Operations and the conjugate

Conjugate, addition, multiplication, powers of 1 ± i, division and the inverse. 8 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Conjugate and complex plane

a) Plot \(z = 3 - 2i\), \(w = -1 + 4i\), \(u = -5i\), \(v = 2\) and their conjugates.

b) Write the conjugates: \(\frac13 - 5i\), \(4i - 7\), \(-i\), \(\sqrt7 + \sqrt2\).

c) Find the pure imaginary \(z\) with \(z - \bar z + \overline{(\bar z)} = i\).

Show solution

a) Points \((3, -2)\), \((-1, 4)\), \((0, -5)\), \((2, 0)\); conjugates are their mirror images in the real axis: \((3, 2)\), \((-1, -4)\), \((0, 5)\), \((2, 0)\).

b) \(\frac13 + 5i\); \(-4i - 7\); \(i\); \(\sqrt7 + \sqrt2\).

c) \(z = bi\): \(bi - (-bi) + bi = 3bi = i\), \(b = \frac13\): \(z = \frac{i}{3}\).

Problem 2 Addition and subtraction

\(z_1 = 3 - i\), \(z_2 = -2 + 5i\). Calculate:

a) \(z_1 + z_2\) b) \(z_1 - z_2\) c) \(2z_1 + 3z_2\) d) \(z_2 - \frac12 z_1\) e) \(4z_1 - 3\bar z_1\) for \(z_1 = 3 - i\)

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a) \(1 + 4i\)
b) \(5 - 6i\)
c) \(6 - 2i - 6 + 15i = 13i\)
d) \(-2 + 5i - \frac32 + \frac12 i = -\frac72 + \frac{11}{2}i\)
e) \(12 - 4i - 9 - 3i = 3 - 7i\)

Problem 3 Multiplication

Calculate.

a) \((2 + 3i)(4 - i)\) b) \((-3 + 2i)(5 + 4i)\) c) \((1 - 5i)(1 + i)\) d) \((5 - 3i)(1 + 2i)\) e) \(\left(\frac12 + \frac13 i\right)(6 - 12i)\)

Show solution

a) \(8 - 2i + 12i + 3 = 11 + 10i\)
b) \(-15 - 12i + 10i - 8 = -23 - 2i\)
c) \(1 + i - 5i + 5 = 6 - 4i\)
d) \(5 + 10i - 3i + 6 = 11 + 7i\)
e) \(3 - 6i + 2i + 4 = 7 - 4i\)

Problem 4 z times its conjugate

\(z\cdot\bar z = a^2 + b^2\).

a) Calculate \(z\bar z\) for \(z = 5 + 12i\), \(4 - 3i\), \(2 + 2i\), \(7i\), \(6\).

b) Find \(z\) with \(z(\bar z + 2) = 13 - 4i\).

Show solution

a) \(169\); \(25\); \(8\); \(49\); \(36\).

b) With \(z = a + bi\): \(z\bar z + 2z = a^2 + b^2 + 2a + 2bi = 13 - 4i\). So \(2b = -4\), \(b = -2\). \(a^2 + 4 + 2a = 13\), \(a^2 + 2a - 9 = 0\), \(a = -1 \pm \sqrt{10}\). So \(z = -1 \pm \sqrt{10} - 2i\).

Problem 5 Powers of 1 ± i

\((1 + i)^2 = 2i\) and \((1 - i)^2 = -2i\).

a) Calculate \((1 + i)^4 + (1 - i)^6\).

b) Calculate \((1 + i)^{20}\) and \(\left(\frac{1}{\sqrt2} + \frac{1}{\sqrt2}i\right)^{2026}\).

c) Calculate \(3(1 + i)^{101} - (1 - i)^{102}\).

Show solution

a) \((2i)^2 + (-2i)^3 = -4 + 8i\).

b) \((2i)^{10} = 1024\cdot i^{10} = -1024\). \(\frac{1 + i}{\sqrt2}\) squared is \(i\), so the power is \(i^{1013} = i\).

c) \((1 + i)^{101} = (2i)^{50}(1 + i) = 2^{50}(-1)(1 + i)\). \((1 - i)^{102} = (-2i)^{51} = -2^{51}i^{51} = -2^{51}(-i) = 2^{51}i\). Result: \(-3\cdot 2^{50}(1 + i) - 2^{51}i = -3\cdot 2^{50} - 5\cdot 2^{50}i\).

Problem 6 Division

Write in standard form.

a) \(\dfrac{1}{4 - 3i}\) b) \(\dfrac{2 + 5i}{2 - 5i}\) c) \(\dfrac{3}{1 - i} - \dfrac{2i}{1 + i}\) d) \(\dfrac{(2 + i)^2}{1 - 2i}\)

Show solution

Multiply by the conjugate of the denominator.

a) \(\frac{4 + 3i}{25} = \frac{4}{25} + \frac{3}{25}i\)
b) \(\frac{(2 + 5i)^2}{29} = \frac{-21 + 20i}{29}\)
c) \(\frac{3(1 + i)}{2} - \frac{2i(1 - i)}{2} = \frac{3 + 3i - 2i - 2}{2} = \frac12 + \frac12 i\)
d) \(\frac{3 + 4i}{1 - 2i} = \frac{(3 + 4i)(1 + 2i)}{5} = \frac{-5 + 10i}{5} = -1 + 2i\)

Problem 7 Inverse and quotients

a) Write the multiplicative inverse of \(z = 2 + 5i\) and of \(w = \frac{\sqrt3}{2} - \frac12 i\).

b) Calculate \(\left(\dfrac{1 + i}{1 - i}\right)^{2023} + \left(\dfrac{1 - i}{1 + i}\right)^{2022}\).

c) Calculate \(2(1 + i)^{-2} + 2(1 - i)^{-2}\).

d) Find \(z\) with \(z - 3 = (2z + 1)i\).

Show solution

a) \(\frac{2 - 5i}{29}\); \(|w| = 1\), so \(w^{-1} = \bar w = \frac{\sqrt3}{2} + \frac12 i\).

b) \(\frac{1 + i}{1 - i} = i\) and \(\frac{1 - i}{1 + i} = -i\). \(i^{2023} = -i\), \((-i)^{2022} = i^{2022} = -1\). Sum: \(-1 - i\).

c) \(\frac{2}{2i} + \frac{2}{-2i} = 0\).

d) \(z - 2zi = 3 + i\), \(z = \frac{3 + i}{1 - 2i} = \frac{(3 + i)(1 + 2i)}{5} = \frac{1 + 7i}{5}\).

Problem 8 Mixed expressions

a) \(\frac1z = \frac{1}{2 + i} + \frac{1}{2 - i}\). Find \(\operatorname{Re}(z)\cdot\operatorname{Im}(z)\).

b) Simplify \(\dfrac{\sqrt3 + i}{\sqrt3 - i} + \dfrac{\sqrt3 - i}{\sqrt3 + i}\).

c) \(z = \dfrac{a + bi}{b - ai}\) (\(a, b\) real, not both 0). Find \(z\).

d) \(z = 2\left(\frac{1 + i}{1 - i}\right)^7 + 5\left(\frac{1 - i}{1 + i}\right)^4\). Write \(\bar z\).

Show solution

a) \(\frac{(2 - i) + (2 + i)}{5} = \frac45\), so \(z = \frac54\): Re \(\cdot\) Im \(= 0\).

b) \(\frac{(\sqrt3 + i)^2 + (\sqrt3 - i)^2}{4} = \frac{2(3 - 1)}{4} = 1\).

c) \(a + bi = i(b - ai)\), so \(z = i\).

d) \(2i^7 + 5(-i)^4 = -2i + 5\), so \(z = 5 - 2i\) and \(\bar z = 5 + 2i\).