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Complex Number Calculator

Complex numbers extend the real number system by incorporating the imaginary unit i, where i² = −1. Engineers, physicists, and mathematicians routinely work with complex arithmetic when analysing AC circuits, wave mechanics, and signal processing. This calculator handles two complex numbers in rectangular form (real + imaginary parts) or polar form (magnitude and phase), then computes their sum, difference, product, quotient, powers, and natural logarithms with full precision control.

Last updated: April 28, 2026

Creators Jasmine J. Mah, BSc

Reviewers Mateusz Mucha and Anna Szczepanek

568 people find this calculator helpful

Understanding Complex Numbers

A complex number combines a real component and an imaginary component. The standard form is z = a + ib, where a and b are real numbers and i is the imaginary unit satisfying i² = −1.

The real part is denoted Re(z) = a, and the imaginary part is Im(z) = b. A purely imaginary number has zero real part; for example, 3i has Re = 0 and Im = 3.

Complex numbers are essential in fields requiring oscillating or rotating quantities: AC electrical circuits use complex impedance, quantum mechanics relies on complex wavefunctions, and control theory employs complex poles and zeros to characterise system stability.

Magnitude and Phase Angle

Every complex number can be expressed in polar form using its magnitude (absolute value) and phase angle (argument). These two representations—rectangular and polar—are equivalent but useful in different contexts.

The magnitude represents the distance from the origin to the point in the complex plane:

|z| = √(a² + b²)

The phase angle (measured in radians) is found using the inverse tangent function:

φ = atan2(b, a)

The polar form is then written as:

z = |z| × e^(iφ)

a — Real part of the complex number
b — Imaginary part of the complex number
|z| — Magnitude (absolute value) of z
φ — Phase angle in radians, ranging from −π to π

Arithmetic Operations on Complex Numbers

Addition and Subtraction are performed component-wise. For z₁ = a + ib and z₂ = c + id:

z₁ + z₂ = (a + c) + i(b + d)
z₁ − z₂ = (a − c) + i(b − d)

Multiplication uses the distributive property, remembering that i² = −1:

z₁ × z₂ = (ac − bd) + i(ad + bc)

Division requires multiplying numerator and denominator by the complex conjugate of the denominator. The conjugate of z₂ = c + id is c − id.

Powers and Logarithms are typically easier in polar form: raise the magnitude to the power and multiply the phase by the exponent, or apply logarithmic rules to magnitude and phase separately.

Common Pitfalls and Practical Tips

Avoid these frequent mistakes when working with complex numbers:

Phase angle quadrant errors — The arctangent function only returns values between −π/2 and π/2. Use <code>atan2(b, a)</code> instead of <code>atan(b/a)</code> to correctly place the angle in the appropriate quadrant based on the signs of both a and b.
Conjugate in division — When dividing complex numbers algebraically, always multiply both numerator and denominator by the conjugate of the denominator. Forgetting this step is the most common algebraic error and produces incorrect results.
Rounding precision loss — Complex number calculations accumulate rounding errors, especially over multiple operations. Store intermediate results with more precision than your final answer requires, then round only at the end. Specify decimal places or significant figures consistently.
Polar vs. rectangular trade-offs — Multiplication and division are simpler in polar form (multiply/divide magnitudes, add/subtract phases), but addition and subtraction require conversion back to rectangular form. Choose your representation based on the operation sequence.

Example: Natural Logarithm of a Complex Number

Finding the natural logarithm of a complex number exemplifies why polar form is powerful. For z = 5 + 7i:

Calculate the magnitude: |z| = √(5² + 7²) = √74 ≈ 8.602
Calculate the phase angle: φ = atan2(7, 5) ≈ 0.951 rad
Apply the complex logarithm formula: ln(z) = ln|z| + iφ
Substitute: ln(5 + 7i) = ln(8.602) + i(0.951) ≈ 2.152 + 0.951i

This method generalises to any complex number and avoids the algebraic complexity of taking logarithms in rectangular form.

Frequently Asked Questions

How do I convert a complex number from rectangular to polar form?

Start with the rectangular form z = a + ib. Calculate the magnitude using |z| = √(a² + b²) and the phase angle using φ = atan2(b, a). The polar form is then z = |z|e^(iφ). This conversion is useful when multiplying or dividing complex numbers, where polar form simplifies calculations considerably.

Why is the imaginary unit i defined as √(−1)?

The imaginary unit was introduced to solve polynomial equations that have no real solutions, such as x² + 1 = 0. By defining i² = −1, mathematicians could extend the real number system consistently. This extension enabled the development of complex analysis, which is fundamental to engineering, physics, and signal processing. Complex numbers capture both magnitude and phase information essential for describing oscillatory and rotational phenomena.

Can I express every real number as a complex number?

Yes. Any real number a can be written as the complex number a + 0i. The imaginary part is zero, so the number lies on the real axis of the complex plane. This shows that the real numbers are a subset of the complex numbers. Even zero itself can be expressed as 0 + 0i, making it both real and complex.

What is the complex conjugate, and why does it matter?

The complex conjugate of z = a + ib is z* = a − ib. Multiplying a complex number by its conjugate always yields a real number: z × z* = a² + b², which equals |z|². This property is crucial for dividing complex numbers algebraically and for simplifying expressions in signal processing and quantum mechanics.

How does raising a complex number to a power work?

In polar form, raising z = |z|e^(iφ) to the power n yields z^n = |z|^n × e^(inφ). You raise the magnitude to the power and multiply the phase angle by the exponent. In rectangular form, the calculation is more involved; for integer powers, use the binomial theorem or repeated multiplication. Polar form is almost always simpler for this operation.

What causes errors when computing complex number operations on a calculator?

Rounding errors accumulate through multiple steps, especially if intermediate results are truncated prematurely. The phase angle from <code>atan2</code> can be misinterpreted if you assume a simple arctangent—always account for all four quadrants. Division errors often stem from forgetting to multiply by the conjugate. To minimise issues, carry extra precision throughout your calculation and round only the final answer to your desired accuracy.

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1st number: z₁ = a + bi

Real part (a)

Imaginary part (b)

Magnitude |z₁|

Phase (φ₁)

2nd number: z₂ = c + di

Real part (c)

Imaginary part (d)

Magnitude |z₂|

Phase (φ₂)

Operations on z₁ and z₂

Rounding type

Precision

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