What does this complex number calculator do?

The calculator works with complex numbers in a+bi form. For two numbers, it can add, subtract, multiply, or divide z₁ and z₂. For a single z₁, it can show rectangular and polar information, modulus, argument, conjugate, principal Log(z), and an integer power.

• Binary operations: addition, subtraction, multiplication, and division.
• Single-number information: rectangular form, polar form, modulus, argument, and conjugate.
• Extra modes: principal Log(z₁) and integer power z₁^n.
• Division by z₂=0 is blocked with an explicit error.
• Fractional powers, complex exponents, and all logarithm branches are outside this calculator’s scope.

How to read a complex number

A complex number is written as z=a+bi. The number a is the real part, b is the coefficient of the imaginary part, and i is the imaginary unit with i²=−1.

This form can be read as a point in the complex plane. Rectangular form uses horizontal and vertical components; polar form uses a distance from the origin and an angle.

How the basic operations work

Addition and subtraction are component-wise. Multiplication uses distribution together with i²=−1. Division uses the conjugate of the denominator so that the denominator becomes real.

1. For addition, add the real parts and imaginary coefficients separately.
2. For subtraction, subtract each component of z₂ from z₁.
3. For multiplication, distribute the terms and replace i² with −1.
4. For division, multiply by the conjugate of z₂ and reject the case z₂=0.

Polar form, modulus, and argument

The modulus is the distance of z from the origin. The argument is the angle measured from the positive real axis. Polar form is useful for understanding multiplication, division, powers, roots, and logarithms.

• The modulus |z| is always non-negative.
• The argument is shown in radians and degrees.
• Polar form represents the same number using distance and angle.
• These quantities are usually displayed as numerical approximations.

Principal Log(z) and integer powers

The complex logarithm is naturally multi-valued. This calculator returns only the principal value Log(z), not every possible branch.

Power mode accepts integer exponents only, with n limited to the range from −50 to 50. This keeps the result bounded and avoids implying support for fractional or complex exponent branches.

Worked examples

Example 1: if z₁=3+2i and z₂=1−4i, then z₁+z₂=4−2i. The real parts add to 4, and the imaginary coefficients add to −2.

Example 2: if z₁=2+i and z₂=1−i, multiplication gives (2+i)(1−i)=3−i after applying i²=−1.

Example 3: for z₁=3+4i, the modulus is 5 and the argument is approximately 53.13°. This is the bridge from rectangular form to polar form.

Example 4: principal Log(z₁) returns one selected principal value. It should not be read as a list of all logarithm branches.

How this differs from nearby calculators

This page is the general complex-number operations page. If you only need a conjugate and z×z̄ check, the conjugate calculator is narrower. If you need all n-th roots, the complex root calculator is the better match.

• Complex conjugate calculator: focused on z̄ and z×z̄.
• Complex root calculator: lists all n-th roots of a complex number.
• Powers of i calculator: focuses only on the repeating i^n cycle.
• Quaternion calculator: works with a+bi+cj+dk, a larger algebraic structure.

Common mistakes and limitations

• Forgetting that i² equals −1.
• Trying to divide by 0+0i.
• Treating the argument as if branch choice never mattered.
• Reading principal Log(z) as all possible logarithm values.
• Entering a fractional or complex exponent even though only integer powers are supported.