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 **x ^{2} = -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.