Goals

• Represent complex numbers as points in the complex plane.
• Convert between rectangular and trigonometric forms of complex numbers.
• Multiply and divide complex numbers in trigonometric form.
• Use DeMoivre’s Theorem to compute the powers and roots of complex numbers.

Complex Numbers and the Complex Plane

• A complex number, usually denoted by z, is a number that has two parts: a real part and an imaginary part.
  • z = x + yi
  • The x is the real part of the complex number and the yi is the imaginary part of the complex number.
• The imaginary unit is defined as i = \sqrt{-1}, where i^2 = -1.
• Complex numbers can be graphed similarly to points on the xy-plane.
  • Example: Representing points (2, -4), (-2, 0), and (5, 3) on both the Cartesian and Complex Planes.

Properties of Complex Numbers

• The modulus or magnitude of a complex number z = x + yi is the distance from the origin to the point (x, y) on the complex plane. It is denoted by |z| and can be calculated using:
  • |z| = \sqrt{x^2 + y^2}
• The conjugate of the complex number z = x + yi is given by:
  • \bar{z} = x - yi
  • The modulus can also be expressed as:
  • |z| = \sqrt{z ar{z}}
  • z \cdot \bar{z} = x^2 + y^2
• Examples: Find the magnitude and conjugates of the following:
  • For 1 - 2i
  • For 3 + 7i
  • For -10i

Operations with Complex Numbers

• Operations such as addition, subtraction, multiplication, and division can be performed just like with regular real numbers.
• Consider z1 = a + bi and z2 = c + di:
  • Addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + i(b + d)
  • Subtraction: z1 - z2 = (a + bi) - (c + di) = (a - c) + i(b - d)
  • Multiplication:
    z1 \cdot z2 = (a + bi)(c + di) = ac + adi + bci + bdi^2
    = (ac - bd) + i(ad + bc)
  • Division:
    \frac{z1}{z2} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2}
• Examples: Find the sum, difference, product, and quotient for:
  • z1 = -1 + 5i, z2 = -4i
  • z1 = \frac{1}{2} + \frac{1}{3}i, z2 = 5 - 2i

Converting Complex Numbers between Rectangular and Polar Form

• Using polar coordinates, a complex number can be converted into its polar form using:
  • x = r \cos \theta and y = r \sin \theta
• Definition: If r \geq 0 and 0 \leq \theta < 2\pi, the complex number z = x + yi can be expressed in polar form as:
  • z = (r \cos \theta) + (r \sin \theta)i = r (\cos \theta + i \sin \theta)
  • The angle \theta is known as the argument of z.
• Note that since r = \sqrt{x^2 + y^2, we also have |z| = r.
• Euler’s Formula: For any real number \theta,
  • e^{i\theta} = \cos \theta + i \sin \theta
  • Hence, the polar form in exponential notation is:
  • re^{i\theta} = r(\cos \theta + i \sin \theta)
• Examples: Express the following complex numbers in polar form:
  • For z = \sqrt{3} - i
  • For z = 4 - 4i
  • For z = -2 + 3i
  • For z = 9\sqrt{3} + 9i

Product and Quotients of Complex Numbers in Polar Form

• The polar form simplifies multiplication and division of complex numbers.
• Theorem: Let z1 = r1(\cos \theta1 + i \sin \theta1) and z2 = r2(\cos \theta2 + i \sin \theta2):
  • Product:
    z1 \cdot z2 = r1r2 [\cos(\theta1 + \theta2) + i \sin(\theta1 + \theta2)]
  • Also, z1 \cdot z2 = r1r2e^{i(\theta1+\theta2)}
  • Quotient:
  • \frac{z1}{z2} = \frac{r1}{r2} [\cos(\theta1 - \theta2) + i \sin(\theta1 - \theta2)]
  • Also, \frac{z1}{z2} = \frac{r1}{r2} e^{i(\theta1-\theta2)} if z_2 \neq 0.
• Examples: If z = 2(\cos \frac{2\pi}{9} + i \sin \frac{2\pi}{9}) and w = 4(\cos \frac{\pi}{9} + i \sin \frac{\pi}{9}), find the product and quotient and express in exponential form.

DeMoivre’s Theorem

• This theorem enables the calculation of powers of complex numbers in polar form.
• DeMoivre’s Theorem: If z = r(\cos \theta + i \sin \theta), then:
  • z^n = r^n [\cos(n\theta) + i \sin(n\theta)] = r^ne^{i(n\theta)} where n \geq 1 is a positive integer.
• Examples: Write the following in polar and exponential form:
  • (2 + 3i)^3
  • (1 + i)^5
  • (1 - 5i)^8