Complex Number Exercises

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Questions and Answers

Convert the equation $arg(z - 1) = -\frac{\pi}{4}$ into Cartesian form.

The equation in Cartesian form is $x + y = 1$, where $z=x+iy$.

Convert the equation $z\bar{z} = 4e^{i\theta}$ into cartesian form.

The equation in cartesian form is $x^2 + y^2 = 4e^{i\theta}$, where $z = x + iy$.

Determine the Cartesian form of the inequality $-\frac{\pi}{3} \leq arg(z - 4) \leq \frac{\pi}{3}$.

The Cartesian form involves inequalities related to the slope of the line connecting $(x, y)$ and $(4, 0)$. Specifically, $-\sqrt{3} \leq \frac{y}{x-4} \leq \sqrt{3}$.

Express the inequality $0 \leq arg(\frac{z-4}{1+i}) \leq \frac{\pi}{6}$ in Cartesian coordinates.

The inequality can be expressed in terms of the arguments of $z-4$ and $1+i$. Recognizing that $arg(1+i) = \frac{\pi}{4}$, the inequality transforms into $-\frac{\pi}{4} \leq arg(z-4) \leq \frac{5\pi}{12}$ Signup and view all the answers

Find the Cartesian representation of the equation $arg(\frac{1-iz}{1+iz}) = \frac{\pi}{4}$, where $z \neq i$.

After applying properties of arguments and complex numbers, the equation simplifies to a linear relationship between $x$ and $y$, specifically $x^2 + y^2 + 2x - 1 = 0$. Signup and view all the answers

Express $\frac{1}{2}arg(z - i) = \frac{\pi}{3} - \frac{1}{2}arg(z + i)$ in Cartesian form.

Using properties of arguments and trigonometric identities, show that the relationship between the real and imaginary parts x and y can be described by $x^2 + (y - \sqrt{3})^2 =4 $. Signup and view all the answers

A particle is at $x = 2 + 3i$ relative to its mean position, and its maximum displacement is $x_{max} = 1 + 4i$. Determine the angle at time $t = 0$, assuming simple harmonic motion, and find the position of the particle at that moment.

The angle $\theta$ when $t = 0$ is $arctan(\frac{3}{2})$, and the position of the particle can be further defined using the maximum displacement information. Signup and view all the answers

Given $x = 2 + 3i$ and $x_{max} = 1 + 4i$ for a particle's position, calculate the frequency of oscillation when $t = 2$.

To calculate the frequency, you will need to determine the angular frequency $\omega$ from the relationships between $x$, $x_{max}$, and $t$. Given the limited information, further context is required to accurately compute the frequency, often requiring the spring constant or mass. Signup and view all the answers

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Flashcards

Polar Form of Complex Numbers

Express a complex number in the form r(cos θ + i sin θ), where r is the magnitude and θ is the argument.

Rectangular Form

Convert a complex number from its polar form back to the form a + bi, where a and b are real numbers.

Cartesian Form

Convert equations involving complex numbers into equations using only real numbers (x, y).

Argument of a Complex Number

The angle between the positive real axis and the line connecting the origin to the point representing the complex number in the complex plane.

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Impedance (Z)

Opposition to alternating current (AC) flow, including both resistance and reactance. Z = E/I

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Study Notes

The following are exercises for complex numbers and related concepts.

Complex Numbers in Polar Form

Convert complex numbers to polar form.
Examples include 2 + i2√3, 3 - i√3, -2 - i2, and (i - 1)/(cos(π/3) + i sin(π/3)).

Complex Numbers in Rectangular Form

Convert complex numbers to rectangular form.
Examples include (cos(π/6) + i sin(π/6))(cos(π/3) + i sin(π/3)) and (cos(π/6) - i sin(π/6)) / (2(cos(π/3) + i sin(π/3))).

Complex Number Product Identity

If (x₁ + iy₁)(x₂ + iy₂)(x₃ + iy₃)...(xₙ + iyₙ) = a + ib, then (x₁² + y₁²)(x₂² + y₂²)(x₃² + y₃²)...(xₙ² + yₙ²) = a² + b².
Σₖ₌₁ⁿ tan⁻¹(yₖ/xₖ) = tan⁻¹(b/a) + 2kπ, where k ∈ Z.

Complex Equation

If (1 + z) / (1 - z) = cos 2θ + i sin 2θ, show that z = i tan θ.

Trigonometric Identity

If cos α + cos β + cos γ = sin α + sin β + sin γ = 0, then:
- cos 3α + cos 3β + cos 3γ = 3cos(α + β + γ).
- sin 3α + sin 3β + sin 3γ = 3sin(α + β + γ).

Complex to Algebraic Form

Write complex numbers in algebraic form.
- √2(cos 315° + i sin 315°)
- 5(cos 210° + i sin 210°)
- 2(cos(3π/2) + i sin(3π/2))
- 4(cos(5π/6) + i sin(5π/6))
- 2(cos(π/6) + i sin(π/6))
- cos 135° + i sin 135°
- 10(cos 50° + i sin 50°)
- √2(cos(3π/4) + i sin(3π/4))
- 4(cos(2π/3) + i sin(2π/3))
- 7√2(cos(5π/4) + i sin(5π/4))
- 10√2(cos(7π/6) + i sin(7π/6))
- 2(cos(5π/3) + i sin(5π/3))
- (1/√2)(cos(π/4) + i sin(π/4))
- 7(cos 180° + i sin 180°)
- 2e^(i(5π/4))
- 3e^(i(π/2))
- 5e^(i(π/3))

Cartesian Form Conversion

Convert equations and inequations to Cartesian form involving complex numbers.
- arg(z - 1) = -π/4
- z z̄ = 4|e^(iθ)|
- -π/3 ≤ arg(z - 4) ≤ π/3
- 0 ≤ arg((z - i) / (1 + i)) ≤ π/6
- arg((1 - z) / (1 + z)) = π/4 ; z ≠ i
- (1/2)arg(z - i) = (π/3) - (1/2)arg(z + i)`

Particle Position Calculation

Calculate the position of a particle from its mean position when the amplitude is 0.004mm and the angle is π/4, π/3, or π/6.

Particle Position and Frequency

Analyze particle position with respect to time.
- Given a particle at x = 2 + 3i and x_max = 1 + 4i, calculate the angle at t = 0 and the particle's position.
- If x = 2 + 3i and x_max = 1 + 4i, calculate the frequency when t = 2.

Impedance Calculation

Compute impedance Z for given voltage (E) and current (I) values.
- E = (-50 + 100i) volts and I = (-6 - 2i) amps.
- E = (100 + 10i) volts and I = (-8 + 3i) amps.

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Description

Exercises coverting complex numbers between polar and rectangular forms. Includes complex number product identity and trigonometric identities. Problems extend to algebraic form.