# Base-2 exponentiation bug on Citizen SR-270X for $2^{255}$?

According to my calculator, $2^{255} = 0.5$.

Also, for the range $2^x, x\in[254.5, 255.5)$ the result spans $[0.3535533906, 0.7071067812]$. Outside of that, the answers seem correct.

Curiosly, $2^{255.4999999} = 0.7071062911 \approx 1/\sqrt{2}$ in the calculator's opinion, how about that!

Can any other owner of the Citizen SR-270X reproduce and confirm this? Besides that, is such behaviour to be expected and acceptable?

• I'd just like to comment that $255$ represented in binary is $11111111$, which is the two's complement representation for $-1$, so that the bug might be that the calculator is using $8$-bit integers for small numbers and misinterprets $255$. Just speculating, though. (See en.wikipedia.org/wiki/Two%27s_complement ) – J. J. Jul 26 '11 at 19:22
• I know, it was that which prompted me to enter it in the first time. – Cecil Dishwasher Jul 26 '11 at 19:29
• The obligatory xkcd link. – J. M. isn't a mathematician Jul 27 '11 at 2:26

Speculating further, perhaps they have a special exponentiation routine for powers of two, which increments the exponent for some reason, and then divides by $2$. But they forgot to check for overflow when incrementing the exponent.