Complex Number Calculator

Having trouble with complicated numbers in engineering, physics, or math? It can take hours of work to fix a +bi or polar sign that you get wrong just once. Because of this, we developed the MathCalc Complicated Number Calculator, a powerful online tool that can quickly and accurately solve complex number problems.You no longer have to perform algebra or complex angle conversions manually. You can type in your numbers and receive step-by-step answers immediately. You can get it online whenever you need to, and it works quickly and efficiently. This calculator can help you save time, reduce errors, and gain a deeper understanding of complex math concepts.

Complex Number Calculator

Perform operations on complex numbers (a + bi)

How to Use
  1. Fill in the Required Values
  2. Click "Calculate" Button
  3. View Step-By-Step Solution

How to use a complex number calculator?

Step by step:

  • Choose a calculation type among (Add, Subtract, Multiply, Divide, complex conjugate, argument phase, and convert to polar form).
  • Enter the needed input: (First number-real part, First Number - Imaginary Part, Second Number - Real Part, and Second Number - Imaginary Part).
  • Click "Calculate," and the results will appear in seconds.

Example 1: Choose to Calculate Addition

Add the two complex numbers (3 + 4i) and (5 – 2i).
Input:
Enter 3 in First Number – Real Part, 4 in First Number – Imaginary Part, 5 in Second Number – Real Part, and -2 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 3 + 4i
  • Second complex number: z₂ = 5 + -2i
  • Addition: z₁ + z₂ = (3 + 5) + (4 + -2)i
  • Result: 8 + 2i

result:

  • 8 + 2i

Example 2: Choose to Calculate Subtraction

Subtract the two complex numbers (3 + 4i) and (5 – 2i).
Input:
Enter 3 in First Number – Real Part, 4 in First Number – Imaginary Part, 5 in Second Number – Real Part, and -2 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 3 + 4i
  • Second complex number: z₂ = 5 + -2i
  • Subtraction: z₁ – z₂ = (3 – 5) + (4 – (-2))i
  • Simplify: z₁ – z₂ = (-2) + (6)i

result:

  • -2 + 6i

Example 3: Choose to Calculate Multiplication

Multiply the two complex numbers (7 + 6i) and (2 + 3i).
Input:
Enter 7 in First Number – Real Part, 6 in First Number – Imaginary Part, 2 in Second Number – Real Part, and 3 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 7 + 6i
  • Second complex number: z₂ = 2 + 3i
  • Multiplication formula: z₁ × z₂ = (a + bi)(c + di) = (ac - bd) + (ad + bc)i
  • = (7×2 - 6×3) + (7×3 + 6×2)i
  • Result: -4 + 33i

result:

  • -4 + 33i

Example 4: Choose to Calculate Division

Divide the two complex numbers (7 + 6i) and (2 + 3i).
Input:
Enter 7 in First Number – Real Part, 6 in First Number – Imaginary Part, 2 in Second Number – Real Part, and 3 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 7 + 6i
  • Second complex number: z₂ = 2 + 3i
  • Division formula: z₁ / z₂ = (7 + 6i)(2 – 3i) / (2 + 3i)(2 – 3i)
  • Simplify denominator: (2 + 3i)(2 – 3i) = 4 + 9 = 13
  • Simplify numerator: (7 + 6i)(2 – 3i) = (7×2) + (7×-3i) + (6i×2) + (6i×-3i) = 14 – 21i + 12i – 18i²
  • Since i² = –1 → = 14 – 9i + 18 = 32 – 9i
  • Divide by the denominator: (32 – 9i) / 13 = 32/13 – 9/13 i

result:

  • (32/13) – (9/13)i ≈ 2.46 – 0.69i

Example 5: Choose to Calculate Complex Conjugate

Complex conjugate the two complex numbers (7 + 6i) and (2 + 3i).
Input:
Enter 7 in First Number – Real Part, 6 in First Number – Imaginary Part, 2 in Second Number – Real Part, and 3 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 7 + 6i
  • Complex conjugate formula: z̄ = a - bi
  • Compute: z̄₁ = 7 - (6)i = 7 - 6i

result:

  • 7 - 6i

Example 6: Choose to Calculate Magnitude (Modulus)

Calculate the magnitude (modulus) of the two complex numbers (7 + 6i) and (2 + 3i).
Input:
Enter 7 in First Number – Real Part, 6 in First Number – Imaginary Part, 2 in Second Number – Real Part, and 3 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 7 + 6i
  • Magnitude formula: |z| = √(a² + b²)
  • Compute: |z₁| = √(7² + 6²) = √85 ≈ 9.2195444572929

result:

  • Magnitude: 9.2195

Example 7: Choose to Calculate Argument (Phase)

Calculate the argument (phase) of the two complex numbers (7 + 6i) and (2 + 3i).
Input:
Enter 7 in First Number – Real Part, 6 in First Number – Imaginary Part, 2 in Second Number – Real Part, and 3 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 7 + 6i
  • Second complex number: z₂ = 2 + 3i
  • Formula for argument (phase): Arg(z) = tan⁻¹(Im/Re)
  • Calculate for z₁: Arg(z₁) = tan⁻¹(6/7) ≈ 0.71 radians (≈ 40.6°)
  • Calculate for z₂: Arg(z₂) = tan⁻¹(3/2) ≈ 0.98 radians (≈ 56.3°)

result:

  • Argument (Phase) of z₁ ≈ 0.71 radians (40.6°)
  • Argument (Phase) of z₂ ≈ 0.98 radians (56.3°)

Example 8: Choose to Calculate “Convert to Polar Form”

Convert to Polar Form the two complex numbers (7 + 6i) and (2 + 3i).
Input:
Enter 7 in First Number – Real Part, 6 in First Number – Imaginary Part, 2 in Second Number – Real Part, and 3 in Second Number – Imaginary Part.

Step by step:

  • Complex Number Calculator
  • First complex number: z₁ = 7 + 6i
  • Polar form: z = r(cos θ + i sin θ) = re^(iθ)
  • Magnitude: r = √(a² + b²) = 9.2195
  • Angle: θ = arctan(b/a) = 40.6° = 0.7086 rad
  • Polar form: 9.2195∠40.6°

result:

  • Polar magnitude: 9.2195
  • Polar angle_deg: 40.6°
  • Polar angle_rad: 0.7086 rad
  • Polar form: 9.2195∠40.6°

Who Can Use This Complex Number Calculator?

Those who require quick and correct answers to complicated numbers should use MathCalc.
• Data scientists can effortlessly verify methods, resolve code issues, and perform calculations involving imaginary numbers.
• Researchers can streamline their efforts by employing robots to handle intricate numerical calculations and conversions that need to be performed repeatedly.
• Students can effortlessly grasp complex numbers with the right support in their algebra, trigonometry, or electrical engineering studies.
• Engineers streamline phasor analysis, AC circuits, and signal processing to ease the burden of complex calculations.

Why Use This Complex Number Calculator?

It can be challenging to work with complex numbers because it’s easy to make mistakes when handling a+bi, polar angles, and complex conjugates. The MathCalc Complex Number Calculator solves that problem. So, why should you use this tool?
• Instant Result: You won’t have to guess your algebra anymore. Always get correct outcomes.
• Step-by-Step Explanations: You can learn while you do calculations. See how each solution is made.
• One tool can perform multiple tasks, including addition and subtraction, finding powers and roots, and converting to polar coordinates, among others.
• It Saves Time: The tool can quickly and accurately resolve complex mathematical problems that would otherwise take hours or minutes to resolve.
• Increases Learning: The clear explanations can help you solve difficult calculations more professionally.

Conclusion

Use the MathCalc Complex Number Calculator for fast and accurate solutions to complex numbers. You won’t find a better calculator online than MathCalc. It will help you learn as you work by providing step-by-step results. It also performs both polar and rectangular forms, as well as basic math functions such as addition, subtraction, multiplication, and division. It is simple to use, provides accurate results, and delivers knowledge. One of the most effective methods for studying complex numbers online is with this tool.
Try the Complex Number Calculator today.

FAQs

Can I find out the roots and powers of complicated numbers?

Yes. Type in your complicated number and pick Power/Root. The tool finds all acceptable answers for exponentiation, square roots, and cube roots.

Is MathCalc’s complex number solver free?

Yes. There is a free online tool called the MathCalc Complex Number tool that you can use.

Does the tool provide detailed answers?

Of course. MathCalc tells you what to do at each step, so you can learn how to do it instead of just copying the answer.

Can I use it to look at AC circuits and phasors?

Yes. Many people in electrical engineering utilize this tool to simplify the analysis of AC circuits, convert phasors, and interpret signals.