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## Computational Errors

When we approximate a number $$x^*$$ by a number $$x$$, there are a few ways to measure errors:

1. Absolute Error: $$|x^*-x|$$
2. Relative Error: $$\displaystyle\frac{|x^*-x|}{|x|}, \; x\neq 0$$

For example, if we approximate $$x^*=1.24$$ by $$x=1.25$$, then the absolute error is $$|x^*-x|=|1.24-1.25|=0.01$$ and the relative error is $$\displaystyle\frac{|x^*-x|}{|x|}= \displaystyle\frac{|1.24-1.25|}{|1.25|}=0.008$$.

The relative error gives us information about the number of decimal digits of $$x^*$$ and $$x$$ match. We approximate $$x^*$$ by $$x$$ to $$n$$ significant digits if $$n$$ is the largest nonnegative integer for which $\displaystyle\frac{|x^*-x|}{|x|}<5\cdot 10^{-n}.$

Since $$\displaystyle\frac{|x^*-x|}{|x|}=0.008<5\cdot 10^{-2}$$ and $$\displaystyle\frac{|x^*-x|}{|x|}=0.008\nless 5\cdot 10^{-3}$$, we have the largest nonnegative integer $$n=2$$. Thus $$x^*=1.24$$ and $$x=1.25$$ agree to $$2$$ significant digits.

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