Introduction
This complex number calculator lets you add, subtract, multiply, divide, and simplify complex numbers with ease. Type any expression using real and imaginary parts, and the tool solves it for you right away. It shows your answer in rectangular, polar, exponential, and cis form so you can use whichever one you need.
You also get a step-by-step solution that walks you through the math, an Argand diagram that plots your result on the complex plane, and a built-in answer checker to verify your own work. Whether you need to find a modulus, compute powers of i, evaluate complex trigonometric functions, or convert between forms, this calculator handles it all. Just enter your expression, hit Calculate, and get your result instantly.
How to Use Our Complex Number Calculator
Enter any complex number expression to get the result in rectangular, polar, exponential, and cis form, along with a step-by-step solution and an Argand diagram.
Complex Expression: Type your complex number expression into the main input field. Use i for the imaginary unit. For example, type (3+2i)*(1-4i) or i^22. You can also use functions like sin(), sqrt(), and exp(), or enter polar form with the angle symbol, like 2∠(pi/3).
Operation Mode: Pick a mode to focus on a specific task. Choose General for any expression, Powers of i for exponents of i, Multiply or Divide for those operations, Conjugate to find the conjugate, Modulus for the absolute value, A+Bi to convert to rectangular form, or Argument to find the angle.
Angle Mode: Select RAD for radians, DEG for degrees, or GRD for gradians. This controls how angles are read and displayed in your results.
Decimal Places: Open the Format panel and drag the slider to set how many decimal places show in your answer. You can pick any number from 0 to 16.
Number Notation: In the Format panel, choose Auto, Fixed, Scientific, or Engineering to change how numbers are written in the output.
Calculate As You Type: When this toggle is on, the calculator updates your result instantly as you type. Turn it off if you want to press the Calculate button yourself.
Show All Result Forms: Turn this on to see the polar, exponential, and cis forms alongside the rectangular result. Turn it off to see only the rectangular form.
Virtual Keypad: Click the Keypad button to open or close the on-screen keypad. Use it to insert numbers, operators, functions, and symbols like π and ∠ directly into the input field.
Quick Actions: Click buttons like Simplify, Find Conjugate, Find Modulus, or Convert to A+Bi to instantly wrap your current expression in the matching function and calculate.
Verify Your Answer: Type your own answer into the verify field and click Check Answer. The calculator will tell you if your answer matches the computed result.
What Are Complex Numbers?
A complex number has two parts: a real part and an imaginary part. It is written in the form a + bi, where a is the real part, b is the imaginary part, and i is the square root of −1. Since no real number multiplied by itself gives −1, mathematicians created i to solve this problem.
How Complex Numbers Work
You can add, subtract, multiply, and divide complex numbers just like regular numbers, but you must keep track of the real and imaginary parts separately. For example, to add (3 + 2i) and (1 + 4i), you add the real parts (3 + 1 = 4) and the imaginary parts (2i + 4i = 6i) to get 4 + 6i.
Ways to Write Complex Numbers
Complex numbers can be written in several forms:
- Rectangular form (a + bi): Shows the real and imaginary parts directly.
- Polar form (r∠θ): Uses a distance r from the origin and an angle θ. This is also called phasor form and is common in engineering.
- Exponential form (reiθ): Based on Euler's formula, which connects exponentials to trigonometry.
- Cis form (r·cis θ): A shorthand for r(cos θ + i sin θ), meaning the same thing as exponential form.
Key Terms
The modulus (or absolute value) of a complex number is its distance from the origin on the complex plane. You find it with the formula r = √(a² + b²), which follows directly from the Pythagorean theorem. The argument is the angle the number makes with the positive real axis, which you can compute using the inverse tangent. The conjugate of a + bi is a − bi: you just flip the sign of the imaginary part. Conjugates are useful when dividing complex numbers because multiplying a number by its conjugate removes the imaginary part from the denominator.
The Complex Plane
Complex numbers can be plotted on a graph called the complex plane or Argand diagram. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Each complex number becomes a point on this plane, and you can draw an arrow (vector) from the origin to that point. This makes it easy to see the modulus and argument visually.
Powers of i
The imaginary unit i follows a repeating pattern when raised to higher powers: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then the cycle starts over. So to find any power of i, just divide the exponent by 4 and look at the remainder.
Where Complex Numbers Are Used
Complex numbers are used in many fields. Engineers use them to study electrical circuits and signals, including applications with parallel resistors and AC analysis. Physicists use them in quantum mechanics. Mathematicians use them to solve equations that have no real solutions, like x² + 1 = 0. They also appear in computer graphics, control systems, and signal processing.