The imaginary unit

Powers of i, real and imaginary parts, equal complex numbers, square roots and quadratic equations. 8 problems with step-by-step solutions.

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

Problem 1 Imaginary unit

a) Solve \(x^2 + 9 = 0\) and \(x^2 + 50 = 0\).

b) Write \(\dfrac{\sqrt{-16}\cdot\sqrt{-3}}{\sqrt{-12}}\) in terms of \(i\).

Show solution

a) \(x = \pm 3i\); \(x = \pm 5\sqrt2\,i\).

b) \(\frac{4i\cdot\sqrt3 i}{2\sqrt3 i} = 2i\).

Problem 2 Powers of i

\(i^{4n} = 1\), \(i^{4n + 1} = i\), \(i^{4n + 2} = -1\), \(i^{4n + 3} = -i\).

a) Simplify \(i^{2026}\), \(i^{1531}\), \(i^{873}\), \(i^{1200}\).

b) Simplify \(i^{-75}\), \(i^{-202}\), \(i^{-3001}\).

c) Simplify \(i^{8n + 6}\) and \(i^{12n - 5}\), \(n \in \mathbb{Z}\).

Show solution

a) \(2026 = 4\cdot 506 + 2\): \(-1\). \(1531 \equiv 3\): \(-i\). \(873 \equiv 1\): \(i\). \(1200 \equiv 0\): \(1\).

b) \(-75 \equiv 1 \pmod 4\): \(i\). \(-202 \equiv 2\): \(-1\). \(-3001 \equiv 3\): \(-i\).

c) \(i^6 = -1\); \(i^{-5} = i^{3} = -i\).

Problem 3 Sums of powers

a) Evaluate \(i + i^2 + i^3 + \dots + i^{50}\).

b) \(P(x) = 1 + x + x^2 + \dots + x^{4n + 1}\). Find \(P(i)\).

c) \(P(x, y) = 3x^{10}y^{6} - 2x^{7}y^{5} + x^4y^9 - 1\). Find \(P(i, -i)\).

Show solution

a) Every 4 consecutive powers add up to 0. \(50 = 48 + 2\): \(i^{49} + i^{50} = i - 1\).

b) The first \(4n\) terms give 0; then \(i^{4n} + i^{4n + 1} = 1 + i\).

c) \(i^{10} = -1\), \((-i)^6 = -1\): \(3\). \(i^7 = -i\), \((-i)^5 = -i\): \((-i)(-i) = -1\), so \(-2(-1) = 2\). \(i^4 = 1\), \((-i)^9 = -i\): \(-i\). Total \(3 + 2 - i - 1 = 4 - i\).

Problem 4 Real and imaginary parts

a) Write \(\operatorname{Re}\) and \(\operatorname{Im}\): \(z = -4 + \sqrt5 i\), \(w = 7i - 2\), \(v = \frac34\), \(u = -3\sqrt2 i\), \(p = 0\).

b) Find \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\) for \(z = 3i^{22} - 4i^{-13} + 2i^{40} - i^{-30}\).

Show solution

a) \(z\): \(-4\) and \(\sqrt5\); \(w\): \(-2\) and \(7\); \(v\): \(\frac34\) and \(0\); \(u\): \(0\) and \(-3\sqrt2\); \(p\): \(0\) and \(0\).

b) \(i^{22} = -1\), \(i^{-13} = i^{3} = -i\), \(i^{40} = 1\), \(i^{-30} = i^{2} = -1\). \(z = -3 + 4i + 2 + 1 = 4i\): Re \(0\), Im \(4\).

Problem 5 Equal complex numbers

a) Find real \(x, y\) with \(x^2 - y^2 + (x - y)i = 12 + 2i\).

b) Find \(x, y\) with \(2x - 3y + (x + y)i = 4 + 7i\).

c) Find \(z\) with \(4z = 3 - iz\).

d) Find \(z\) with \(2i - z = 5 + 3iz\).

Show solution

a) \(x - y = 2\) and \(x^2 - y^2 = (x - y)(x + y) = 12\), so \(x + y = 6\): \(x = 4\), \(y = 2\).

b) \(x + y = 7\), \(2x - 3y = 4\): \(x = 5\), \(y = 2\).

c) \(z(4 + i) = 3\), \(z = \frac{3(4 - i)}{17} = \frac{12}{17} - \frac{3}{17}i\).

d) \(z(1 + 3i) = -5 + 2i\), \(z = \frac{(-5 + 2i)(1 - 3i)}{10} = \frac{1 + 17i}{10}\).

Problem 6 Square roots

Find the square roots of a) \(5 + 12i\) b) \(-7 - 24i\) c) \(8i\).

Show solution

Let \((a + bi)^2 = a^2 - b^2 + 2abi\).

a) \(a^2 - b^2 = 5\), \(ab = 6\): \(a = 3, b = 2\). Roots \(\pm(3 + 2i)\).

b) \(a^2 - b^2 = -7\), \(ab = -12\): \(a = 3, b = -4\). Roots \(\pm(3 - 4i)\).

c) \(a^2 = b^2\), \(2ab = 8\): \(a = b = 2\). Roots \(\pm(2 + 2i)\).

Problem 7 Quadratic equations

Solve.

a) \(x^2 - 6x + 13 = 0\) b) \(x^2 + x + 1 = 0\) c) \(x^3 - 2x^2 + 5x = 0\) d) \(x^3 + 27 = 0\) e) \(x^4 - 16 = 0\)

Show solution

a) \(\Delta = -16\): \(x = 3 \pm 2i\)
b) \(x = \frac{-1 \pm \sqrt3 i}{2}\)
c) \(x(x^2 - 2x + 5) = 0\): \(x = 0\), \(x = 1 \pm 2i\)
d) \((x + 3)(x^2 - 3x + 9) = 0\): \(x = -3\), \(x = \frac{3 \pm 3\sqrt3 i}{2}\)
e) \((x^2 - 4)(x^2 + 4) = 0\): \(x = \pm 2\), \(x = \pm 2i\)

Problem 8 Equation from roots

a) A quadratic equation with real coefficients has the root \(3 - 2i\). Write the equation.

b) The roots are \(1 + \sqrt3 i\) and \(1 - \sqrt3 i\). Write the equation.

c) The roots are \(2i + 3\) and \(2i - 3\). Write the equation.

d) Solve \(2x^2 - 3ix - 1 = 0\).

Show solution

a) The other root is \(3 + 2i\). Sum 6, product \(9 + 4 = 13\): \(x^2 - 6x + 13 = 0\).

b) Sum 2, product 4: \(x^2 - 2x + 4 = 0\).

c) Sum \(4i\), product \((2i)^2 - 9 = -13\): \(x^2 - 4ix - 13 = 0\).

d) \(\Delta = (3i)^2 + 8 = -1\), \(\sqrt\Delta = i\): \(x = \frac{3i \pm i}{4}\), so \(x = i\) or \(x = \frac{i}{2}\).