Question 1

What is a complex number?

Accepted Answer

A complex number is written as a + bi where a is the real part, b is the imaginary part, and i is the imaginary unit with the property i^2 = -1. Examples include 3 + 4i and -2 - i.

Question 2

How do you multiply two complex numbers?

Accepted Answer

Use the distributive law and the fact that i^2 = -1. For (a+bi)(c+di) the result is (ac - bd) + (ad + bc)i. This calculator shows the expanded step-by-step calculation automatically.

Question 3

What are the modulus and argument of a complex number?

Accepted Answer

The modulus (or absolute value) of a+bi is sqrt(a^2 + b^2), which is the distance from the origin. The argument is the angle atan2(b, a) measured in radians from the positive real axis.

Question 4

Can this calculator find square roots of negative numbers?

Accepted Answer

Yes. Because complex numbers extend the real line, the square root of a negative real number is purely imaginary. For example sqrt(-9) = 3i. The calculator handles real, imaginary and fully complex inputs.