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# Imaginary numbers

Imaginary numbers are complex numbers having no real part; that is, if $xi$ is an imaginary number ($i$ is the imaginary unit), then $\Re (xi)=0$ . For example, $-7i$ 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, $(7i)^{2}=-49$ , and conversely ${\sqrt {-49}}=7i$ .
The square root of a "positive" imaginary number is the same as that of half the corresponding positive real number but multiplied by $1+i$ , e.g., ${\sqrt {7i}}={\sqrt {\frac {7}{2}}}+i{\sqrt {\frac {7}{2}}}$ . Likewise for the square root of a "negative" imaginary number, but with a factor of $1-i$ rather than $1+i$ : ${\sqrt {-7i}}={\sqrt {\frac {7}{2}}}-i{\sqrt {\frac {7}{2}}}$ . 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.