## Complex Numbers / Imaginary Numbers

What are **complex numbers**? Complex numbers are *ordered pairs* of real numbers (a, b), where a is called the **real part** and b is called the **imaginary part**. Instead of denoting a complex number with an ordered pair, it is customary to combine the pairs with a plus sign and denote the resulting complex number as a+b**i**, where i has the property that i^{2}= -1. in other words, i = √-1. Because there is no real number with negative square, i is called the **imaginary unit**. **Imaginary numbers** are complex numbers whose real parts are zero.

Note: Both the real part and the imaginary part are *Real numbers*. The imaginary part is so called because it is the coefficient of the imaginary unit i.

To do arithmetic/algebra with complex numbers, all the laws and rules, including the commutative, associative and distributive laws, which we use in conjunction with real number system, are also applicable to the complex number system. This makes the basic binary operations of addition, subtraction, multiplication and division easy to do.

### Adding & Subtracting Complex Numbers / Imaginary Numbers

To add or subtract two complex numbers a+bi & c+di just add or subtract the corresponding real and imaginary parts of them. That is, (a+bi) + (c+di) = (a+c) + (b+d)i (a+bi) - (c+di) = (a-c) + (b-d)i

### Multiplying Complex Numbers / Imaginary Numbers

To multiply two complex numbers a+bi & c+di use the the usual laws of algebra keeping in mind that i*i = -1. (a+bi)(c+di) = (ac - bd) + (ad + bc)i

### Dividing Complex Numbers / Imaginary Numbers

To divide two complex numbers a+bi & c+di, first multiply by (c-di)/(c-di) and use the usual laws of algebra keeping in mind that i*i = -1.
(a+bi)/(c+di) = (a+bi)(c-di)/(c^{2} + d^{2})
= ((ac + bd) + (-ad + bc)i)/(c^{2} + d^{2})