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Complex Number Calculator.

Calculate modulus, argument, conjugate, and modulus squared from a + bi.

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Modulus: 5

Modulus

5.000000
Argument: 0.927295 rad

Argument

0.927295rad
Argument: 53.130102 °

Argument

53.130102°
Conjugate real part: 3

Conjugate real part

3.000000
Conjugate imaginary part: -4

Conjugate imaginary part

-4.000000
Modulus squared: 25

Modulus squared

25.000000

Results update automatically as you type.

Use Cases

Engineering and Physics Calculations

Use modulus and argument to represent phasors or impedances in AC circuit analysis and signal processing.

Example: Find the magnitude and phase of an impedance 3 + 4j ohms.

Mathematics Homework and Verification

Quickly check your manual calculations for modulus, argument, conjugate, or modulus squared of complex numbers.

Example: Verify that the conjugate of 2 - 5i is 2 + 5i.

Frequently Asked Questions

What does the modulus of a complex number represent?
The modulus (or absolute value) of a complex number a + bi is its distance from the origin in the complex plane, calculated as sqrt(a^2 + b^2). It is always a non-negative real number.
How is the argument of a complex number determined?
The argument is the angle (in radians or degrees) the complex number makes with the positive real axis. It is computed using atan2(b, a), which accounts for the correct quadrant.
What is the conjugate of a complex number?
The conjugate of a + bi is a - bi. It has the same real part but the imaginary part has the opposite sign. It is useful for simplifying division and finding magnitudes.

Tips & Common Mistakes

Tips

  • Enter the real part and imaginary part as integers or decimals. For example, for 3 - 4i, enter 3 and -4.
  • The argument is typically given in radians; you can convert to degrees by multiplying by 180/π.
  • Modulus squared is simply a^2 + b^2, which is always a non-negative real number.
  • Use the conjugate to rationalize denominators when dividing complex numbers.

Common Mistakes to Avoid

  • Forgetting to include the sign of the imaginary part. For -2 - 3i, enter -3 for the imaginary part, not 3.
  • Confusing the argument with the angle in degrees when the calculator outputs radians.
  • Assuming the modulus squared is the same as the modulus; they differ by a square root.

Last updated: August 13, 2026