Complex Numbers and i is the imaginary unit Numbers in the form a + bi are called complex numbers a is the real part b is the imaginary part

Examples a) b) c) d) e)

Example: Solving Quadratic Equations Solve x = √-25 Take the square root on both sides. The solution set is {±5i}.

Another Example Solve: x2 + 54 = 0 The solution set is

Example: Products and Quotients Multiply: Divide:

Addition and Subtraction of Complex Numbers For complex numbers a + bi and c + di, Examples (10 − 4i) − (5 − 2i) = (10 − 5) + [−4 − (−2)]i = 5 − 2i (4 − 6i) + (−3 + 7i) = [4 + (−3)] + [−6 + 7]i = 1 + i

Multiplication of Complex Numbers For complex numbers a + bi and c + di, The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i2 = −1.

Examples: Multiplying (2 − 4i)(3 + 5i) (7 + 3i)2

Powers of i i1 = i i5 = i i9 = i i2 = −1 i6 = −1 i10 = −1 and so on.

Simplifying Examples i17 i4 = 1 i17 = (i4)4 • i = 1 • i = i i−4

Property of Complex Conjugates For real numbers a and b, (a + bi)(a − bi) = a2 + b2. The product of a complex number and its conjugate is always a real number. Example

Relationships Among x, y, r, and θ

Trigonometric (Polar) Form of a Complex Number The expression is called the trigonometric form or (polar form) of the complex number x + yi. The expression cos θ + i sin θ is sometimes abbreviated cis θ. Using this notation

Example Express 2(cos 120° + i sin 120°) in rectangular form. Notice that the real part is negative and the imaginary part is positive, this is consistent with 120 degrees being a quadrant II angle.

Converting from Rectangular Form to Trigonometric Form Step 1 Sketch a graph of the number x + yi in the complex plane. Step 2 Find r by using the equation Step 3 Find θ by using the equation choosing the quadrant indicated in Step 1.

Example Example: Find trigonometric notation for −1 − i. First, find r. Thus,

Product Theorem If are any two complex numbers, then In compact form, this is written

Example: Product Find the product of

Quotient Theorem If are any two complex numbers, where then

Example: Quotient Find the quotient.

De Moivre’s Theorem If is a complex number, and if n is any real number, then In compact form, this is written

Example: Find (−1 − i)5 and express the result in rectangular form. First, find trigonometric notation for −1 − i Theorem

nth Roots For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if

nth Root Theorem If n is any positive integer, r is a positive real number, and θ is in degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly n distinct nth roots, given by where

Example: Square Roots    

Example: Fourth Root Find all fourth roots of Write the roots in rectangular form. Write in trigonometric form. Here r = 16 and θ = 120°. The fourth roots of this number have absolute value  

Example: Fourth Root continued There are four fourth roots, let k = 0, 1, 2 and 3. Using these angles, the fourth roots are

Example: Fourth Root continued Written in rectangular form The graphs of the roots are all on a circle that has center at the origin and radius 2.

Polar Coordinate System The polar coordinate system is based on a point, called the pole, and a ray, called the polar axis.

Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r, θ), then these coordinates are related as follows.

Example Plot the point on a polar coordinate system. Then determine the rectangular coordinates of the point. P(2, 30°) r = 2 and θ = 30°, so point P is located 2 units from the origin in the positive direction making a 30° angle with the polar axis.

Example continued Using the conversion formulas: The rectangular coordinates are

Example Convert (4, 2) to polar coordinates.

The following slides are extension work for Complex Numbers …..

Rectangular and Polar Equations To convert a rectangular equation into a polar equation, use and and solve for r. For the linear equation you will get the polar equation

Example Convert x + 2y = 10 into a polar equation. x + 2y = 10

Example Graph r = −2 sin θ 1 1.414 2 -1 -1.414 r 330 315 270 180 150 -1 -1.414 r 330 315 270 180 150 135 θ -1.732 120 -2 90 60 45 30

Example Graph r = 2 cos 3θ −1.41 −2 1.41 2 r 90 75 60 45 30 15 θ

Example Convert r = −3 cos θ − sin θ into a rectangular equation.

Circles and Lemniscates

Limacons

Rose Curves