How do you calculate imaginary numbers?

Answer 1

Usually you don't use pure imaginary numbers, but complex numbers - numbers that have a real and an imaginary part.

To add and subtract complex numbers, you add and subtract the components (just like any other vector). That is, you add (or subtract) the real part and the imaginary part separately.

To multiply them, you multiply the components, just like you would multiply any two polynomials - multiply each part of the first number by each part of the second number. Remember that i2 = -1. Combine the real and the imaginary parts.

To divide, consider the division as a fraction, and multiply top and bottom by the complex conjugate. For example, if you are dividing by (2 + 3i), multiply numerator and denominator by (2 - 3i). This will convert the denominator into a real number.

Another way to multiply is to convert the complex numbers into polar coordinates (absolute value, i.e., length, and angle). Multiply the absolute values (which are real numbers) together, and simply add the angles. For example, (3 angle 30°) x (4 angle 20°) = (12 angle 50°). You may need to convert the result back to rectangular coordinates in the end. Note that scientific calculators usually have an option to quickly convert from rectangular to polar, or polar to rectangular, coordinates.