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# Imaginary numbers

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