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10.3 Polar Form of Complex Numbers

10.3 Polar Form of Complex Numbers. We have explored complex numbers as solutions. Now we connect to both the rectangular and polar planes. Every complex number can be represented in the form a + bi. horizontal axis = real axis vertical axis = imaginary axis.

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10.3 Polar Form of Complex Numbers

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  1. 10.3 Polar Form of Complex Numbers

  2. We have explored complex numbers as solutions. Now we connect to both the rectangular and polar planes. Every complex number can be represented in the form a + bi horizontal axis = real axis vertical axis = imaginary axis real part corresponds to x-axis imaginary part corresponds to y-axis P a + bi  (a, b)  (r, θ) argument θ r a = rcosθ & b = rsinθ b a + bi = rcosθ + irsinθ = r(cosθ + isinθ) θ a / absolute value of a + bi modulus

  3. Ex 1) Graph each complex number and find the modulus. A) 2 + 3i B) –2i (0, –2) (2, 3) modulus: modulus: A B

  4. The expression r(cosθ + isinθ) is often abbreviated rcisθ. This is the polar form of the complex number. (a + bi is the rectangular form) We need to be able to convert between the forms. Ex 2) Express the complex number in rectangular form. A) B)

  5. Ex 3) Express each complex number in polar form. Use θ [0, 2π) A) z = 2 – 2i in QIV On your own B) in QII C) • remember: cos (–π) = cosπ • sin (–π) = –sin π needs to be positive!

  6. Ex 4) Describe the polar form of real number a. a = a + 0i this means values on the x-axis for positive x-axis values, a needs to be positive for negative x-axis values, a needs to be negative (and y-value needs to be 0) so where θ = 0 if a > 0 & θ = π if a < 0, plus if a = 0, θ can be anything

  7. Homework #1004 Pg 506 #1-45 odd, 46-50 HW hint: If (conjugate)

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