Free Complex Number to Rectangular Form Calculator

When you encounter a complex number written in polar form—characterized by a magnitude $r$ and an argument $\phi$ (or angle)—you may need to convert it to the standard rectangular (also called Cartesian or A+Bi) notation. This task appears frequently in fields like engineering, physics, and applied mathematics. The polar to rectangular conversion is straightforward with the help of a dedicated complex number converter. Whether you refer to it as a Polar Complex Number Converter or an A+Bi Calculator, the tool instantly supplies the real and imaginary components.

The Rectangular Form of a Complex Number

The rectangular form of a complex number is expressed as $z = a + bi$, where $a$ represents the real part and $b$ the imaginary part. Geometrically, this corresponds to the point $(a, b)$ on the complex plane: move $a$ units along the horizontal (real) axis and $b$ units along the vertical (imaginary) axis. This representation is intuitive for algebraic operations such as addition, subtraction, and multiplication.

Conversion from Polar to Rectangular Form

Given a polar (exponential) number $z = r e^{i\phi}$ (or $z = r \angle \phi$), the conversion relies on basic trigonometry. From the definitions of cosine and sine relative to the right triangle formed by the magnitude and the axes:

$$\cos\phi = \frac{a}{r}, \qquad \sin\phi = \frac{b}{r}.$$

Solving for the rectangular coordinates yields:

$$a = r \cos\phi, \quad b = r \sin\phi.$$

Thus, the rectangular form becomes $z = r\cos\phi + i\, r\sin\phi$, which simplifies to $a + bi$. This pair of formulas forms the core of every polar to rectangular form conversion, and it is exactly what the online calculator uses behind the scenes.

Example: For the complex number $z = e^{i\pi/3}$ (magnitude $1$, argument $\pi/3$):

$$a = 1 \cdot \cos\frac{\pi}{3} = \frac{1}{2}, \qquad b = 1 \cdot \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}.$$

Hence, its rectangular form is $\frac{1}{2} + i\frac{\sqrt{3}}{2}$. The same logic applies for any magnitude and argument.

How to Use the Polar to Rectangular Converter

Operating this complex number converter is straightforward. Enter the magnitude (or modulus) into the first field and the argument (phase angle) into the second field. You can choose between radians and degrees for the angle—the converter accommodates both. As soon as the values are supplied, the tool displays the real part $a$ and the imaginary part $b$ in the familiar $a+bi$ format. No manual calculation is necessary; the A+Bi calculator provides the result immediately.

This online polar complex number converter eliminates errors and saves time, whether you are learning complex numbers or solving professional problems. Once you have the rectangular form, you can proceed with further arithmetic operations with confidence.