(12C) Signum Function

[Thomas Klemm]

(03-01-2019 06:29 AM)Gamo Wrote:  

Code:

X=0
GTO 00
ENTER
ENTER
x
√x
÷

This doesn't work for \(\left | x \right |\leqslant 10^{-50}\) or \(\left | x \right | \geqslant 10^{50}\) due to under- or overflow.
Obviously it works for numbers like -113 or 47.

But what happens with a number that uses all the 10 digits like 12.78974364?
When we square this number we get 163.5775424 which is rounded to 10 digits.
The exact value is 163.5775423769204496.

Taking the square root of the rounded value 163.5775424, why can we be sure that we get the original number?
If we received something off by 1 in the last digit, such as 12.78974363 or 12.78974365, we'd end up with 1.000000001 or 0.999999999 instead.
Any thoughts why this can't happen?

The reverse isn't true as we know: taking the square root of 3 gives 1.732050808.
But the square of this number is rounded to 3.000000001.
The exact value is 3.000000001493452864.

Kind regards
Thomas