Real and Complex Numbers: Algebra 2 Math Worksheet – PDF Download

Problem 1: Classify numbers

List every set (natural, integer, rational, irrational, real) that each number belongs to.

1. \(9\)
2. \(-14\)
3. \(\frac{3}{8}\)
4. \(\sqrt{13}\)
5. \(0\)

Problem 2: True or false

Decide whether each statement is true or false.

1. Every integer is a rational number.
2. Every irrational number is a real number.
3. \(\sqrt{49}\) is irrational.
4. \(0.125\) is irrational.
5. Every real number is rational.

Problem 3: Match points on a number line

The points A, B, C, and D are marked on the number line. Match the numbers \(-\sqrt{5}\), \(-\frac{1}{2}\), \(\frac{3}{2}\), and \(\sqrt{7}\) to the points, then name the two that are rational.

Problem 4: Regions of the number diagram

In the diagram, name the set of numbers that each region A, B, C, and D represents. Then name the smallest region that holds each number: \(12\), \(-5\), \(\frac{2}{7}\), and \(\sqrt{11}\).

Problem 5: Square roots of negative numbers

Write each expression in the form \(bi\) or \(bi\sqrt{c}\) with \(c\) as small as possible.

1. \(\sqrt{-49}\)
2. \(\sqrt{-18}\)
3. \(\sqrt{-75}\)
4. \(-\sqrt{-81}\)

Problem 6: Equations with imaginary solutions

Solve each equation over the complex numbers.

1. \(x^{2}=-64\)
2. \(x^{2}+45=0\)
3. \(2x^{2}+50=0\)

Problem 7: Real and imaginary parts

State the real part and the imaginary part of each number.

1. \(6+9i\)
2. \(-4-7i\)
3. \(12i\)
4. \(15\)
5. \(\frac{1}{2}-i\)

Problem 8: Read points in the complex plane

Write the complex number represented by each of the points P, Q, R, and S.

Problem 9: Plot complex numbers

Plot these numbers in the complex plane and label them: \(A=3+2i\), \(B=-4+i\), \(C=-2-3i\), and \(D=5-i\).

Problem 10: Add complex numbers

Find each sum.

1. \((3+5i)+(2-8i)\)
2. \((-4+i)+(6+9i)\)
3. \((7-2i)+(-7+2i)\)

Problem 11: Subtract complex numbers

Find each difference.

1. \((9+4i)-(3+6i)\)
2. \((2-5i)-(-3+i)\)
3. \((-1+8i)-(-1+3i)\)

Problem 12: Add arrows in the plane

The arrows show \(z_1\) and \(z_2\). Write each as \(a+bi\), then find \(z_1+z_2\) and \(z_1-z_2\).

Problem 13: Multiply by an imaginary number

Multiply and write the product as \(a+bi\).

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

Problem 14: Multiply two binomials

Find each product.

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

Problem 15: Multiply conjugates

Find each product.

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

Problem 16: Powers of i

Simplify each power.

1. \(i^{6}\)
2. \(i^{13}\)
3. \(i^{30}\)
4. \(i^{47}\)
5. \(i^{64}\)

Problem 17: Simplify with powers of i

Simplify each expression.

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

Problem 18: Verify a solution

Show that \(1+2i\) is a solution of \(x^{2}-2x+5=0\).

Problem 19: Impedance in a circuit

Electrical engineers use complex numbers for impedance, in ohms. A resistor and a coil are wired in series. Their impedances are \(3+4i\) and \(2-i\) ohms. The total impedance is the sum. Find it.

Problem 20: Voltage across a circuit

Maya measures a current of \(2+i\) amps through a part with impedance \(3-2i\) ohms. Voltage equals current times impedance. Find the voltage in volts.

Problem 21: Find the error

Noah writes \(\sqrt{-4}\cdot\sqrt{-9}=\sqrt{36}=6\). Explain his mistake and find the correct value.

Problem 22: Moving on a game map

A video game map is the complex plane. Diego’s character starts at \(3+2i\). It moves by \(-5+4i\), and then by \(2-7i\). Find the final position.

Problem 23: Rotating a drone

A drone starts at \(4+3i\) on a map in the complex plane. Each time its position is multiplied by \(i\), the drone moves to a new point. Find its position after one, two, and three multiplications, and describe what each multiplication does to the point.

Problem 24: Make a product real

Find the real number \(k\) that makes \((k+3i)(2-i)\) a real number, and give the product.

See every chapter on the Algebra 2 math page.