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Imaginary numbers

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Imaginary numbers are complex numbers having no real part; that is, if is an imaginary number ( is the imaginary unit), then . For example, is an imaginary number. The term "imaginary" is no longer a comment on the metaphysical quality of whether these numbers exist outside the imagination of those who first proposed their use; as the initial prejudice was overcome, the term was retained purely as a matter of convenience.

Imaginary numbers are the square roots of negative numbers: for example, , and conversely .

The square root of a "positive" imaginary number is the same as that of half the corresponding positive real number but multiplied by , e.g., . Likewise for the square root of a "negative" imaginary number, but with a factor of rather than : . In a nutshell, this means that the complex numbers are closed under the operation of square root, and thus there is no need to invent "super-imaginary" numbers to account for their square roots.