Free Online Complex Number Calculator
Add, subtract, multiply, divide, and convert complex numbers.
Enter numbers as a+bi, such as 3+4i, 2-i, 5
or -2.5i. Spaces are ignored. The imaginary unit is the letter i.
Step-by-step working
- Enter two complex numbers to see the calculation.
About the Complex Number Calculator
Complex numbers extend the real number line with the imaginary unit i, defined by i² = −1. A complex number has a real part and an imaginary part and is written a + bi. This calculator parses two complex numbers, performs addition, subtraction, multiplication or division, and reports the result, magnitudes, the conjugate of the first number and the polar form of both the input and the result. It is a quick companion for algebra, signal processing and electrical engineering.
Parsing the a + bi form
The parser accepts ordinary decimal notation, negative signs, a missing real part, a missing imaginary coefficient and even a lone i. For example, 3+4i has real part 3 and imaginary part 4; 2-i means 2 − 1i; 5 is a real number with zero imaginary part; and -2.5i is purely imaginary. Exponents such as 1e-3 are supported. Invalid text is rejected with a clear message rather than being silently treated as zero. Spaces are removed before parsing, so “3 + 4i” and “3+4i” behave the same.
The four arithmetic operations
Addition and subtraction combine real parts and imaginary parts separately. Multiplication uses the distributive law and the identity i² = −1, giving (ac − bd) + (ad + bc)i. Division multiplies numerator and denominator by the conjugate of the denominator, which makes the denominator real: (a + bi)/(c + di) = ((ac + bd) + (bc − ad)i) / (c² + d²). If the denominator is zero, division is undefined and the calculator stops with an error. Each step lists the substituted values.
Magnitude, conjugate and polar form
The magnitude of a + bi is √(a² + b²), the distance from the origin in the complex plane. The conjugate flips the sign of the imaginary part: the conjugate of 3 + 4i is 3 − 4i. Multiplying a number by its conjugate gives the square of its magnitude. Polar form expresses the same point as r(cos θ + i sin θ), where r is the magnitude and θ is the angle from the positive real axis, measured in degrees here. This form makes multiplication and division intuitive: magnitudes multiply or divide and angles add or subtract.
Practical interpretation
In electrical engineering, complex numbers represent impedance, with resistance as the real part and reactance as the imaginary part. In signal processing, they represent amplitude and phase. In mathematics, they provide the roots that real numbers cannot, such as the square root of a negative value. The angle is reported in degrees with atan2 semantics, so it lands between −180 and 180 degrees; a negative angle means the point lies below the real axis. Rounding is applied only for display.
Frequently asked questions
Can I write j instead of i?
No, this parser expects the letter i. Engineers often use j for current, but converting is simple: replace every j with i before entering the number.
What happens when I divide by zero?
Division by the complex number 0 + 0i is undefined because the denominator c² + d² becomes zero. The calculator reports the problem and leaves the result blank instead of returning infinity.
Why are there two angle conventions?
Angles can be measured in degrees or radians, and the range can be 0 to 360 or −180 to 180. This tool uses degrees and the −180 to 180 range from atan2, which is the most common convention in engineering calculators.