I really dislike the expression "*the* square root of minus one".

Like all non-zero numbers, #-1# has two square roots, which we call #i# and #-i#.

If #x# is a Real number then #x^2 >= 0#, so we need to look beyond the Real numbers to find a square root of #-1#.

Complex numbers can be thought of as an extension of Real numbers from a line to a plane. The unit in the #x# direction is the number #1#. The unit in the #y# (imaginary) direction is the number #i#. So #i# is called the *imaginary unit*.

#i# has the property that #i^2 = -1#.

If #a >= 0# then #sqrt(a)# means the non-negative square root of #a#, which lies on the part of the Real line at and to the right of the origin #0#.

If #a < 0# then we define #sqrt(a)# to be the principal square root of #a#, lying on the positive part of the imaginary (#y#) axis.

So #sqrt(-1) = i# and #-sqrt(-1) = -i#.

This all looks like it is working well, but some things break down for square roots of negative numbers. For example, the identity #sqrt(ab) = sqrt(a)sqrt(b)# does not hold in general:

#1 = sqrt(1) = sqrt(-1 * -1) != sqrt(-1) * sqrt(-1) = -1#