Complex Number Worksheet for Students

Complex Number 

Before you start doing the maths, it's necessary to know what you are looking at. The traditional way to write a complex number is a + bi.

• 'a' represents the real part.

• 'bi' represents the imaginary part.

• The letter 'i' is defined as the square root of -1 (i² = -1).

When you work with complex numbers, the first thing you normally do is figure out what these two pieces are. In the number 5 + 3i, for instance, 5 is real, and 3i is not. This difference is what everything else you do will be based on.

Core Components of Complex Values

To make things easier to visualise, here's a quick breakdown of how these numbers are structured:

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Operations on Complex Numbers Worksheet: Addition and Subtraction

Performing operations on complex numbers worksheet tasks is very similar to combining like terms in algebra. You merely put the real and imaginary numbers together.

• Addition Rule: (a + bi) + (c + di) = (a + c) + (b + d)i

• Subtraction Rule: (a + bi) - (c + di) = (a - c) + (b - d)i

Practice Problems with Solutions

Here are some samples of what you might discover on a complex number worksheet.

1. Question: Simplify (2 + 3i) + (4 + 8i)

• Solution: Group the real parts: 2 + 4 = 6. Group the imaginary parts: 3i + 8i = 11i.

• Result: 6 + 11i.

2. Question: Simplify (10 - 5i) - (3 + 2i)

• Solution: Subtract the real parts: 10 - 3 = 7. Subtract the imaginary parts: -5i - 2i = -7i.

• Result: 7 - 7i.

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Multiplication in your Complex Number Worksheet

Use the FOIL method (First, Outer, Inner, Last) to multiply complex numbers. The most crucial thing to remember is that i² = -1. When you see i², change it to -1 to make the phrase easier to grasp.

Multiplication Example

Question: Multiply (3 + 2i)(1 + 4i)

• First: 3 * 1 = 3

• Outer: 3 * 4i = 12i

• Inner: 2i * 1 = 2i

• Last: 2i * 4i = 8i²

• Simplify i²: Since i² = -1, 8i² becomes 8(-1) = -8.

• Combine: (3 - 8) + (12i + 2i) = -5 + 14i.

By following these steps, you can solve any complex number problem. 

Complex Number Worksheet: Division and Conjugates

Division is generally the hardest part. To divide, you have to get rid of the imaginary unit in the bottom number. To do this, we multiply both the top and bottom by the complex number's conjugate.

If the denominator is a + bi, the conjugate is a - bi.

Division Practice

Question: Simplify 5 / (2 + i)

• Step 1: Find the conjugate of the denominator (2 - i).

• Step 2: Multiply the numerator and denominator by (2 - i).

• Step 3 (Numerator): 5 * (2 - i) = 10 - 5i.

• Step 4 (Denominator): (2 + i)(2 - i) = 4 - 2i + 2i - i². This simplifies to 4 - (-1) = 5.

• Step 5 (Final Divide): (10 - 5i) / 5 = 2 - i.

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Complex Number Worksheet for Students

This part is a full complex number worksheet that will test your understanding of all the operations. To be sure you know everything there is to know about complex numbers, work through these complex numbers practice questions. 

Questions

1. Identify Parts: For the number -15 + 4i, identify the real part and the imaginary part.

2. Basic Addition: Simplify the expression (6 + 2i) + (8 - 5i).

3. Basic Subtraction: Simplify the expression (12 + 7i) - (5 + 3i).

4. Square Roots: Find the value of √(-49).

5. Multiplication (FOIL): Multiply (2 + 3i) and (1 - 2i).

6. Conjugate Logic: Write the complex conjugate for the number 9 - 6i.

7. Complex Division: Simplify the fraction (3 + i) / (2 - i).

8. Powers of i: What is the simplified value of i^10?

Detailed Solutions

1. Solution: Real part = -15. Imaginary part = 4i.

2. Solution: (6 + 8) + (2i - 5i) = 14 - 3i.

3. Solution: (12 - 5) + (7i - 3i) = 7 + 4i.

4. Solution: Since √(-1) = i and √49 = 7, the result is 7i.

5. Solution: 2(1) + 2(-2i) + 3i(1) + 3i(-2i) = 2 - 4i + 3i - 6i². Replace i² with -1: 2 - i + 6 = 8 - i.

6. Solution: To find the conjugate, change the sign of the imaginary part. Result = 9 + 6i.

7. Solution: Multiply by (2 + i) / (2 + i). Numerator: 6 + 3i + 2i + i² = 5 + 5i. Denominator: 4 - i² = 5. Final result: (5 + 5i) / 5 = 1 + i.

8. Solution: Divide the power by 4. 10 / 4 leaves a remainder of 2. Since i² = -1, the answer is -1.

Why Practice with a Complex Numbers Worksheet?

Consistent practice is the only way to get comfortable with the "imaginary" aspect of these numbers. Using a complex numbers worksheet for students helps you recognise patterns, such as the power of i and the fact that conjugates always result in a real number when multiplied.

If you're downloading a complex numbers worksheet PDF to study at home or working on issues in class, paying attention to the step-by-step logic will help you avoid making little sign mistakes. These small mistakes are the most common reason students lose marks in mental maths and algebra.

Complex Numbers: Chapter Overview

Here’s a quick and easy summary to doing math with complex numbers step by step: 

Working through a complex number worksheet might seem daunting at first, but it is just a new way of looking at numerical relationships. Once you understand the four basic operations and the idea of the conjugate, these problems will be as easy as basic addition. Keep practicing these complex numbers practice questions to get better.