What is the concise form of physical constants? (What is the number in the parentheses?)

Question

According to nist :

Proton mass $(m_p)$ :

Numerical value $: \mathrm{1.672 \ 621 \ 923 \ 69 \ × \ 10^{-27} \ kg}$

Concise form $\ \ \ \ \ \ \ \ \ \ : \mathrm{1.672 \ 621 \ 923 \ 69(51) \ × \ 10^{-27} kg}$

I want to store the constant of Proton Mass to use in calculation , but I'm confused which number to store : the $1.672 \ 621 \ 923 \ 69 \ × \ 10^{-27}$ or the $1.672 \ 621 \ 923 \ 69(51) \ × \ 10^{-27}$?

So what is Concise form? and what is the number inside the parentheses (e.g $51$), is it just 2 more number of the constant for more precision? I couldn't find the explanation anywhere else.

Answer 1

The numbers inside the parentheses represent the uncertainty.

If you wanted to explicitly include the uncertainty in the proton's mass, you would write it as $$m_p=(1.672\ 621\ 923\ 69 \pm 0.000\ 000\ 000\ 51) \times 10^{-27}\text{kg}$$ Or in "concise form" $$m_p=1.672\ 621\ 923\ 69(51)\times 10^{-27}\text{kg}$$

Unless the problem(s) you are solving state that you must include uncertainties, you would use $$m_p=1.672\ 621\ 923\ 69 \times 10^{-27}\text{kg}$$ for the mass in your calculations.

Answer 2

It's the "concise form" for writing uncertainties. I.e. $1.672\, 621\, 923\, 69(51)\times 10^{-27}$ kg instead of $(1.672\, 621\, 923\, 69\pm0.000\, 000\, 000\, 51)\times 10^{-27}$ kg.