Complex Numbers

Question 1

Dividing by complex numbers

Answer 1

Write the division as a fraction
Multiply the numerator and denominator by the conjugate of the denominator
Separate into real and imaginary parts and simplify

Question 2

Solving equations with complex numbers

Answer 2

Separate and equate real and imaginary parts
Solve the equations simultaneously (without the i’s)
Substitute back into equation
Combine the x and y to get z = x + yi

Question 3

Complex roots of polynomials

Answer 3

If a polynomial has all real coefficients, its complex roots come in pairs of conjugates.

Question 4

Argand diagrams

Answer 4

Argand diagrams are used to represent complex numbers as points. They are similar to the Cartesian coordinate system, except the x axis is the real axis and the y axis is the imaginary axis. To add or subtract on an Argand diagram, you represent the complex numbers as vectors and add/subtract the vectors. The modulus of a complex number z = x + yi is √(x² + y²) (the length of the vector). The argument is the angle between the positive real axis and the vector.

Question 5

Modulus - Argument form

Answer 5

z = r(cosθ + i sinθ)
z₁z₂ = r₁r₂(cos(θ₁+θ₂) + i sin(θ₁+θ₂))
z₁ / z₂ = r₁/r₂(cos(θ₁-θ₂) + i sin(θ₁-θ₂))