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# Accuracy and rounding

Rounding causes a loss of accuracy.

If a number is rounded to a scale (unit, tenth, etc.), the original number could have differed by up to half a scale above or below the rounded value.

A number $n$ rounded to the nearest ten to $\Tblue{125}0$ must be between $\Tblue{124}5$ (included) and $\Tblue{125}5$ (excluded) $$\Tblue{124}5\le n \lt \Tblue{125}5$$

The rounding error increases when we add or multiply rounded values. So, to avoid the rounding errors accumulating, only round values at the end of your calculation.

Take two numbers $n_1$ and $n_2$ rounded to the nearest $10$. Their possible values are covered by the ranges: $$\Tred{5}\le n_1\lt\Torange{15},\quad \Tred{5}\le n_2\lt\Torange{15}.$$ We get \begin{align*} \Tred{10}=5+5\le n_1 &+n_2\lt 15+15=\Torange{30},\\ \Tred{25}=5\times5\le n_1 &\times n_2\lt15\times15=\Torange{225}. \end{align*} The biggest possible rounding error for the product ($\Torange{225}-\Tred{25}=200$) is bigger than the error for the sum ($\Torange{30}-\Tred{10}=20$).