Same r and θ, new shorthand: There's a third way to write a complex number: exponential (Euler) form, z = r eiθ.
It uses the same modulus r and argument θ as polar form, because eiθ = cosθ + i sinθ (Euler's formula).
IB-style question — into exponential form
Write z = 4 + 4i in exponential form.
Step by step
- Modulus.
- Argument (quadrant 1).
- Drop r and θ into r eiθ.
Final answer
4√2 eiπ/4.
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Just add the exponents: Because it's an exponential, multiplying uses the index law eiθ₁ × eiθ₂ = ei(θ₁+θ₂) — exactly 'multiply moduli, add arguments' again, but now it looks like ordinary powers.
IB-style question — multiply in exponential form
Find (2 eiπ/4)(3 eiπ/12), giving your answer in exponential form.
Step by step
- Multiply the numbers in front; add the exponents.
- Add the angles.
- Combine.
Final answer
6 eiπ/3.
The most famous equation in maths: Put θ = π into eiθ = cosθ + i sinθ: eiπ = cos π + i sin π = −1. So eiπ + 1 = 0 — Euler's identity, linking e, i, π, 1 and 0.