What does this complex number calculator do?

This complex number calculator lets you add, subtract, multiply and divide two complex numbers. It also converts between rectangular form a + bi and polar form r ∠ θ, and computes magnitude, argument and complex conjugate. For each mode it shows both rectangular and polar representations and explains the intermediate steps.

What is the magnitude and argument of a complex number?

For a complex number z = a + bi, the magnitude (or modulus) is |z| = sqrt(a^2 + b^2), which is the distance from the origin in the complex plane. The argument is the angle θ such that tan θ = b / a, usually computed with a two argument arctangent function to place it in the correct quadrant. The principal argument is often taken in the interval −π to π radians, or −180° to 180°.

What is the difference between rectangular and polar form?

Rectangular form writes a complex number as a + bi using horizontal and vertical components on the complex plane. Polar form writes the same number as r ∠ θ, where r is the magnitude and θ is the angle from the positive real axis. Addition and subtraction are more natural in rectangular form, while multiplication and division become especially simple in polar form.

How accurate are the results of this complex number calculator?

The calculator uses standard double precision floating point arithmetic, which is accurate for most educational and engineering purposes. Results are rounded for display using a sensible number of significant figures. For very sensitive numerical work, you should always cross check with a computer algebra system or arbitrary precision library.

What happens to the argument for the zero complex number?

For the zero complex number 0 + 0i, the magnitude is zero and the argument is mathematically undefined because there is no specific direction from the origin. The calculator reports magnitude zero and indicates that the argument is not defined for the zero complex number.

Complex Number Calculator – Rectangular & Polar Form, Magnitude, Operations

Q: What is a complex number?

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i^2 = −1. The part a is called the real part and b is the imaginary part. Complex numbers extend the real number line to the complex plane and are essential in advanced algebra, calculus, engineering, and signal processing.

Work with complex numbers in rectangular and polar form. Add, subtract, multiply, divide, convert between forms, and compute magnitude, argument and conjugate – with clear numeric output and step-by-step explanation.

What is a complex number?

A complex number is a number of the form

\[ z = a + bi, \]

where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with \(i^2 = -1\). The real part is \(\Re(z) = a\), and the imaginary part is \(\Im(z) = b\).

Complex numbers can be represented as points or arrows (vectors) on the complex plane: the horizontal axis corresponds to the real part, and the vertical axis corresponds to the imaginary part.

Rectangular form and polar form

The form \(z = a + bi\) is called rectangular (or Cartesian) form. For many operations, especially addition and subtraction, rectangular form is the most natural.

Complex numbers can also be written in polar form:

\[ z = r (\cos \theta + i \sin \theta) = r \angle \theta, \]

where the modulus (or magnitude) \(r = |z|\) and the argument \(\theta = \arg(z)\).

Using Euler's formula \(e^{i\theta} = \cos \theta + i \sin \theta\), we can also write

\[ z = r e^{i\theta}. \]

Magnitude and argument

If \(z = a + bi\), then:

\[ |z| = \sqrt{a^2 + b^2} \] \[ \theta = \arg(z) = \operatorname{atan2}(b, a) \]

Here \(\operatorname{atan2}(b,a)\) is the two-argument arctangent that automatically selects the correct quadrant. The principal argument is usually taken in the interval \((-\pi, \pi]\) (in radians) or \((-180^\circ, 180^\circ]\) (in degrees).

For the special case \(z = 0\), we have \(|z| = 0\) and the argument is undefined, since the zero vector has no direction.

Basic operations with complex numbers

Addition and subtraction

For \(z_1 = a_1 + b_1 i\) and \(z_2 = a_2 + b_2 i\):

\[ z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i \] \[ z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i \]

Geometrically, addition corresponds to vector addition in the complex plane.

Multiplication

In rectangular form:

\[ z_1 z_2 = (a_1 + b_1 i)(a_2 + b_2 i) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1)i. \]

In polar form, multiplication is especially simple. If \(z_1 = r_1 \angle \theta_1\) and \(z_2 = r_2 \angle \theta_2\), then

\[ z_1 z_2 = (r_1 r_2) \angle (\theta_1 + \theta_2). \]

Division

For non-zero \(z_2\), division in rectangular form is:

\[ \frac{z_1}{z_2} = \frac{a_1 + b_1 i}{a_2 + b_2 i} = \frac{(a_1 + b_1 i)(a_2 - b_2 i)}{a_2^2 + b_2^2} = \frac{a_1 a_2 + b_1 b_2}{a_2^2 + b_2^2} + \frac{b_1 a_2 - a_1 b_2}{a_2^2 + b_2^2} i. \]

In polar form:

\[ \frac{z_1}{z_2} = \left(\frac{r_1}{r_2}\right) \angle (\theta_1 - \theta_2), \]

again highlighting why polar form is natural for division and phasor calculations.

Complex conjugate

The complex conjugate of \(z = a + bi\) is

\[ \overline{z} = a - bi. \]

The product \(z \overline{z} = a^2 + b^2 = |z|^2\) is always real and non-negative, which is used in the formula for division and in many engineering derivations.

Worked example: rectangular to polar

Take \(z = -3 + 4i\).

Compute the magnitude: \[ |z| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5. \]
Compute the argument using \(\operatorname{atan2}\): \[ \theta = \operatorname{atan2}(4, -3) \approx 2.2143\ \text{rad} \approx 126.87^\circ. \]
Write the polar form: \[ z = 5 \angle 126.87^\circ. \]

Good practice when using complex number calculators

Always double-check the sign of the imaginary part when entering rectangular form.
Be clear about whether angles are in radians or degrees; converting incorrectly is a common source of error.
For engineering phasors, prefer degrees if that is the convention in your field, but radians in advanced mathematics and physics.
When results matter for safety or design (e.g. control systems, power systems), validate with a second method or software package.

Related complex and algebra tools

For more advanced work with algebra and complex quantities, explore:

Complex Number Calculator

Interactive complex number calculator

z₁ = a₁ + b₁i and z₂ = a₂ + b₂i

First complex number z₁

Second complex number z₂

Rectangular form z = a + bi

Polar form z = r ∠ θ

Results