Questions: Complex Exponential Form and Euler's Formula
3 questions to test your understanding
Score: 0 / 3
Question 1 Multiple Choice
Using Euler's formula, what is e^(iπ/2)?
A-1
Bi
C-i
D1
e^(iπ/2) = cos(π/2) + i sin(π/2) = 0 + i(1) = i. Geometrically, multiplying by e^(iπ/2) rotates a point 90° counterclockwise on the unit circle, mapping 1 to i. The famous identity e^(iπ) = -1 corresponds to a 180° rotation, landing at -1 on the real axis.
Question 2 True / False
e^(iπ) and e^(i·3π) represent the same complex number.
TTrue
FFalse
Answer: True
Since 3π = π + 2π, we have e^(i·3π) = e^(i(π + 2π)) = e^(iπ) · e^(i·2π) = e^(iπ) · 1 = e^(iπ) = -1. The 2π periodicity of the complex exponential means adding any multiple of 2π to the argument returns the same point on the unit circle. This periodicity is a key difference from the real exponential, which is injective.
Question 3 Short Answer
Two complex numbers z₁ = 2e^(iπ/6) and z₂ = 5e^(iπ/4) are multiplied. What are the modulus and argument of z₁z₂, without converting to rectangular form?
Think about your answer, then reveal below.
Model answer: The modulus of z₁z₂ is 2 · 5 = 10, and the argument is π/6 + π/4 = 5π/12.
In exponential form, multiplication is: (r₁ e^(iθ₁))(r₂ e^(iθ₂)) = r₁r₂ e^(i(θ₁+θ₂)). Moduli multiply and arguments add. This is the primary computational advantage of exponential form — multiplication that would require expanding (a+bi)(c+di) in rectangular form reduces to multiplying two real numbers and adding two angles.