# Raising to half power = PRINCIPAL square root?

by perishingtardi
Tags: complex, power, principal, raising, square
 P: 21 This may seem like a very elementary question...but here goes anyway. When a positive number is raised to the power 1/2, I have always assumed that this is defined as the PRINCIPAL (positive) square root, e.g. $$7^{1/2} = \sqrt{7},$$. That is, it does not include both the positive and negative square rootsL $$7^{1/2} \neq -\sqrt{7} = -7^{1/2}.$$ In complex analysis, however, this doesn't seem to be the case? E.g. we write $$(-1)^{1/2} = \pm i.$$ Have I understood these conventions correctly? I have also been thinking about a similar situation: how in real analysis we think of every positive number as having a single natural logarithm, e.g. $$\ln 2 = 0.693\dotsc,$$ when in fact there are actually infinitely many: $$\ln 2 = 0.693\dotsc + 2\pi n i \qquad (n=0,\pm1,\pm2,\dots).$$