Free Complex Number to Polar Form Calculator

Overview

The Complex Number to Polar Form Calculator is a free online tool designed to convert any complex number from its rectangular form $a + bi$ into the equivalent polar representation. Whether you’re a student learning about complex analysis or an engineer who needs to simplify phasor calculations, this rectangular to polar form calculator serves as a reliable complex number converter and a magnitude and phase calculator combined. It eliminates the manual work of applying square roots and trigonometric functions, giving you the modulus and argument in seconds.

Understanding the Polar Representation

A complex number in rectangular form has two components: the real part $a$ and the imaginary part $b$. In the polar form, these are replaced by two other quantities:

• The modulus $r$ (also called the magnitude or absolute value) – the distance from the origin $(0,0)$ to the point $a+bi$ in the complex plane.
• The argument $\phi$ (or phase) – the angle measured from the positive real axis to the line segment connecting the origin to the point.

The polar representation can be expressed trigonometrically as

$$z = r(\cos\phi + i\sin\phi),$$

or, more compactly, using Euler’s formula:

$$z = r e^{i\phi}.$$

Both forms are equivalent and widely used.

Why Choose the Polar Representation?

Although addition and subtraction are simpler in rectangular form, multiplication, division, and power operations become far more intuitive in polar form. For instance, the product of two complex numbers is simply

$$(r_1 e^{i\phi_1})(r_2 e^{i\phi_2}) = r_1 r_2 \, e^{i(\phi_1+\phi_2)},$$

and division likewise reduces to magnitude division and angle subtraction. This is why polar form is the standard tool for analyzing alternating current circuits, waves, and oscillations.

How to Convert Rectangular to Polar

The conversion relies on two fundamental relations derived from right‑triangle trigonometry and the Pythagorean theorem:

$$r^2 = a^2 + b^2 \quad\Longrightarrow\quad r = \sqrt{a^2 + b^2},$$ $$\tan\phi = \frac{b}{a}.$$

Because the arctangent alone cannot uniquely determine the quadrant of $\phi$, the conversion uses the two‑argument function $\operatorname{atan2}(b, a)$. The following table details how $\phi$ is computed based on the signs of $a$ and $b$:

Condition	Argument $\phi$
$a > 0$	$\arctan(b/a)$
$a < 0,\; b \ge 0$	$\arctan(b/a) + \pi$
$a < 0,\; b < 0$	$\arctan(b/a) - \pi$
$a = 0,\; b > 0$	$\dfrac{\pi}{2}$
$a = 0,\; b < 0$	$-\dfrac{\pi}{2}$
$a = 0,\; b = 0$	undefined

The polar calculator implements exactly this logic, ensuring that the resulting argument always lies in the correct quadrant.

Using the Calculator

Using this a+bi to polar form calculator could not be easier. You enter the real part $a$ and the imaginary part $b$ in the designated input fields. The calculator immediately displays the magnitude $r$ and the argument $\phi$ (selectable in radians or degrees). The result can be copied or used directly in further calculations. If you need the inverse conversion – from polar to rectangular – simply apply $a = r\cos\phi$ and $b = r\sin\phi$; several companion tools are also available for that purpose.

Example: For the complex number $3 + 4i$,

$$r = \sqrt{3^2 + 4^2} = 5,\qquad \phi = \operatorname{atan2}(4,3) \approx 0.9273\ \text{rad} \;(53.13^\circ).$$

Thus the polar form is $5 e^{i0.9273}$ or $5(\cos 0.9273 + i\sin 0.9273)$.

Special Cases and Notes

• Imaginary unit $i$: Since $i = 0 + 1i$, we have $r = 1$ and $\phi = \pi/2$; therefore $i = e^{i\pi/2}$.
• The number zero: $0 = 0+0i$ has magnitude $0$ but its argument is not defined. Because the point lies at the origin, any angle could be assigned, and the polar representation is not unique.
• Negative real numbers: For a negative real number such as $-1$ (i.e., $-1 + 0i$), the magnitude is $1$ and the argument is $\pi$ (or $180^\circ$), so $-1 = e^{i\pi}$.

These examples highlight the ability of the polar coordinates calculator to handle a variety of inputs accurately.

Related Tools

Complex numbers are a cornerstone of mathematics, and various online utilities complement this converter. You may find it useful to also use a general complex number calculator for arithmetic operations, a complex number to trigonometric form converter, or an exponential form tool. Together, these resources cover most tasks you’ll encounter when working with imaginary and complex numbers.