i Calculator

How i Calculator Works?

The i Calculator performs arithmetic on two complex numbers written in rectangular form. A complex number is expressed as $z=a+bi$, where $a$ is the real part, $b$ is the imaginary coefficient, and $i^2=-1$. The calculator accepts the real and imaginary parts of two numbers, $z_1=a+bi$ and $z_2=c+di$, and applies the selected operation.

The resulting rectangular form is assembled from the calculated components:

The modulus, or magnitude, measures the distance of the result from the origin in the complex plane:

The argument is the principal angle between the positive real axis and the result. The calculator determines it using the two-argument arctangent so that the signs of both components identify the correct quadrant:

The calculator also returns the argument in radians and degrees, along with the complex conjugate. For $z_r=x+yi$, the conjugate is:

Practical Use Cases

Students: Students can verify exercises involving addition, subtraction, multiplication, and division of complex numbers. For example, entering $3+4i$ and $1+2i$ with addition produces $4+6i$, while the modulus and argument help connect rectangular calculations with geometric interpretation.

Electrical and electronics engineers: Engineers can use the calculator for intermediate phasor and impedance calculations represented by real and imaginary components. Multiplication and division are especially useful when combining complex quantities, while the modulus, argument, and conjugate provide important checks on the computed result.

Software developers: Developers implementing signal processing, graphics, simulations, or scientific routines can compare program output with independently calculated real parts, imaginary parts, modulus, argument, and conjugate values.

Researchers and scientists: Researchers can quickly inspect complex-valued expressions, test algebraic transformations, and identify the magnitude and phase orientation of a computed result without manually repeating each component calculation.

Accuracy and Limitations

The calculator uses standard double-precision numerical arithmetic for inputs from -1,000,000,000 to 1,000,000,000 and reports the result components, modulus, argument, and conjugate from the selected operation. Results are subject to ordinary floating-point rounding, so very small residual values may appear after cancellation or division. Division is undefined when both components of the second number are zero because $c^2+d^2=0$. The argument is a principal angle determined by $\mathrm{atan2}$, and the argument of the zero complex result is not mathematically defined.

Quick Examples