Why does the SQRT function in MATLAB return different answers when I supply numbers that appear to be the same?

I create the following vectors, which appear to be the same:

x = -[1; i]

y = [-1; -i]

When I compare these two vectors:

x == y

it reports that the corresponding elements in the two vectors to be equal. However, when I compute the square root of the two vectors:

sx = sqrt(x)

sy = sqrt(y)

I find that the first element is different.

 sx =

         0 - 1.0000i

    0.7071 - 0.7071i

 sy =

         0 + 1.0000i

    0.7071 - 0.7071i

Both answers are correct, but it seems inconsistent that the following does not hold.

 x==y    ==>    sqrt(x)==sqrt(y)

Answer 1

[MathWorks Support Team]

The SQRT function is returning different roots because the first elements of "x" and "y" are not in fact the same. The difference can be observed by using hexadecimal format:

format hex

x,y

which produces:

 x =

    bff0000000000000   8000000000000000i

    8000000000000000   bff0000000000000i

 y =

    bff0000000000000   0000000000000000i

    8000000000000000   bff0000000000000i

The first row of "x" is the hexadecimal representation of "-1-0i", while the first row of "y" represents "-1+0i". Although the "==" incorrectly indicates these two values are identical, situations such as this indicate the difference between these two values.

The following is another situation where the use of +0 and -0 returns different results:

 1/0  ==> Inf

 1/(-0) ==> -Inf

If you would like to remove all -0 parts of each element of a matrix (real and imaginary), use the following command:

z = z.^1;