Complex Number Calculator
Perform arithmetic operations on complex numbers, convert between rectangular and polar forms, calculate powers using De Moivre's theorem, and find all nth roots.
Complex Arithmetic
Add, subtract, multiply, divide complex numbers or find conjugate
How to Use the Complex Number Calculator
Enter complex numbers in rectangular form (a + bi) by providing the real and imaginary parts separately. Select the operation type and click Calculate to see results in multiple formats.
Calculator Modes
Arithmetic Operations
Perform basic operations on two complex numbers:
- Addition: (a + bi) + (c + di) = (a+c) + (b+d)i
- Subtraction: (a + bi) - (c + di) = (a-c) + (b-d)i
- Multiplication: (a + bi)(c + di) = (ac-bd) + (ad+bc)i
- Division: Uses the conjugate to rationalize the denominator
- Conjugate: The conjugate of (a + bi) is (a - bi)
Form Conversion
Convert between three representations of complex numbers:
- Rectangular: a + bi (Cartesian form)
- Polar: r ∠ θ where r is magnitude and θ is angle
- Exponential: re^(iθ) using Euler's formula
Powers (De Moivre's Theorem)
Raise a complex number to any integer power. De Moivre's theorem states: (r∠θ)^n = r^n ∠ nθ. This makes power calculations much simpler in polar form.
Roots
Find all n distinct nth roots of a complex number. A complex number has exactly n nth roots, evenly spaced around a circle in the complex plane.
What is a Complex Number?
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1). Complex numbers extend the real numbers and are essential in many areas of mathematics, physics, and engineering.
Common Applications
- Electrical engineering (AC circuit analysis)
- Signal processing and Fourier transforms
- Quantum mechanics
- Control systems
- Fractal geometry (Mandelbrot set)
Key Formulas
- Magnitude: |z| = √(a² + b²)
- Angle: θ = arctan(b/a)
- Euler's formula: e^(iθ) = cos(θ) + i·sin(θ)
Worked Example: Dividing Complex Numbers
To compute (3 + 2i) ÷ (1 − i), multiply numerator and denominator by the conjugate of the denominator, (1 + i). The numerator becomes (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 1 + 5i, and the denominator becomes (1 − i)(1 + i) = 1 − i² = 2. So the result is 0.5 + 2.5i. The calculator performs this conjugate step automatically and shows the answer in rectangular, polar, and exponential form.
Frequently Asked Questions
How do I enter a complex number into the calculator?+
Type the real part and the imaginary part into their separate fields — for 3 − 4i, enter 3 as the real part and −4 as the imaginary part. Then pick the operation (add, subtract, multiply, divide, conjugate, convert, power, or roots) and click Calculate to see the result in rectangular, polar, and exponential form.
How do I convert a complex number to polar form?+
Use the Form Conversion mode. The calculator computes the magnitude r = √(a² + b²) and the angle θ = arctan(b/a), adjusted for the correct quadrant. For example, 1 + i converts to √2 ∠ 45°, or equivalently √2·e^(iπ/4) in exponential form.
How does the calculator find the nth roots of a complex number?+
It applies De Moivre's theorem in reverse: the n roots of r∠θ have magnitude r^(1/n) and angles (θ + 360°k)/n for k = 0 to n−1. All n roots are listed, evenly spaced around a circle in the complex plane — for instance, the three cube roots of 8 are 2, −1 + 1.732i, and −1 − 1.732i.
What is i squared, and why does it matter?+
By definition i² = −1, which is what lets complex numbers solve equations like x² + 1 = 0 that have no real solutions. Powers of i cycle every four steps (i, −1, −i, 1), a pattern the calculator exploits when raising complex numbers to integer powers.
Is this complex number calculator free and does it work on mobile?+
Yes. It is free, requires no account, and all computation happens locally in your browser — nothing is uploaded. The layout is responsive, so it works well on phones and tablets for engineering coursework, circuit analysis homework, or quick checks during an exam review.
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