In my recent working paper I needed to calculate

\[v=c\sqrt{1-\frac{l_p^2}{\bar{\lambda}_e^2}}\]

well c is known so basically I needed a numerical value for \(\sqrt{1-\frac{l_p^2}{\bar{\lambda}_e^2}}\)

where \(l_p\) is the Planck length and \(\bar{\lambda}_e\) is the reduced Compton wavelength of the electron.

First I tried

N [Sqrt[1 (1.616 * 10^ (-33))^ 2/(3.862* 10^ (-13))^ 2], 50]

This did not give me the precision needed, but when I changed to write it on the following form, then mathematica gave me the precision needed. I played around with a few other ways that confirmed the result.

N [Sqrt[1 (1616*10^ (-36))^ 2/(3862 *10^ (-16))^ 2], 50]

Still I wonder why the N [Sqrt[1 (1.616 * 10^ (-33))^ 2/(3.862* 10^ (-13))^ 2], 50] not could handle it? It return 1 while the answer should be .99999999999999999999999999999999999999999999912416

I could naturally have increased precision further by adding more known digits to the Planck length and the reduced Compton wavelength of the electron, but I was mostly interested in when the first few digits goes down from 9 to see if LHC was anything close to these velocities needed in according to my theory. Well here a electron, in my paper also a proton.

Well my main question why can N [Sqrt[1 (1.616 * 10^ (-33))^ 2/(3.862* 10^ (-13))^ 2], 100] not handle it? only returns 1.

any other simple ways to do this in mathematica?