The Geometry of Complex Numbers Section 9.1

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Complex Numbers  * z = x + yi  * C represents the set of complex numbers Real part (RE) Imaginary part (IM)

Complex Plane

Example 1  Find the polar coordinates [r, θ] of i with r ≥ 0 and 360˚≤ θ ≤ 720˚  Rectangular coordinate: (-8, 11) 2 nd quadrant! r = or θ = ˚ + 180˚+360˚ [13.6, ˚]

Trigonometric Form  If (x, y) = [r, θ] r is called the modulus, will ALWAYS be positive θ is called the argument  Then, x = rcos θ and y = rsin θ  Substituting back in…  x + yi  rcos θ + (rsin θ)i r(cos θ + i sin θ)

Example 2  Write the complex number 6 + 6i in trigonometric form. (6,6)  1 st quadrant r = or 8.49 θ = π/4 [8.49, π/4]  8.49 (cos π/4 + i sin π/4)

Multiplying Complex Numbers  If z = [r, θ] and w = [s, φ] then zw = [rs, θ + φ] Complex Conjugates: z = a + bi then z = a – bi

Example 3  Prove: complex numbers z, z – z is an imaginary number.  Let a + bi = z  Then, z = a – bi a + bi – (a – bi) a + bi – a + bi 2bi  Imaginary!

Example 4  Perform the indicated complex number operation and express the answer in the same form as the given numbers.

Homework Pages 526 – – 9,