# Imaginary number

**Imaginary numbers** are numbers that are made from combining a real number with the imaginary unit, called *i*, where *i* is defined as [math]i^2=-1[/math]. They are defined separately from the negative real numbers in that they are a square root of a negative real number instead of a positive real number. This is not possible with real numbers, as there is no real number that will multiply by itself to get a negative number (e.g. 3*3 = 9 and -3*-3 = 9).

One way of thinking about them is to say that **imaginary numbers** are to **negative numbers** what negative numbers are to **positive numbers**. If I say "go east by -1 mile" it is the same as if I had said "go west by 1 mile". If I say "go east by i miles" it means the same thing as if I had said "go north by 1 mile". If I say "go east by -i mile" it means the same as if I had said "go south by 1 mile".

**Adding** is easy too. If I say "go east by 1 + i miles" it means the same as if I had said "go east by one mile and north by one mile".

**Multiplying** two imaginary numbers is a lot like multiplying a **positive number** with a **negative number**. If I say "go east by 2*-3 miles" it means "rotate all of the way around (so that you are now facing west) and go 2*3 = 6 miles". Imaginary numbers work the same, except that you can rotate part way. If I say go "east by 2*3i miles", it means the same as if I had said "rotate until you are facing north, and then go 2*3 = 6 miles"

Subtracting 5 - 9 used to be impossible until negative numbers were invented. After they were, taking the square root of a negative number used to be impossible until imaginary numbers were invented. The square root of 9 is 3, but the square root of −9 is not −3. This is because −3 x −3 = +9, not −9. For a long time it seemed as though there was no answer to the square root of −9.

This is why mathematicians invented the imaginary number, *i*, and said that it is the square root of −1. The square root of −1 is not a real number, so this definition creates a new type of number, just like fractions create numbers like 2/3 that are not counting numbers like 4 or 10, and negative numbers let us have numbers less than 0. Sometimes, mathematicians seem rather comfortable using a number that is so unusual, but the name *imaginary* should not fool you because *i* is as valid a number as 3 or 145,379.

Many branches of science and engineering have found uses for this number. Sometimes electrical engineers need *i* to understand how an electric circuit will work when they are designing it (electrical engineers use *j* instead of *i* to avoid confusion with the symbol for the current). Certain branches of physics such as quantum physics and high energy physics use *i* as often as they use any other regular number. Many equations in the world simply cannot be solved without *i*.

Imaginary numbers can be mixed with numbers we are more familiar with. For example a **real number** such as *2* can be added to an **imaginary number** such as *3i* to create *2+3i*. These kinds of mixed numbers are known as complex numbers.