7 = –i i8 = 1 i9 = i i10 = –1 i11 = –i i12 = 1 i3 i2 • i2 = (–1)(–1) = 1 i4 • i2 = (1)(–1) = –1 i6 • i2 = (–1)(–1) = 1 82 ÷ 4 = 20 remainder 2

4. i Addition: Subtraction: Multiplication: Division: Properties of i 4i + 3i = 7i 5i – 4i = i (6i)(2i) = 12i2 = –12

5. Definition: real numbers pure imaginary number Any number can be expressed as a complex number: a + bi Complex Numbers 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 7 + 0i = 7 0 + 2i = 2i

6. -i Real Numbers Complex Numbers Rational Numbers i i i3 Integers Irrational Numbers i i75 -i Whole Numbers Counting Numbers i9 i i -i47 i The Number System 2 + 3i -6 – 3i 1/2 – 12i

7. 5i 4i 3i 2i 1 -5 -4 -3 -2 -1 2 3 4 5 6 -2i -3i -4i -5i -6i Graphing Complex Numbers (x, y) a + bi Complex Number Plane pure imaginaries (4 + 5i) (0 + 3i) i (0 + 0i) reals 0 -i (3 – 2i) (–5 – 2i) (0 – 4i)

8. P (3 + 4i) OP OP Vectors Vector - a directed line segment that represents directed force notation: pure imaginaries 5i 4i 3i 2i i reals O 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 The length of vectors is found by using the Pythagorean Theorem & is always positive. -i -2i The length of a vector representing a complex number is -3i -4i -5i -6i

9. Add: Multiply: Simplify: Model Problems Express in terms of i and simplify: = 10i = 4/5i Write each given power of i in simplest terms: i49 = i i54 = -1 i300 = 1 i2001 = i

10. yi 5i 4i (1) 3i 2i i x (3) 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 -i -2i -3i -4i (4) (2) -5i -6i Model Problems Which number is included in the shaded region? 1) (-1.5 + 3.5i) 2) (1.5 – 3.5i) 3) (3.5) 4) (4.5i)

11. Fractals Fractals are self-similar designs. Functions that generate fractals have the form f(z) = z2 + 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 Fractal Sierpinski Triangle

12. first output second output Model Problems Fractals are self-similar designs. Functions that generate fractals has the form f(z) = z2 + 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) = z2 + i, 0 is the first input value. = i f(z) = z2 + i f(0) = 02 + i = -1 + i f(i) = i2 + i f(-1 + i) = (-1 + i)2 + i = [(-1)2 + (-1)(i) + (-1)(i) -(i)2] + i = -i = (1– 2i – 1) + i third output