Euler's form (exponential form) of a complex number z is given by:
A
z = r e^(iθ)
B
z = r(cos θ + i sin θ)
C
z = a + ib
D
z = r(cos θ - i sin θ)
Analysis & Theory
Euler's formula: e^(iθ) = cos θ + i sin θ, so z = r e^(iθ).
In Euler's form z = r e^(iθ), r represents:
A
Modulus of z
B
Argument of z
C
Real part of z
D
Imaginary part of z
Analysis & Theory
r = |z| is the modulus of the complex number.
In Euler's form z = r e^(iθ), θ represents:
A
Argument of z
B
Modulus of z
C
Real part of z
D
Imaginary part of z
Analysis & Theory
θ = arg(z) is the angle made by the line joining origin to z with the positive x-axis.
Euler's form of z = 1 + i is:
A
√2 e^(i π/4)
B
√2 e^(i π/2)
C
1 e^(i π/4)
D
2 e^(i π/4)
Analysis & Theory
z = 1 + i ⇒ r = √2, θ = π/4, so Euler form: √2 e^(i π/4).
The rectangular form z = a + ib can be written in Euler's form as:
A
z = r e^(iθ), where r = √(a² + b²), θ = arctan(b/a)
B
z = r e^(iθ), where r = a + b, θ = arctan(b/a)
C
z = r e^(iθ), where r = √(a² + b²), θ = a/b
D
z = r e^(iθ), where r = a² + b², θ = arctan(a/b)
Analysis & Theory
Convert rectangular to Euler form: r = |z| = √(a² + b²), θ = arctan(b/a).
Multiplication of two complex numbers z₁ = r₁ e^(iθ₁) and z₂ = r₂ e^(iθ₂) in Euler form is:
A
z₁z₂ = r₁r₂ e^(i(θ₁ + θ₂))
B
z₁z₂ = r₁r₂ e^(i(θ₁ - θ₂))
C
z₁z₂ = r₁/r₂ e^(i(θ₁ + θ₂))
D
z₁z₂ = r₁r₂ e^(iθ₁) + e^(iθ₂)
Analysis & Theory
Multiply moduli and add arguments in exponential form.
Division of two complex numbers z₁ = r₁ e^(iθ₁) and z₂ = r₂ e^(iθ₂) in Euler form is:
A
z₁/z₂ = (r₁/r₂) e^(i(θ₁ - θ₂))
B
z₁/z₂ = (r₁/r₂) e^(i(θ₁ + θ₂))
C
z₁/z₂ = r₁ r₂ e^(i(θ₁ - θ₂))
D
z₁/z₂ = (r₁ - r₂) e^(i(θ₁ - θ₂))
Analysis & Theory
Divide moduli and subtract arguments in exponential form.
Euler's form z = r e^(iθ) is equivalent to which trigonometric form?
A
z = r(cos θ + i sin θ)
B
z = a + ib
C
z = r(cos θ - i sin θ)
D
z = r(cos θ + sin θ)
Analysis & Theory
By Euler's formula: e^(iθ) = cos θ + i sin θ.
If z = 2 e^(i π/3), then z in rectangular form a + ib is:
A
1 + √3 i
B
2 + √3 i
C
1 - √3 i
D
√3 + i
Analysis & Theory
z = 2 e^(i π/3) = 2(cos π/3 + i sin π/3) = 2(1/2 + i√3/2) = 1 + √3 i.
Euler's form of a purely imaginary number z = ib is:
A
|b| e^(i π/2) if b > 0, |b| e^(-i π/2) if b < 0
B
b e^(i 0)
C
ib e^(i π/4)
D
b e^(i π)
Analysis & Theory
For z = ib, r = |b|, θ = π/2 if b > 0, θ = -π/2 if b < 0.