6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem Objectives Fine absolute value of a complex # Write complex # in polar form Convert a complex # from polar to rectangular form Plot complex numbers in the complex plane Find products & quotients of complex numbers in polar form Find powers of complex # in polar form Find roots of complex # in polar form

Complex number = z = a + bi a is a real number bi is an imaginary number Together, the sum, a+bi is a COMPLEX # Complex plane has a real axis (horizontal) and an imaginary axis (vertical) 2 – 5i is found in the 4th quadrant of the complex plane (horiz = 2, vert = -5) Absolute value of 2 – 5i refers to the distance this pt. is from the origin (continued)

Find the absolute value Since the horizontal component = 2 and vertical = -5, we can consider the distance to that point as the same as the length of the hypotenuse of a right triangle with those respective legs

z = 2 + 3i Plot the above point and find the absolute value.

z = -2 + -3i Plot the above point and find the absolute value.

Expressing complex numbers in polar form z = a + bi

Express z = -5 + 3i in complex form

Express z = -1 + i in complex form

Express z = -6 + 3i in complex form

Product & Quotient of complex numbers

Product & Quotient of complex numbers

Product & Quotient of complex numbers

Product & Quotient of complex numbers

Product & Quotient of complex numbers

Multiplying complex numbers together leads to raising a complex number to a given power If r is multiplied by itself n times, it creates If the angle, theta, is added to itself n times, it creates the new angle, (n times theta) THUS,

Multiplying complex numbers together leads to raising a complex number to a given power Find

Multiplying complex numbers together leads to raising a complex number to a given power Find

Multiplying complex numbers together leads to raising a complex number to a given power Find

Multiplying complex numbers together leads to raising a complex number to a given power Find

Taking a root (DeMoivre’s Theorem) Radian Degree Where k = 0,1,2,…n-1

If you’re working with degrees add 360/n to the angle measure to complete the circle. Example: Find the 6th roots of z= -2 + 2i Express in polar form, find the 1st root, then add 60 degrees successively to find the other 5 roots.

Example: Find the 4th roots of z= -1 + 1i

Example: Find the 4th roots of

Example: Find the cube roots of 8, in degrees.

Example: Find the cube roots of 27, in radians.