1
vote
2answers
37 views

Approximating 'big' ratio with 'small' ratio

Given a ratio $ \frac{m}{n}, p \in N, q \in N $ where either $m$ or $n$ (or both) is a very big number, how can we find a ratio $ \frac{p}{q}, p \in N, q \in N $ which estimates $ \frac{m}{n} $ up to ...
2
votes
3answers
72 views

Factoring/approximating an apparently simple formula

Does anyone know if the following formula can be factorized or approximated: $a^3 + b^3 + c^3 + a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2 + abc$ It looks a lot like $(a + b + c)^3$, except for the ...
0
votes
2answers
95 views

Aproximate calculation in decimals

I am trying to refresh on precision of calculations. If we have the decimal fractions: $.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals ...
21
votes
9answers
3k views

What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$ n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n. $$ I wonder what benefit can be got from it? From computational perspective (I admit I don't ...
5
votes
2answers
2k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...