In my calculator why does $\sqrt4 -2=-8.1648465955514287168521180122928e-39?$

I have tried this on my Windows 7 calculator with $\sqrt9 -3$ it too gives some weird answer- ie$1.1546388020691628168216106791278e-37$. And so for any $n$(positive) $\sqrt n^2-n= wierd_ .answer$
Why does this happen?

• Commented Oct 6, 2013 at 15:44
• Floats only store up to 8 bits of data iirc Commented Oct 6, 2013 at 15:46
• So your calculator is close, but no cigar. Thus use a more reliant calculator. Commented Oct 6, 2013 at 15:47

• I don't think it is a bug, roots are calculated afaik via iterations so the errors may grow extremely. If you have double floating point precision, shouldn't the largest machine number smaller than 2 be something like $2-10^{-16}$ and not something like $2-10^{-39}$ ? Commented Oct 6, 2013 at 16:02
• I understand why it happens, but it's still wrong - it does not meet the specification. The makers probably didn't care and don't consider it a bug, though, but there I'm guessing. As for the the $2 - 10^{-16}$ vs. $2 - 10^{-39}$, it's probably extended double instead of double (128 bits instead of 64), but I didn't check. Commented Oct 6, 2013 at 16:05
The weird answer $-8.16\ldots e-39$ stands for $-8.16\ldots \cdot 10^{-39}$. It happens because a calculator just calculates numerical, so the maximal precision is the so called machine precision, which is usally something about $10^{-16}$