# Formulation of “significantly” close/far

I am writing a tech document and I would like to formulate "significantly" close/far. For example a certain suboptimal solution $x^{\dagger}$ is "significantly close" to the optimal solution $x^{*}$.

I was tempted to write something like $x^{*} - x^{\dagger} \simeq 0$ however this is not correct because the "closeness" is relative to the problem and its parameters. In other words the difference might be 0.01, 0.1, 1, 5, 10 and still it would be considered to be "significantly close".

Thank you

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This seems way too vague. Is $99\%$ very far from $100\%$, or very close? If your patient lives $99\%$ of the way through an operation, it's far. If a plane travels $99\%$ of the way to its destination, that's far from $100%$. – Douglas Zare Mar 29 '11 at 12:22
You may also want to have a look at this: en.wikipedia.org/wiki/Closeness_%28mathematics%29 – ShyPerson Mar 29 '11 at 19:13

How about $\frac{x^{*} - x^{\dagger}}{x^{*}} \simeq 0$? The denominator is a stab in the dark but often the notion of 'relative' closeness is conveyed as a ratio of the distance to the solution normalized by some function of the parameters.
You could also try something like $f(x^{*}) - f(x^{\dagger}) \simeq 0$ if there is a function $f$ that you can define for your problem to convey the notion of a 'value' of a solution.