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Mathematics LibreTexts

2.E: Exercises for Chapter 2

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    Calculational Exercises

    1. Express the following complex numbers in the form \(x + yi\) for \(x, y \in \mathbb{R}:\)

    (a) \((2 + 3i) + (4 + i)\)

    (b) \((2 + 3i)^2 (4 + i)\)

    (c) \(\frac{2+3i}{4+i}\)

    (d) \(\frac{1}{i}+\frac{3}{1+i}\)

    (e) \((−i)^{−1}\)

    (f) \((−1 + i \sqrt{3})^3\)

    2. Compute the real and imaginary parts of the following expressions, where \(z\) is the
    complex number \(x + yi\) and \(x, y \in \mathbb{R}:\)

    (a) \(\frac{1}{z^2}\)

    (b) \(\frac{1}{3z+2}\)

    (c) \(\frac{z+1}{2z-5}\)

    (d) \(z^3\)

    3. Find \(r > 0\) and \(\theta \in [0, 2\pi) \) such that \((1 − i)/ 2 = re^{i \theta}.\)

    4. Solve the following equations for \(z\) a complex number:
    (a) \(z^5 − 2 = 0\)
    (b) \(z^4 + i = 0\)
    (c) \(z^6 + 8 = 0\)
    (d) \(z^3 − 4i = 0\)

    5. Calculate the
    (a) complex conjugate of the fraction \((3 + 8i)^4 /(1 + i)^10 .\)
    (b) complex conjugate of the fraction \((8 − 2i)^10 /(4 + 6i)^5 .\)
    (c) complex modulus of the fraction \(i(2 + 3i)(5 − 2i)/(−2 − i).\)
    (d) complex modulus of the fraction \((2 − 3i)^2 /(8 + 6i)^2 .\)

    6. Compute the real and imaginary parts:
    (a) \(e^{2+i}\)
    (b) \(sin(1 + i)\)
    (c) \(e^{3−i}\)
    (d) \(cos(2 + 3i)\)

    7. Compute the real and imaginary part of \(e^{e^{z}}\) for \(z \in \mathbb{C}.\)

    Proof-Writing Exercises

    1. Let \(a \in \mathbb{R}\) and \(z, w \in \mathbb{C}.\) Prove that
    (a) \( Re(az) = aRe(z)\) and \( Im(az) = aIm(z).\)
    (b) \( Re(z + w) = Re(z) + Re(w)\) and \( Im(z + w) = Im(z) + Im(w).\)

    2. Let \(z \in \mathbb{C}.\) Prove that \( Im(z) = 0\) if and only if \( Re(z) = z.\)

    3. Let \(z, w \in \mathbb{C}.\) Prove the parallelogram law \(|z − w|^2 + |z + w|^2 = 2(|z|^2 + |w|^2).\)

    4. Let \(z, w \in \mathbb{C}\) with \(\bar{z}w \neq 1\) such that either \(|z| = 1\) or \(|w| = 1.\) Prove that \( \left| \frac{z−w}{1 − \bar{z}w} \right| =1. \)

    5. For an angle \(\theta \in [0, 2\pi),\) find the linear map \(f_\theta : \mathbb{R}^2 \rightarrow \mathbb{R}^2\), which describes the rotation by the angle \(\theta\) in the counterclockwise direction.

    Hint: For a given angle \(\theta\), find \(a, b, c, d \in \mathbb{R}\) such that \(f_\theta (x_1 , x_2 ) = (ax_1 +bx_2 , cx_1 +dx_2 ).\)