I tried doing just the absolute value of i by itself on a calculator and I got "one", so wouldn't that mean "i" has a value of 1 or negative one, and not the square root of negative 1?

 

And then I even did the square root of i by itself and got ".7071067812+.70..."

 

What's going on here?

Edited December 24, 201114 yr by questionposter

The absolute value of a complex number [math]z = x + iy[/math] is defined as [math]|z| = \sqrt{x^2 + y^2}[/math], so the absolute value of i is [math]|i| = |0 + 1i| = \sqrt{0 + 1^2} = 1[/math].

The value you got for the square root of i looks correct, assuming the part you elided ends in i.

The exact value is (1 + i)/√2.

The value you got for the square root of i looks correct, assuming the part you elided ends in i.

The exact value is (1 + i)/√2.

 

There is no convention regarding square roots of complex numbers as there is for square roots of positive real numbers (i.e. the square root is taken to be the the positive one in most cases), so the square root of i is not well-defined and either

 

(1 + i)/√2

 

or

 

 

(-1 - i)/√2

 

are roots that fit the bill.