Multiplying & dividing in polar form
In polar form multiplication becomes beautifully simple: moduli multiply and arguments add, so (r₁e^(iθ₁))(r₂e^(iθ₂)) = r₁r₂e^(i(θ₁+θ₂)). Division divides the moduli and subtracts the arguments. These are just the index laws, and they turn geometrically opaque Cartesian multiplication into a rotation-and-scaling statement.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Convert both numbers to polar formGet r and θ for each, with quadrant checks.
- Multiply: r₁r₂ and θ₁ + θ₂Moduli multiply, arguments add.
- Divide: r₁/r₂ and θ₁ − θ₂Moduli divide, arguments subtract.
- Bring the argument back into rangeIf the result leaves (−π, π], add or subtract 2π.
Worked example
With z = 2e^(iπ/3) and w = 3e^(iπ/6), find zw and z/w.
- Product modulus: 2 × 3 = 6. Product argument: π/3 + π/6 = π/2.
- So zw = 6e^(iπ/2) = 6i.
- Quotient modulus: 2/3. Quotient argument: π/3 − π/6 = π/6.
- So z/w = (2/3)e^(iπ/6) = (2/3)(√3/2 + i/2).
Answer. zw = 6i and z/w = (2/3)e^(iπ/6) = √3/3 + i/3.
Where marks get dropped
These are the specific errors that cost credit on multiplying & dividing in polar form questions — QED's rubric penalises each of them separately.
- Adding the moduli and multiplying the arguments — exactly backwards.
- Leaving the final argument outside the principal range when a principal value is required.
- Converting to Cartesian to multiply when polar is available. The polar route is far shorter and less error-prone.
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Multiplying & dividing in polar form — frequently asked questions
Why do arguments add?
Because e^(iθ₁)·e^(iθ₂) = e^(i(θ₁+θ₂)) by the index law. Geometrically, multiplying by a complex number rotates by its argument.
What does multiplying by i do?
It rotates by π/2 anticlockwise, since i = e^(iπ/2) with modulus 1. Multiplying by i four times returns you to the start.
How does this lead to De Moivre?
Multiplying z by itself n times multiplies the modulus n times and adds the argument n times, giving (re^(iθ))ⁿ = rⁿe^(inθ).
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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