12.A.E: Exercises for Complex Numbers

Exercise \(\PageIndex{1}\)

Solve each of the following for the real number \(x\).

1. \(x-4i = (2-i)^2\)
2. \((2+xi)(3-2i) = 12+5i\)
3. \((2+xi)^2=4\)
4. \((2+xi)(2-xi)=5\)

Answer

2. \(x = 3\)
4. \(x = \pm 1\)

Exercise \(\PageIndex{2}\)

Convert each of the following to the form \(a + bi\).

2. \((2-3i)-2(2-3i)+9\)
3. \((3-2i)(1+i)+|3+4i|\)
4. \(\frac{1+i}{2-3i} + \frac{1-i}{-2+3i}\)
5. \(\frac{3-2i}{1-i} + \frac{3-7i}{2-3i}\)
6. \(i^{131}\) \((2 - i)^{3}\) \((1 + i)^{4}\)
7. \((1 - i)^{2}(2 + i)^{2}\)
8. \(\frac{3\sqrt{3}-i}{\sqrt{3}+i} + \frac{\sqrt{3}+7i}{\sqrt{3}-i}\)

Answer

2. \(10 + i\)
4. \(\frac{11}{26} + \frac{23}{26}i\)
6. \(2 - 11i\)
8. \(8 - 6i\)

Exercise \(\PageIndex{3}\)

In each case, find the complex number \(z\).

1. \(iz - (1 + i)^{2} = 3 - i\)
2. \((i + z) - 3i(2 - z) = iz + 1\)
3. \(z^{2} = -i\)
4. \(z^{2} = 3 - 4i\)
5. \(z(1+i) = \overline{z} + (3+2i)\)
6. \(z(2-i) = (\overline{z}+1)(1+i)\)

Answer

2. \(\frac{11}{5} + \frac{3}{5}i\)
4. \(\pm(2 - i)\)
6. \(1 + i\)

Exercise \(\PageIndex{4}\)

In each case, find the roots of the real quadratic equation.

1. \(x^{2} - 2x + 3 = 0\)
2. \(x^{2} - x + 1 = 0\)
3. \(3x^{2} - 4x + 2 = 0\)
4. \(2x^{2} - 5x + 2 = 0\)

Answer

2. \(\frac{1}{2} \pm \frac{\sqrt{3}}{2}i\)
4. \(2\), \(\frac{1}{2}\)

Exercise \(\PageIndex{5}\)

Find all numbers \(x\) in each case.

1. \(x^{3} = 8\)
2. \(x^{3} = -8\)
3. \(x^{4} = 16\)
4. \(x^{4} = 64\)

Answer

1. \(-2\), \(1 \pm \sqrt{3}i\)
4. \(\pm 2\sqrt{2}\), \(\pm 2\sqrt{i}\)

Exercise \(\PageIndex{6}\)

In each case, find a real quadratic with \(u\) as a root, and find the other root.

1. \(u = 1 + i\)
2. \(u = 2 - 3i\)
3. \(u = -i\)
4. \(u = 3 - 4i\)

Answer

2. \(x^{2} - 4x + 13\); \(2 + 3i\)
4. \(x^{2} - 6x + 25\); \(3 + 4i\)

Exercise \(\PageIndex{7}\)

Find the roots of \(x^{2} - 2\cos \theta x + 1 = 0\), \(\theta\) any angle.

Exercise \(\PageIndex{8}\)

Find a real polynomial of degree \(4\) with \(2 - i\) and \(3 - 2i\) as roots.

Answer

\(x^{4} - 10x^{3} + 42x^{2} - 82x + 65\)

Exercise \(\PageIndex{9}\)

Let \(\text{re }z\) and \(im \;z\) denote, respectively, the real and imaginary parts of \(z\). Show that:

\(im \;(iz) = \text{re }z\) \(\text{re}(iz) = -im \;z\) \(z + \overline{z} = 2 \text{re}z\) \(z - \overline{z} = 2i im \; z\) \(\text{re}(z + w) = \text{re }z + \text{re }w\), and \(\text{re}(tz) = t \cdot \text{re }z\) if \(t\) is real \(im \;(z + w) = im \;z + im \;w\), and \(im \;(tz) = t \cdot im \;z\) if \(t\) is real

Exercise \(\PageIndex{10}\)

In each case, show that \(u\) is a root of the quadratic equation, and find the other root.

1. \(x^{2} - 3ix + (-3 + i) = 0\);
2. \(u = 1 + i\) \(x^{2} + ix - (4 - 2i) = 0\)
3. \(u = -2\) \(x^{2} - (3 - 2i)x + (5 - i) = 0\)
4. \(u = 2 - 3i\) \(x^{2} + 3(1 - i)x - 5i = 0\)
5. \(u = -2 + i\)

Answer

1. \((-2)^{2} + 2i - (4 - 2i) = 0\); \(2 - i\)
4. \((-2 + i)^{2} + 3(1 - i)(-1 + 2i) - 5i = 0\); \(-1 + 2i\)

Exercise \(\PageIndex{11}\)

Find the roots of each of the following complex quadratic equations.

1. \(x^{2} + 2x + (1 + i) = 0\)
2. \(x^{2} - x + (1 - i) = 0\)
3. \(x^{2} - (2 - i)x + (3 - i) = 0\)
4. \(x^{2} - 3(1 - i)x - 5i = 0\)

Answer

2. \(-i\), \(1 + i\)
4. \(2 - i\), \(1 - 2i\)

Exercise \(\PageIndex{12}\)

In each case, describe the graph of the equation (where \(z\) denotes a complex number).

1. \(|z| = 1\)
2. \(|z - 1| = 2\)
3. \(z = i \overline{z}\)
4. \(z = -\overline{z}\)
5. \(z = |z|\)
6. \(im \;z = m \cdot \text{re }z\), \(m\) a real number

Answer

2. Circle, centre at \(1\), radius \(2\)
4. Imaginary axis
6. Line \(y = mx\)

Exercise \(\PageIndex{13}\)

1. Verify \(|zw| = |z||w|\) directly for \(z = a + bi\) and \(w = c + di\).
2. Deduce (a) from properties C2 and C6.

Exercise \(\PageIndex{14}\)

Prove that \(|z+w| = |z|^2 + |w|^2 + w\overline{z} + \overline{w}z\) for all complex numbers \(w\) and \(z\).

Exercise \(\PageIndex{15}\)

If \(zw\) is real and \(z \neq 0\), show that \(w = a \overline{z}\) for some real number \(a\).

Exercise \(\PageIndex{16}\)

If \(zw = \overline{z}v\) and \(z \neq 0\), show that \(w = uv\) for some \(u\) in \(\mathbb{C}\) with \(|u| = 1\).

Exercise \(\PageIndex{17}\)

Show that \((1 + i)^{n} + (1 - i)^{n}\) is real for all \(n\), using property C5.

Exercise \(\PageIndex{18}\)

Express each of the following in polar form (use the principal argument).

1. \(3 - 3i\)
2. \(-4i\)
3. \(-\sqrt{3} + i\)
4. \(-4 + 4\sqrt{3}i\)
5. \(-7i\)
6. \(-6 + 6i\)

Answer

1. \(4e^{-\pi i/2}\)
4. \(8e^{2\pi i/3}\)
6. \(6\sqrt{2}e^{3\pi i/4}\)

Exercise \(\PageIndex{19}\)

Express each of the following in the form \(a + bi\).

1. \(3e^{\pi i}\)
2. \(e^{7\pi i/3}\)
3. \(2e^{3 \pi i/4}\)
4. \(\sqrt{2}e^{-\pi i/4}\)
5. \(e^{5\pi i/4}\)
6. \(2\sqrt{3}e^{-2\pi i/6}\)

Answer

2. \(\frac{1}{2} + \frac{\sqrt{3}}{2} i\)
4. \(1 - i\)
6. \(\sqrt{3} - 3i\)

Exercise \(\PageIndex{20}\)

Express each of the following in the form \(a + bi\).

1. \((-1 + \sqrt{3}i)^2\)
2. \((1 + \sqrt{3}i)^{-4}\)
3. \((1 + i)^8\) \((1 - i)^{10}\)
4. \((1 - i)^{6}(\sqrt{3} + i)^{3}\)
5. \((\sqrt{3} - i)^{9}(2 - 2i)^{5}\)

Answer

2. \(-\frac{1}{32} + \frac{\sqrt{3}}{32}i\)
4. \(-32i\)
6. \(-2^{16}(1 + i)\)

Exercise \(\PageIndex{21}\)

Use De Moivre’s theorem to show that:

1. \(\cos 2\theta = \cos^{2} \theta - \sin^{2} \theta\)
2. \(\sin 2\theta = 2 \cos \theta \sin \theta\)

Answer

2. \(\cos 3\theta = \cos^{3} \theta - 3 \cos \theta \sin^{2} \theta\);
   \(\sin 3\theta = 3 \cos^2 \theta \sin \theta - \sin^3 \theta\)

Exercise \(\PageIndex{22}\)

1. Find the fourth roots of unity.
2. Find the sixth roots of unity.

Exercise \(\PageIndex{23}\)

Find all complex numbers \(z\) such that:

1. \(z^{4} = -1\)
2. \(z^{4} = 2(\sqrt{3}i - 1)\)
3. \(z^{3} = -27i\)
4. \(z^{6} = -64\)

Answer

2. \(\pm \frac{\sqrt{2}}{2}(\sqrt{3}+i)\), \(\pm \frac{\sqrt{2}}{2}(-1 + \sqrt{3}i)\)
4. \(\pm 2i\), \(\pm (\sqrt{3} +i)\), \(\pm (\sqrt{3}-i)\)

Exercise \(\PageIndex{24}\)

If \(z = re^{i\theta}\) in polar form, show that:

\(\overline{z} = re^{-i\theta}\) \(z^{-1} = \frac{1}{r} e^{-i\theta}\) if \(z \neq 0\)

Exercise \(\PageIndex{25}\)

Show that the sum of the \(n\)th roots of unity is zero.

Exercise \(\PageIndex{26}\)

1. Let \(z_{1}\), \(z_{2}\), \(z_{3}\), \(z_{4}\), and \(z_{5}\) be equally spaced around the unit circle. Show that \(z_{1} + z_{2} + z_{3} + z_{4} + z_{5} = 0\). [Hint: \((1 - z)(1 + z + z^{2} + z^{3} + z^{4}) = 1 - z^{5}\) for any complex number \(z\).]
2. Repeat (a) for any \(n \geq 2\) points equally spaced around the unit circle.
3. If \(|w| = 1\), show that the sum of the roots of \(z^n = w\) is zero.

Answer

2. The argument in (a) applies using \(\beta = \frac{2\pi}{n}\). Then \(1 + z + \cdots + z^{n-1} = \frac{1-z^n}{1-z}=0\).

Exercise \(\PageIndex{27}\)

If \(z^n\) is real, \(n \geq 1\), show that \((\overline{z})^{n}\) is real.

Exercise \(\PageIndex{28}\)

If \(\overline{z}^2 = z^{2}\), show that \(z\) is real or pure imaginary.

Exercise \(\PageIndex{29}\)

If \(a\) and \(b\) are rational numbers, let \(p\) and \(q\) denote numbers of the form \(a + b\sqrt{2}\). If \(p = a + b\sqrt{2}\), define \(\tilde{p} = a-b\sqrt{2}\) and \([p] = a^{2} - 2b^{2}\). Show that each of the following holds.

Answer

\(a + b\sqrt{2} = a_{1} + b_{1}\sqrt{2}\) only if \(a = a_{1}\) and \(b = b_{1}\) \(\widetilde{p \pm q} = \tilde{p} \pm \tilde{q}\) \(\widetilde{pq} = \tilde{p}\tilde{q}\) \([p] = p \tilde{p}\) \([pq] = [p][q]\) If \(f(x)\) is a polynomial with rational coefficients and \(p = a + b\sqrt{2}\) is a root of \(f(x)\), then \(\tilde{p}\) is also a root of \(f(x)\).