High precision square roots

[BOAS]

Hello,

i'm having trouble evaluating my gamma factor for my special relativity homework, because I need to compute 1 minus a very small number (8.57*10^-13). My calculator treats this value as simply 1, as does Mathematica. Although I don't know much about it, and maybe there's a way to force it to consider the small number.

Are there any methods that I could do by hand?

(I am also looking for a computer method elsewhere, but I figured this might be interesting).

Thanks!

 

[SteamKing]

Evaluating such a difference will be a problem with a standard calculator. Mathematica however should be able to handle floating point calculations to such a precision that you don't get a value of 1 for the difference.

http://reference.wolfram.com/language/ref/SetPrecision.html

 

[BOAS]

Evaluating such a difference will be a problem with a standard calculator. Mathematica however should be able to handle floating point calculations to such a precision that you don't get a value of 1 for the difference.

http://reference.wolfram.com/language/ref/SetPrecision.html

Thanks, that solves the problem at hand.

 

[Vanadium 50]

(1+x)^(1/2) is approximately 1 + x/2.

 

[CellCoree]

Hello,

I can understand the difficulty in evaluating your gamma factor with such a small number. In situations like this, it is important to use high precision methods for calculating square roots. One method you could use by hand is the Babylonian method, also known as the Heron's method, which involves taking an initial guess and iteratively refining it until you reach the desired level of precision. However, this may be quite time consuming and may not be feasible for your homework assignment.

For computer methods, you can try using libraries or functions specifically designed for high precision calculations, such as the GMP library or the mpmath module in Python. These tools allow you to specify the desired level of precision and can handle very small numbers without rounding them off.

I hope this helps. Best of luck with your homework!