Mastering Real and Non-Real Numbers- Free Practice Worksheets

What You're Getting Into

This isn't another fluffy math article. We're covering real numbers, non-real numbers, and giving you actual practice worksheets you can use right now. If you're a student, teacher, or someone trying to remember high school math, this is the page you need.

No registration. No email required. Just scroll down, grab the worksheets, and get to work.

Real Numbers: The Basics

Real numbers include everything you can plot on a number line. Positive, negative, zero, decimals, fractions—all of it.

Two Types of Real Numbers

Rational numbers are numbers you can write as a fraction. This includes integers, finite decimals, and repeating decimals.

• 5 is rational (5/1)
• 0.75 is rational (3/4)
• 0.333... is rational (1/3)
• -8 is rational (-8/1)

Irrational numbers cannot be written as a fraction. Their decimal expansions go on forever without repeating.

• π (pi) ≈ 3.14159...
• √2 ≈ 1.41421...
• e (Euler's number) ≈ 2.71828...

Non-Real Numbers: Where It Gets Weird

Non-real numbers exist outside the number line. The most common are imaginary numbers, built around the imaginary unit i, where:

i = √(-1)

You can't take the square root of a negative number in the real number system. That's why mathematicians invented i.

• √(-1) = i
• √(-4) = 2i
• √(-9) = 3i

Complex Numbers: The Combination

Most of the time, you'll encounter complex numbers, which combine real and imaginary parts:

a + bi

where a is the real part and bi is the imaginary part.

• 3 + 4i
• 7 - 2i
• -1 + i

Real vs. Non-Real: Quick Comparison

Free Practice Worksheets

Print these out or copy them onto paper. Show your work. Answers are at the bottom.

Worksheet 1: Identifying Real vs. Non-Real Numbers

Classify each number as Real or Non-Real:

1. √25
2. √(-16)
3. 0.333...
4. π
5. 7 + 3i
6. -5/8
7. √(-49)
8. 0.121121112...

Worksheet 2: Simplifying Imaginary Numbers

Simplify each expression:

1. √(-36)
2. √(-7)
3. √(-144)
4. 3√(-25)
5. -2√(-81)

Worksheet 3: Complex Number Operations

Solve each problem. Write answers in the form a + bi.

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

How to Use These Worksheets

Here's what actually works:

1. Attempt without looking at answers first. Sounds obvious, but people skip this step constantly.
2. Show every step. Math isn't about getting the answer—it's about understanding the process.
3. Check your work immediately. Don't wait until you finish all problems. Check each one as you go.
4. Redo wrong problems. Don't just read the correct answer. Work through it again from scratch.
5. Time yourself. Once you understand the material, practice for speed. Real tests don't give unlimited time.

Common Mistakes to Avoid

• Forgetting that √(-1) = i. This is the foundation. Memorize it.
• Dropping the negative sign. √(-9) = 3i, not -3i.
• Confusing rational with irrational. If it can be written as a fraction, it's rational.
• Forgetting to distribute in complex multiplication. (a + b)(c + d) ≠ ac + bd. Use FOIL correctly.

Worksheet Answers

Worksheet 1 Answers

1. Real (5)
2. Non-Real (4i)
3. Real (1/3)
4. Real (irrational)
5. Non-Real (complex)
6. Real (-5/8)
7. Non-Real (7i)
8. Real (1/8, actually rational)

Worksheet 2 Answers

1. 6i
2. i√7
3. 12i
4. 15i
5. -18i

Worksheet 3 Answers

1. 4 + 6i
2. 3 - 2i
3. -4 + 7i
4. 17 (this is called a conjugate pair—imaginary parts cancel)