Complex Numbers: math practice, Grade 11 – download the PDF

1 Square roots of negatives ★★★

Write each number in the form \(bi\) (simplify the radical when possible).

1. \(\sqrt{-36}\)
2. \(\sqrt{-50}\)
3. \(-\sqrt{-81}\)
4. \(\sqrt{-12}\)

2 Powers of i ★★★

Simplify. (a) \(i^{15}\) (b) \(i^{22}\) (c) \(i^{40}\) (d) \(i^{101}\)

3 Add and subtract ★★★

Compute and write the answer in the form \(a + bi\).

1. \((5 + 2i) + (3 - 7i)\)
2. \((4 - i) - (9 + 3i)\)
3. \((-2 + 6i) - (-2 - i)\)

4 Multiply and distribute ★★★

Compute. (a) \(3i(4 - 5i)\) (b) \((1 + 3i)(2 - i)\)

5 Conjugates ★★★

Write the conjugate of each number, then compute the product of \(7 - 3i\) with its conjugate. (a) \(7 - 3i\) (b) \(-4i\) (c) \(9\)

6 Reading the complex plane ★★★

The figure shows four points in the complex plane. Write the complex number represented by each point, and give its real part and imaginary part.

7 Absolute values ★★★

Find the absolute value of each number. (a) \(3 + 4i\) (b) \(5 - 12i\) (c) \(-8i\) (d) \(-6 + 8i\)

8 Matching parts ★★★

Find the real numbers \(x\) and \(y\) such that \((x + 2) + (3y - 1)i = 7 + 5i\).

9 Products and squares ★★★

Compute. (a) \((3 - 2i)(4 + 5i)\) (b) \((1 + i)^2\) (c) \((2 - 3i)^2\)

10 Dividing complex numbers ★★★

Write each quotient in the form \(a + bi\). (a) \(\dfrac{8 + i}{2 - i}\) (b) \(\dfrac{6}{1 + i}\)

11 Pure imaginary solutions ★★★

Solve each equation over the complex numbers. (a) \(x^2 + 49 = 0\) (b) \(x^2 + 20 = 0\) (c) \(2x^2 + 18 = 0\)

12 Quadratic formula ★★★

Solve \(x^2 - 6x + 13 = 0\) with the quadratic formula.

13 Circuit impedance ★★★

In an alternating-current circuit, engineers describe opposition to current with complex numbers called impedances (in ohms). Two components wired in series have impedances \(Z_1 = 3 + 4i\) and \(Z_2 = 2 - i\). In series the impedances add.

1. Find the total impedance \(Z_1 + Z_2\).
2. The size of the impedance is its absolute value. Find it, rounded to the nearest hundredth.

14 True or false? ★★★

Decide whether each statement is true or false, and justify.

1. \(\sqrt{-4}\cdot\sqrt{-9} = \sqrt{36} = 6\).
2. For \(z = 2 + 3i\) and \(w = 1 - 4i\), \(\overline{z + w} = \overline{z} + \overline{w}\).

15 Distance in the plane ★★★

A treasure map uses the complex plane with one unit equal to one yard. The start is at \(2 + i\) and the chest is at \(-1 + 5i\). How many yards apart are they?

16 Complex roots with coefficients ★★★

Solve \(x^2 + 4x + 9 = 0\) and check your answer with the sum and the product of the roots.

17 Build the equation ★★★

Find a quadratic equation with real coefficients whose solutions are \(2 + 3i\) and \(2 - 3i\).

18 Higher powers ★★★

Compute. (a) \((2 + i)^3\) (b) \((1 - i)^4\)

19 A square root of a complex number ★★★

Show that \(2 + 3i\) is a square root of \(-5 + 12i\), then name the other square root.

20 Reciprocal with the conjugate ★★★

Let \(z = 8 - 15i\).

1. Compute \(z\cdot\overline{z}\) and deduce \(|z|\).
2. Use part (a) to write \(\dfrac{1}{z}\) in the form \(a + bi\).

21 Quarter turns ★★★

A drone at the point \(z = 4 + i\) of the complex plane is rotated a quarter turn counterclockwise about the origin by multiplying by \(i\). Find its position after one, two, three, and four such moves.