# 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 ($$\Torange{30}-\Tred{10}=20$$).