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Mathematically, What is the difference between asking for the answer rounded and approximated?

i.e. If we have x = 2.4564, and we want x to the nearest two decimal places. What's the best way to ask? ... rounding your answer to the nearest two decimal places? or ... approximating your answer to the nearest two decimal places?

and which one is more common?

I've tried to search about it and didn't find any source talking about it.

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Rounding is to take the nearest number with a given number of digits. To approximate is to give any number nearby, not necessarily with a finite number of digits.

For example, x=2.4564 is rounded to 2.46 with two decimal places, and $\frac {211} {90} = 2.455555...$ is an approximation of x within 0.2%.

So every rounding is an approximation (in the set of numbers with a finite number of digits), but not all approximation are rounding. And if you round to the nearest n-decimal place, you usually approximate near a fraction within a certain percentage.

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It's a bit of a soft question, but "Rounding" is the word you are looking for.

The operationo of rounding is the simple operation of taking some significant digits and ditching the remaining digits. Approximation is more a process of looking for a number that is close enough to your target number.

In a sense, "rounding" is a very simple type of approximation. $2.456$ is an "approximation" of $2.4564$, but it is an approximation that was reached through a process of rounding.

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