i

The Imaginary Number $i$ is defined as $i\equiv\sqrt{-1}$. However, for some reason engineers and physicists prefer the symbol j to $i$. Numbers of the form $z=x+iy$ where $x$ and $y$ are Real Numbers are called Complex Numbers, and when $z$ is used to denote a Complex Number, it is sometimes (in older texts) called an ``Affix.''

The Square Root of $i$ is

$$\sqrt{i}=\pm {i+1\over\sqrt{2}},$$ (1)

since

$$\left[{{1\over\sqrt{2}} (i+1)}\right]^2 = {\textstyle{1\over 2}}(i^2+2i+1) = i.$$ (2)

This can be immediately derived from the Euler Formula with $x=\pi/2$,

$$i=e^{i\pi/2}$$ (3)

$$\sqrt{i}=\sqrt{e^{i\pi/2}} = e^{i\pi/4} = \cos({\textstyle{1... ...4}}\pi)+i\sin({\textstyle{1\over 4}}\pi) = {1+i\over\sqrt{2}}.$$ (4)

The Principal Value of $i^i$ is

$$i^i = (e^{i\pi/2})^i = e^{i^2\pi/2} = e^{-\pi/2}= 0.207879\ldots.$$ (5)

See also Complex Number, Imaginary Identity, Imaginary Number, Real Number, Surreal Number

References

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 89, 1996.