Complex Number Algebra Calculator

Calculator Inputs

z1 Real Part

z1 Imaginary Part

z2 Real Part

z2 Imaginary Part

Operation

Power n

Root Order

Polar Magnitude r

Polar Angle θ

Angle Unit

Decimal Precision

Example Data Table

z1	z2	Operation	Expected Result
3 + 4i	1 - 2i	Addition	4 + 2i
3 + 4i	1 - 2i	Multiplication	11 - 2i
3 + 4i	1 - 2i	Division	-1 + 2i
3 + 4i	None	Modulus	5

Formula Used

For z1 = a + bi and z2 = c + di, addition is (a + c) + (b + d)i. Subtraction is (a - c) + (b - d)i.

Multiplication is (ac - bd) + (ad + bc)i. Division is ((ac + bd) / (c² + d²)) + ((bc - ad) / (c² + d²))i.

Modulus is |z| = √(a² + b²). Argument is θ = atan2(b, a). Polar form is r(cos θ + i sin θ).

Power uses De Moivre formula. If z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos nθ + i sin nθ).

How to Use This Calculator

Enter the real and imaginary parts for z1. Enter z2 when your operation needs two complex numbers. Select the required operation. Add a power, root order, angle unit, or polar input when needed. Press calculate. The result appears above the form. Use the download buttons to save the result.

Complex Number Algebra Guide

Complex numbers extend ordinary numbers with an imaginary part. A value such as 3 + 4i has two parts. The real part is 3. The imaginary part is 4. The symbol i means the square root of negative one. This idea lets algebra handle roots, rotations, waves, circuits, signals, and many geometry problems.

Why This Tool Helps

This calculator reduces manual work. It handles addition, subtraction, multiplication, division, conjugates, modulus, arguments, powers, roots, distances, and polar conversion. Each option follows standard algebra rules. You can choose decimal precision. You can also export results for records or classroom use.

Rectangular and Polar Views

Rectangular form uses a + bi. It is simple for addition and subtraction. Polar form uses magnitude and angle. It is better for powers and roots. The magnitude shows distance from zero. The angle shows direction from the positive real axis. Both forms describe the same point on the complex plane.

Operations Explained

Addition combines matching parts. Subtraction removes matching parts. Multiplication expands brackets and replaces i squared with negative one. Division uses a conjugate, which removes the imaginary term from the denominator. This makes the final quotient easier to read.

Powers and Roots

Powers are faster in polar form. The magnitude is raised to the selected power. The angle is multiplied by that power. Roots reverse the process. The calculator lists all roots because complex roots are usually multiple values. These roots appear around a circle with equal angle spacing.

Practical Uses

Complex algebra appears in electronics, physics, control systems, signal processing, and advanced mathematics. Students use it for homework and exam checking. Teachers use it to prepare examples. Engineers use it to model phase, impedance, and oscillation. Clean exports make results easy to share.

Best Practice

Always check which angle unit your problem uses. Degrees and radians give different displayed values. Keep enough precision during long work. Round only the final result. Use the example table to compare expected answers. Review the formulas when a result seems unexpected or unclear.

FAQs

1. What is a complex number?

A complex number has a real part and an imaginary part. It is written as a + bi, where i represents the square root of negative one.

2. Can this calculator divide complex numbers?

Yes. It divides complex numbers by multiplying with the conjugate of the denominator. Division by zero is blocked because it is undefined.

3. What is the modulus of a complex number?

The modulus is the distance from zero on the complex plane. For a + bi, it equals the square root of a squared plus b squared.

4. What is the argument?

The argument is the angle of the complex number from the positive real axis. You can display it in degrees or radians.

5. Why do roots show many answers?

Complex roots often have multiple values. The number of answers usually equals the selected root order, unless the original number is zero.

6. What does conjugate mean?

The conjugate changes the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

7. Can I convert polar form to rectangular form?

Yes. Select the polar to rectangular option. Enter magnitude, angle, and angle unit. The calculator returns the matching a + bi form.

8. What do the downloads include?

The CSV and PDF downloads include the selected operation, input numbers, final answer, and key calculation details for later review.