Polar & exponential form re^{iθ}
Every non-zero complex number can be written as r(cos θ + i sin θ), abbreviated to re^(iθ) by Euler’s formula. The exponential form is not just shorthand: it makes multiplication, powers and roots almost trivial, because the ordinary index laws then do all the work on the moduli and arguments.
✓ Unlimited questions · marked criterion by criterion · no card needed
Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Compute r and θ from a and br = √(a²+b²) and θ = arg z with a quadrant check.
- Write the polar formz = r(cos θ + i sin θ) = re^(iθ). Both notations describe the same number.
- Convert back with the trig valuesa = r cos θ and b = r sin θ.
- Prefer exponential form for multiplicationIt turns products into sums of arguments and products of moduli.
Worked example
Write z = 1 + i in exponential form, and convert 4e^(iπ/3) to a + bi.
- For 1 + i: r = √2 and the point is in Q1 with arctan(1) = π/4.
- So 1 + i = √2 e^(iπ/4).
- For 4e^(iπ/3): a = 4cos(π/3) = 4 × 1/2 = 2.
- b = 4sin(π/3) = 4 × √3/2 = 2√3.
Answer. 1 + i = √2 e^(iπ/4), and 4e^(iπ/3) = 2 + 2√3 i.
Where marks get dropped
These are the specific errors that cost credit on polar & exponential form re^{iθ} questions — QED's rubric penalises each of them separately.
- Writing r as negative. The modulus is non-negative by definition; a negative sign belongs in the argument, since −re^(iθ) = re^(i(θ+π)).
- Mixing degrees into the exponential form. e^(iθ) requires θ in radians.
- Forgetting the quadrant when converting to polar, which puts the number in the wrong half of the plane.
Practise this until it is automatic
Unlimited fresh questions
QED generates new polar & exponential form re^{iθ} problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.
Marked like an examiner
Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.
Answer in real notation
A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.
Saved to your library
Every question you generate is kept and re-takeable as a timed exam, and your Complex Numbers mastery is tracked so you know when this is exam-ready.
Polar & exponential form re^{iθ} — frequently asked questions
Why is e^(iθ) = cos θ + i sin θ?
Compare the Maclaurin series: substituting iθ into the series for eˣ and separating real and imaginary parts gives exactly the series for cosine and sine.
What is e^(iπ) + 1 = 0?
Euler’s identity, linking e, i, π, 1 and 0. It is the case θ = π of Euler’s formula, since cos π = −1 and sin π = 0.
When should I use polar rather than Cartesian?
Polar for multiplication, division, powers and roots; Cartesian for addition and subtraction. Converting is usually worth it for anything involving powers.
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
Ready to make polar & exponential form re^{iθ} exam-proof?
Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.
Start practising free →