This is an example of a complex number: 3 + 4i. It means take 3 and add 4 times i. The letter i is the symbol for the square root of -1 or √(-1). In other words, the complex number 3 + 4i means 3 plus the quantity 4 times the square root of -1.

A complex number has two parts: an ordinary part and a part that includes the letter i. For example, the complex number 3 + 4i includes the ordinary part, 3. Ordinary numbers are called “real numbers.” These are all the numbers that could be used to express distances measured by a ruler, as examples:

0            1            4½        7.9543       and       -1          -2¾        -5.6947

Complex Numbers Include Imaginary Numbers

The number 3 + 4i also includes an “imaginary” part. An imaginary number is one that includes the symbol i. As noted above, the symbol i represents the square root of –1, written √(-1). What number could be the square root of -1? It would have to be a number that when squared, that is, multiplied by itself, equals -1. No ordinary number fits the bill. Nevertheless, we can write it and calculate with it to solve equations. For example, the equation x² = -4 has the solution 2√(-1), that is, 2i.

An entire set of imaginary numbers can be created by multiplying i times a real number, for example:

                                        3i          4½ i        -7.35 i

Since the early 1900’s, complex numbers have played an important role in the calculations of quantum mechanics.