Complex Number Calculator – Rectangular & Polar Forms

Complex Number Calculator : Free Complex Number Calculator. Add, subtract, multiply, and divide complex numbers. Instantly view results in rectangular and polar forms with an interactive Argand diagram.”

Complex Number Calculator – Rectangular & Polar Forms | DigitalCalculator

Complex Numbers

Enter two complex numbers ($z_1$ and $z_2$) in standard rectangular form ($a + bi$) and select an operation.

First Number (Z₁)

+
i

Second Number (Z₂)

+
i

Calculated Result (Rectangular)

0 + 0i

Re Im
Auto-scaled Plane
Magnitude (|z| = r)
0
Phase/Argument (θ)
0°
0 rad
Polar Form
0 ∠ 0°
Provided by DigitalCalculator.co.in. Hover over input sections to highlight vectors on the plane.

Master Complex Math with Our Complex Number Calculator

Whether you’re studying advanced mathematics, electrical engineering (AC circuits), or physics, dealing with imaginary numbers can be tedious. Our Complex Number Calculator instantly solves arithmetic and converts between rectangular and polar forms, providing a clear visual Argand diagram to help you intuitively grasp the result.

What is a Complex Number?

A complex number is an extension of the real number system that includes the imaginary unit i (where i² = -1). It is typically written in rectangular form as z = a + bi, where:

  • a is the Real part (plotted on the horizontal X-axis).
  • b is the Imaginary part (plotted on the vertical Y-axis).

By combining real numbers and imaginary numbers, we can solve algebraic equations that have no real solutions, opening the door to powerful concepts in modern physics and signal processing.

Formulas Cheat Sheet

  • Addition (+): (a+c) + (b+d)i
  • Subtraction (-): (a-c) + (b-d)i
  • Multiplication (×): (ac - bd) + (ad + bc)i
    Notice how multiplying two imaginary parts creates a real number!
  • Magnitude (r): √(a² + b²)
    The length of the vector using the Pythagorean theorem.

Frequently Asked Questions

1. What is the difference between Rectangular and Polar forms?
Rectangular form (a + bi) describes a complex number by its horizontal (real) and vertical (imaginary) coordinates. Polar form (r ∠ θ) describes the exact same point, but uses the distance from the center (magnitude, r) and the angle from the positive real axis (phase, θ). Polar form makes multiplication and division incredibly easy!
2. How do I interpret the Argand Diagram?
An Argand diagram is just a standard 2D graph, but instead of X and Y, the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Every complex number is a single point on this plane, drawn as a vector (arrow) starting from the center (0,0). Our calculator highlights Z1 in Blue, Z2 in Green, and the Result in Purple.
3. What does it mean to divide by zero with complex numbers?
Just like in standard arithmetic, dividing by a complex number equal to exactly zero (0 + 0i) is undefined. The denominator’s magnitude is zero, resulting in infinity. Our calculator catches this error to prevent crashing and provides a warning message.
4. What is the complex conjugate?
The complex conjugate of a number a + bi is simply a – bi (you flip the sign of the imaginary part). Conjugates are extremely important when dividing complex numbers by hand, as multiplying a complex number by its conjugate always results in a pure, positive real number (a² + b²).

Related Tags:

#ComplexNumberCalculator #ImaginaryNumbers #PolarForm #RectangularForm #ArgandDiagram #ElectricalEngineering #MathTools
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