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AQA A-Level Further Maths > Complex Numbers > Flashcards

Flashcards in Complex Numbers Deck (5)
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1

Dividing by complex numbers

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

2

Solving equations with complex numbers

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

3

Complex roots of polynomials

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

4

Argand diagrams

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.

5

Modulus - Argument form

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))