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AQA A-Level Further Maths > Complex Numbers > Flashcards

Flashcards in Complex Numbers Deck (5)
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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


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


Complex roots of polynomials

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


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.


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))