The letter** i** is the symbol for the square root of **-1**. In other words **i** **= √-1**. The symbol **i** often appears in the equations of quantum mechanics.

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. So, it’s called an “imaginary number” or “impossible number” and symbolized as** i**.

What if we’re calculating, and **√-1** shows up as the answer to an algebra problem? Could happen. For example, the equation **x ^{2} = -4** has the solution

**2(√(-1))**, that is,

**2i**.

For students in many algebra classes, it would be a reason to check for a calculation error. Algebra teachers generally won’t give students problems that involve imaginary numbers. And if the student finds that she has done all her calculations correctly, and she still gets such an answer, she may be told that this answer isn’t allowed. Just as dividing by **0** isn’t allowed.

But in more advanced math, mathematicians and physicists work with **i** without blinking an eye (so to speak). Over the last couple of centuries, both mathematicians and physicists have found **i** to be useful when solving equations. It is, for example, useful in solving **x ^{2} = -4**, even if the solution

**2(√(-1))**challenges conceptualization.

An entire set of imaginary numbers can be created by multiplying **i **times an ordinary number, for example:

**3i 4½ i -7.35 i**

## Imaginary Numbers and Real Numbers

Ordinary numbers are called “real numbers.” These are all the numbers that could be used to express distances measured by a ruler, for example:

**0 1 4½ 7.9543 ** or **-1 -2¾ -5.6947**

A real number can be combined with an imaginary number, for example:

**3 + i -14 + 4i 5 – 2.7i**

## Imaginary Numbers and Complex Numbers

A combination of both real and imaginary numbers is called a “complex number.” Thus, **3 + i **is a complex number.** **Many of the calculations of quantum mechanics involve complex numbers.