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Originally posted by Lugia
But shouldn't the square root of 1 be only 1 ? If it says root of 1^2 then it would be +/-1. Cause root(x^2) = +/- x but root(x) means only the positive square root of x^2 which is x.
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If x is over the complex field, then the story changes and you must take both roots since there is no notion of positive or negative in the complex plane.
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Originally posted by spice girl
Well, actually, the definition of |x| = sqrt(x^2)
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This definition of |x| is absolutely correct if x is to be real (over the real field). However, if we allow x to be complex (over the complex field) then that rule doesn't hold anymore. If for example x=i. Then:
LHS = |i| = 1, and
RHS = sqrt(i^2) = sqrt(-1) = i
The reason why |i| is 1 is because the absolute value of a number is really its modulus when in the form z = rcis. i = 1cis(pi/2) and consequently |i| = 1.
The real definition of the "absolute value" is the distance of that number from the origin when represented on an Argand diagram.
