**Imaginary numbers** are numbers that are made from combining a real number with the imaginary unit, called *i*, where *i* is defined as {\displaystyle i^{2}=-1}. 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. {\displaystyle 3\times 3=9} and {\displaystyle -3\times -3=9}). The set of imaginary numbers are sometimes denoted by the letter {\displaystyle \mathbb {I} }.

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

Adding is easy too. If we say "go east by 1 + i miles" it means the same as if we 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 we 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 we say go "east by 2×3i miles", it means the same as if we had said "rotate until you are facing north, and then go 2×3 = 6 miles"

Subtractions such as 5 - 9 used to be impossible until negative numbers were invented, just as 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 main 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 create numbers that are 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. For example. 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). As another example. certain branches of physics such as quantum physics and high energy physics use *i* as often as they use any other regular number. Many of the 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.

Related pages
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* Complex plane
* Gerolamo Cardano