Complex Number Calculator

Having trouble with complicated numbers in engineering, physics, or math? One small mistake with a + bi or a polar conversion can throw off your entire solution. To simplify the process, we developed the Complex Number Calculator, a powerful online tool designed to quickly and accurately solve complex numbers.No more manual algebra or conversions. Just enter your values, and the calculator provides instant, step-by-step solutions. It works online, anytime you need it, and saves hours of calculation while reducing human errors. Whether you’re a student, engineer, or researcher, this complex number calculator will make problem-solving easier and smarter.

Complex Number Calculator

Perform operations on complex numbers (a + bi)

How to Use the Complex Number Calculator

  • Choose a calculation type: addition, subtraction, multiplication, division, complex conjugate, magnitude (modulus), argument (phase), or convert to polar form.
  • Enter the required input: First Number – Real Part First Number – Imaginary Part Second Number – Real Part Second Number – Imaginary Part
  • Click “Calculate” and see your result instantly.

Examples

Example 1: Complex Number Addition

Add (3 + 4i) and (5 – 2i).
z₁ = 3 + 4i
z₁ = 3 + 4i

  • This shows how the complex number addition calculator provides accurate solutions in seconds.

Step by step:

  • (3 + 5) + (4 - 2)i = 8 + 2i

result:

  • 8 + 2i

Example 2: Complex Number Multiplication

Multiply (7 + 6i) and (2 + 3i).
z₁ = 7 + 6i
z₂ = 2 + 3i

  • Perfect for quickly handling multiplications using the complex number multiplication calculator.

Step by step:

  • (7×2 - 6×3) + (7×3 + 6×2)i
  • = -4 + 33i

result:

  • -4 + 33i

Example 3: Division in Complex Numbers

Divide (7 + 6i) by (2 + 3i).

  • This makes handling division in complex numbers straightforward and accurate.

Step by step:

  • (a + bi) ÷ (c + di) = ((ac + bd) + (bc - ad)i) / (c² + d²)
  • Substitute values: ((7×2 + 6×3) + (6×2 - 7×3)i) / (2² + 3²)
  • = (32 - 9i) / 13
  • = 2.46 - 0.69i

result:

  • 2.46 - 0.69i

Example 4: Complex Conjugate

Find the conjugate of (7 + 6i).

  • Quick and easy with the complex conjugate calculator.

Step by step:

  • z̄ = a - bi
  • = 7 - 6i

result:

  • 7 - 6i

Example 5: Magnitude (Modulus)

For z = 7 + 6i,

  • Instantly find the modulus using the complex number magnitude calculator.

Step by step:

  • |z| = √(a² + b²) = √(7² + 6²) = 9.2195

result:

  • Magnitude = 9.2195

Example 6: Argument (Phase)

For z = 7 + 6i,

  • The complex number argument calculator shows the phase angle instantly.

Step by step:

  • θ = arctan(b/a) = arctan(6/7) = 40.6°

result:

  • Argument (Phase) = 40.6°

Example 7: Convert to Polar Form

For z = 7 + 6i,

  • The convert to polar form complex number calculator makes it effortless.

Step by step:

  • Magnitude r = √(7² + 6²) = 9.2195
  • θ = 40.6° (0.7086 rad)
  • Polar form: z = 9.2195∠40.6°

result:

  • z = 9.2195∠40.6°

Who Can Use This Complex Number Calculator?

  • Students: Master algebra, trigonometry, and electrical engineering concepts.
  • Engineers: Simplify AC circuit analysis, phasor conversions, and signal processing.
  • Researchers & Data Scientists: Speed up repeated complex calculations in applied problems

Why Use AllCalculator Complex Number Calculator?

  • Instant Results: Always accurate, no manual errors.
  • Step-by-Step Explanations: Learn while solving.
  • Multiple Functions in One Tool: Addition, subtraction, conjugates, arguments, and polar forms.
  • Saves Time: Complex problems solved in seconds.
  • Boosts Understanding: Clear explanations support learning and professional use.

FAQs

Can I solve powers and roots of complex numbers with this tool?

Yes. Enter your complex number and choose Power/Root to find square roots, cube roots, and higher powers.

Is the AllCalculator Complex Number Calculator free?

Absolutely. It is a free, online complex number calculator for unlimited use.

Does it show detailed solutions?

Yes. Each step is explained, making it easy to solve complex numbers while learning the method.

Can I use it for division in AC circuits and phasor analysis?

Yes. Engineers rely on it to simplify division in complex numbers, phasor conversions, and AC analysis.

Can I convert rectangular form to polar form with this tool?

Yes. Use the convert to polar form complex number calculator to instantly find magnitude and phase angle.