Polar Vs Rectangular Converter

Math Algebra • Complex Numbers

Written by STEM Calculators Team Published December 20, 2025 Updated February 24, 2026

2. Polar Vs Rectangular Converter

Convert complex numbers between rectangular form \(a+bi\) and polar/exponential form \(r e^{i\theta}\), with principal argument and an Argand plot.

Mode: Angle unit (input θ):

Rectangular input

Re = a Im = b

Interprets \(z=a+bi\).

Polar / exponential input

Modulus r Angle θ

Interprets \(z=r e^{i\theta}\) with \(r\ge 0\) (normalized if needed).

Show principal argument \((- \pi < \theta \le \pi)\) Equal axis units (square grid) Label the plotted point Show angle arc

Ready

Drag to pan • wheel/pinch to zoom • Output includes \(r, \theta\) (rad & deg), plus \(r(\cos\theta+i\sin\theta)\) and \(r e^{i\theta}\).

Multi-conversion table (list → table)

Bulk mode: List input:

Bulk: idle

Bulk results will appear here.

Argand plot

Vector from origin to \(z=(a,b)\). Shows angle \(\theta\) from +Re axis (when enabled).

Re: 0, Im: 0 sx: 70, sy: 70

Enter values and click Convert.

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Theory: Rectangular ↔ Polar / Exponential Form

A complex number can be represented in rectangular form \(z=a+bi\) or in polar/exponential form \(z=re^{i\theta}\), where \(r\ge 0\) is the modulus and \(\theta\) is an argument (an angle).

1) Rectangular → Polar

Given \(z=a+bi\), the modulus is the distance from the origin:

\[ r = |z| = \sqrt{a^2+b^2}. \]

The argument is obtained using a quadrant-safe angle:

\[ \theta = \operatorname{atan2}(b,a). \]

Then you may write either trig form or exponential form:

\[ z = r(\cos\theta + i\sin\theta) = r e^{i\theta}. \]

If \(z=0\), then \(r=0\) and the argument is undefined (many conventions use \(\theta=0\) for display).

2) Polar → Rectangular

Starting from \(z=re^{i\theta}\), the rectangular components follow from cosine and sine:

\[ a = r\cos\theta,\qquad b = r\sin\theta. \]

Standard polar form uses \(r\ge 0\). If a negative \(r\) is entered, you can convert it by flipping the sign and shifting the angle: \( (-r)e^{i\theta} = r e^{i(\theta+\pi)}\).

3) Principal argument and all arguments

A complex number has infinitely many arguments:

\[ \theta = \theta_0 + 2k\pi,\quad k\in\mathbb{Z}. \]

The principal argument \(\operatorname{Arg}(z)\) is the unique angle chosen in the interval \((- \pi, \pi]\). The calculator displays this when the “principal argument” option is enabled.

4) Worked example

Convert \(z = 3 + 4i\) to polar form.

\[ r=\sqrt{3^2+4^2}=\sqrt{9+16}=5. \]

\[ \theta=\operatorname{atan2}(4,3)\approx 0.9273\ \text{rad} \quad (\approx 53.13^\circ). \]

\[ z = 5(\cos\theta+i\sin\theta) = 5e^{i\theta} \approx 5e^{i\cdot 0.9273}. \]

5) Multi-conversion table idea

For practice problems, you often need to convert many numbers. The bulk mode accepts a list and produces a table of \(r\), principal \(\theta\) (in rad/deg), and the converted form.

University extension: phase “unwrapping” chooses a continuous sequence of angles rather than repeatedly jumping by \(2\pi\).

Frequently Asked Questions

How do I convert a+bi to polar form?

Use r = sqrt(a^2 + b^2) and theta = atan2(b, a). The polar/exponential form is r e^(i theta), and the trig form is r(cos theta + i sin theta).

How do I convert polar form r e^(i theta) to rectangular form?

Compute a = r cos(theta) and b = r sin(theta). The rectangular form is a + b i.

What is the principal argument of a complex number?

The principal argument Arg(z) is the unique angle chosen in the interval (-pi, pi]. A complex number has infinitely many arguments that differ by 2k pi.

Why does the calculator use atan2(b, a) instead of arctan(b/a)?

atan2(b, a) selects the correct quadrant automatically and handles cases where a is 0. This prevents incorrect angles that can occur with b/a alone.

Can I convert many complex numbers at once?

Yes, use the bulk mode to paste a list (rectangular or polar depending on the selected bulk option) and run a list-to-table conversion to get results for each entry.