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Imaginary number

An imaginary number is one that when squared gives a negative result.

Normally, with real numbers, when you square them, you always get a positive result.
For example

22 = 4
and
(–3)2 = 9
Recall: A negative times a negative is positive.

With imaginary numbers, when you square them, the answer is negative. They are written like a real number, but with the letter i after them, like this:

23iThe letter i means it is an imaginary number.

What i really means

The letter i is a number, which when multiplied by itself gives -1. This means that

This makes imaginary numbers very useful when we need to find the square root of a real negative number. For example we normally cannot find the square root of say –16.   But using imaginary numbers we can: We understand this imaginary number result as "4 times the square root of negative one".

Remember: real and imaginary numbers are not "like" quantities. You cannot say, add a real to an imaginary. They are separate types of number.

They are used in complex numbers

If you pair a real number with an imaginary number, you get a thing called a complex number, which can be plotted on a two-dimensional plane. They look like this: See Complex Numbers for more.

Pages referring to 'imaginary-number'

Animated definition of complex numbers and how they can be plotted on a two-dimensional plane
An overview of the types of numbers that are used in math. Links to other pages explaining each type in depth. Explains also that some numbers are not numbers at all.
A real number is a value that represents a quantity along a number line
Definition and meaning of the math word i. Includes a list of other pages that refer to this word
Definition and meaning of the math word j. Includes a list of other pages that refer to this word

Other number topics

Scalar numbers

Counting numbers

Numbers that have factors

Special values