Complex Number Modulus Calculator

Complex Number Modulus Calculator — Calculate |z| with Step-by-Step Explanation

Calculate the modulus of any complex number with step-by-step explanation. Learn the Pythagorean formula and geometric interpretation. For related tools, see our complex conjugate calculator.

Table of Contents

1. 1What Is the Modulus?
2. 2The Modulus Formula
3. 3Geometric Interpretation
4. 4Step-by-Step Examples
5. 5More Modulus Examples
6. 6Connection to the Conjugate
7. 7Properties of the Modulus
8. 8Modulus and Polar Form
9. 9Common Mistakes to Avoid
10. 10Real-World Applications
11. 11Practice Problems with Solutions
12. 12Tips for Mastering the Modulus
13. 13Final Thoughts
14. 14FAQ: Common Questions

Calculate the Modulus

Enter a complex number in the form a + bi. Click Calculate for step-by-step working.

Enter a Complex Number

Modulus will appear here.

Note: The modulus of z = a + bi is |z| = √(a² + b²). It's the distance from the origin on the complex plane.

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Complex Number Modulus – Complete Explanation

The modulus measures a complex number's distance from the origin using the Pythagorean theorem. It generalizes absolute value to the complex plane.

Modulus Definition

For z = a + bi:  |z| = √(a² + b²)

Always a non-negative real number.

1. What Is the Modulus?

The modulus answers: "How far is this point from the origin?"

2. The Modulus Formula Explained

The formula comes from the Pythagorean theorem applied to the right triangle formed by a, b, and the modulus.

3. Geometric Interpretation

Geometric Meaning

|z| = distance from origin to (a, b)
    = length of the position vector
    = radius in polar coordinates

4. Step-by-Step Examples

Example 1: |3 + 4i|

a=3, b=4
a²=9, b²=16
9+16=25
|3+4i| = √25 = 5 (3-4-5 triple!)

Example 2: |-5 + 12i|

a=-5, b=12
25+144=169
|-5+12i| = √169 = 13 (5-12-13 triple!)

5. More Modulus Examples

|1+i| = √2 ≈ 1.414

|2-3i| = √13 ≈ 3.606

|7| = 7 (real → absolute value)

|4i| = 4 (imaginary → coefficient)

|0| = 0

6. Connection to the Conjugate

Modulus via Conjugate

|z|² = z × z̄

7. Properties of the Modulus

• |z| ≥ 0, |z|=0 only if z=0

• |z̄| = |z|

• |z₁×z₂| = |z₁|×|z₂|

• |z₁/z₂| = |z₁|/|z₂|

• Triangle Inequality: |z₁+z₂| ≤ |z₁|+|z₂|

8. Modulus and Polar Form

Polar Form

z = r(cos θ + i sin θ) where r = |z|

9. Common Mistakes to Avoid

• Forgetting to square b (the coefficient, not bi).
• Adding before squaring: |z| ≠ a + b.
• Negative modulus (always non-negative).
• Dropping the square root.

10. Real-World Applications

• Electrical Engineering: Impedance magnitude.
• Signal Processing: Fourier amplitude.
• Control Systems: Bode magnitude plots.
• Physics: Probability amplitudes.

11. Practice Problems with Solutions

Problem 1: |2+2i| = 2√2

Problem 2: |-6+8i| = 10

Problem 3: |z|=5, Re=3 → b = ±4

Problem 4: |(3+4i)(1+i)| = 5√2 ✓

12. Tips for Mastering the Modulus

• Memorize |z| = √(a²+b²).
• Recognize Pythagorean triples.
• Real → absolute value. Imaginary → coefficient.
• Use z×z̄ = |z|² to verify.
• Always non-negative.

13. Final Thoughts

The modulus bridges algebra and geometry for complex numbers. Use this calculator to verify your work, but practice until the formula becomes automatic.

14. Frequently Asked Questions

The modulus |z| = √(a² + b²) is the distance of z = a + bi from the origin on the complex plane. It's always non-negative.

The real part a and imaginary part b form a right triangle with the modulus as hypotenuse. By the Pythagorean theorem: |z|² = a² + b².

For a real number z = a (imaginary part = 0), |z| = √(a² + 0) = |a|. The modulus reduces to ordinary absolute value.

For z = bi (real part = 0), |z| = √(0 + b²) = |b|. The modulus is the absolute value of the imaginary coefficient.

|z|² = z × z̄. Multiplying a complex number by its conjugate always gives a² + b² = |z|².

No. The modulus is a distance, so it's always ≥ 0. If you get a negative result, there's an error.

In polar form, z = r(cos θ + i sin θ), the modulus r = |z| is the radius. It tells how far the point is from the origin.

Explore Math Topics

Continue your learning journey with our other math guides

🔢 Basic Arithmetic 📐 Pre-Algebra 📊 Algebra 📏 Geometry 📈 Statistics

Related Calculators

Explore these other math tools that complement the modulus:

• 🔹 Complex Conjugate Calculator — Find z̄.
• 🔹 Complex Polar Form Calculator — Convert to polar form.
• 🔹 Complex Number Multiplication — Multiply complex numbers.
• 🔹 Complex Number Division — Divide complex numbers.
• 🔹 Complex Number Addition — Add complex numbers.

Conclusion

The modulus bridges algebra and geometry for complex numbers. Use this calculator to verify your work, but practice until the formula becomes automatic.

Readers who wish to continue practicing and reinforce what they've learned can explore the companion learning resource for guided exercises.