Approximation

An approximation (commonly represented in mathematics with the symbol ≈ 'almost equal to') is the term used for when two things are close to being equal but are not exactly equal. It is often necessary to use approximations in mathematics, especially when dealing with irrational numbers (common examples are e, π, √2, etc., see rounding) and repeating decimals).

Logarithmic scale approximations are approximations of a logarithmic scale, where certain primes are approximated on the scale to then approximate rationals.

Typically for two things to be approximately equal there is a pre-defined tolerance/error that is acceptable for an approximation. For example the approximation $\sin(x) = x$ around $x = 0$ has a percentage error of $1\%$ at $x=0.244$, which gets worse further from 0, and better closer to 0.

Suppose we have a complicated function $f(x)$, which we would like to approximate with $g(x)$. Then for our approximation to be "valid", we would pick an $\varepsilon > 0$, and look for where $|f(x)-g(x)| < \varepsilon$, either analytically or numerically. This would then give a domain on which the approximation is valid. In some cases, the function $g(x)$ may simply be a taylor polynomial of $f(x)$, in which case it may be denoted $f_n (x)$ for the order of the polynomial.

Then

$\lim_{n \rightarrow \infty} f_n(x) = f(x)$

and stopping for some finite $n$ gives an approximation.