Section 8.1 Complex Numbers. MAT 182 Chapter 8 Section 8.1 Complex Numbers. Given. i to an odd power is always equal to i, just need to determine the sign. i to an even power is always equal to 1, just need to determine the sign. Notice a pattern! Time to determine the sign. Odd number of negatives multiplied together is negative and an Even number of negatives is a positive. Find the number of i2 are in in. EVEN Whole number = + ODD Whole number = –

Simplify radicals with negatives.

Definition of Complex Numbers. Real Number imaginary Number Complex Numbers. Real Numbers. Imaginary Numbers. When the directions read, “Leave the answers in a + bi form.” The answer will have to include a zero if there is no real number or imaginary number. For example. If the answer is 2, then we write the answer as 2 + 0i. If the answer is -5i, then we write the answer as 0 – 5i.

Combine Like Terms. Treat i like a variable. Distribute the minus sign.

Complex Conjugate Product Rule. Rationalize the Denominator above rule! The denominator is a single term, just multiply by i top and bottom.

Solving Quadratic equations that create complex solutions. Solve for x.

Solving Quadratic equations that create complex solutions. Solve for x.

SECTION 8.2 Complex Numbers in Polar Form A complex number a + bi is represented as a point (a, b) in a coordinate plane. The horizontal axis of the coordinate plane is called the real axis. The vertical axis is called the imaginary axis. The coordinate system is called the complex plane. When we represent a complex number as a point in the complex plane, we say that we are plotting the complex number. We plot (a, b) as if it were (x, y).

Polar notations. A complex number in the form a + bi is said to be in rectangular form. The expression is called the polar form of a complex number. The number r is the modulus of a + bi, and is called an argument of a + bi. A shortcut notation for

Writing a Complex number (Rect.) into Polar Form. Convert – 2 – 2i into Polar Form. 1st Plot the point to determine the angle. imaginary real 2nd Find r.

Writing a Complex number (Rect.) into Polar Form. Convert into Polar Form. 1st Plot the point to determine the angle. imaginary real 2nd Find r.

Writing a Complex number (Polar) into Rectangular Form. Convert into Rectangular Form. 1st Find the exact values for the cosine and sine. Convert into Rectangular Form. 1st Find the values for the cosine and sine with the calculator.

Section 8.3 Multiplication and Division of Complex Numbers. Given two complex numbers in trigonometric form. and The product is L F O I Use the sum formulas for sine and cosine,

Find the product of the complex numbers. and Find the product of the complex numbers.

Section 8.3 Multiplication and Division of Complex Numbers. Given two complex numbers in trigonometric form. and The quotient is Multiply top and bottom by the conjugate of the denominator.

Find the quotient of the complex numbers. and Find the quotient of the complex numbers.

Power of Complex Numbers. Given a complex number in trigonometric form. The pattern leads to DeMoivre’s Theorem , where n is a positive integer.

Find and write the result in rectangular form.

Working De Moivre’s Theorem backwards. Find the 3 cube roots of . This implies that r = 2, and must represent a coterminal angle with . k is any integer.

Finding nth Roots of a Complex Numbers. Given a complex number in trigonometric form and n is a positive integer, has exactly n distinct roots given by , where k = 0, 1, 2, 3, … n – 1. imaginary Find all complex fourth roots of . real ; k = 0, 1, 2, 3

Find all complex roots of x5 – 1 = 0. There is one real solution, x = 1, but there are 5 complex solutions. The first one is rewriting 1 in trigonometric form, where r = 1. for k = 0, 1, 2, 3, 4.

SECTION 8.5 The foundation of the polar coordinate system is a horizontal ray that extends to the right. The ray is called the polar axis. The endpoint of the ray is called the pole. A point P in the polar coordinate system is represented by an ordered pair of numbers . We refer to the ordered pair as the polar coordinates of P. r is a directed distance from the pole to P. is an angle from the polar axis to the line segment from the pole to P. This angle can be measured in degrees or radians. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the polar axis. 90o 120o 60o 150o 30o 180o 2 4 6 0o Plot the following points. Find 3 different ways to rewrite the coordinates of point A. 210o 330o 240o 300o 270o

Relations between Polar and Rectangular Coordinates

Find the rectangular coordinates of the points with the following polar coordinates:

Find the polar coordinates of the points with the following rectangular coordinates:

Convert each rectangular equation to a polar equation. Replace x with r cos and y with r sin . Simplify and solve for r.

Convert the polar equation to rectangular equations. We will need the following equations. B. A.

Convert the polar equation to rectangular equations. We will need the following equations. D. C.

Convert the polar equation to rectangular equations. We will need the following equations.