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Let us say that we have estimated the value of some constant $S$ as a number $T$ with $|S-T|<0.001$. Does this mean that the number $T$ is correct to $3$ decimal places? My textbook seems to indicate so.

My concern is that the numbers at the first $3$ decimal places of $S$ might not be the same as the first three decimal places of $T$ even though $|S-T|<0.001$.

Consider the following example. $S=3.14159...$ , $T=3.1409$. In this case $|S-T|<0.001$. The third decimal places of $S$ and $T$ do not match. Yet do we say that $T$ is correct to $3$ decimal places?

The Cryptic Cat
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2 Answers2

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Note that the counting of decimal places starts with the first nonzero digit. Therefore the two approximations of $\pi$ you give have three matching decimal digits alright.

You have to be aware that "accuracy to three decimal places" is not an exact mathematical notion. I'm usually interpreting it in the following sense: "The relative error is $\approx 0.001$." Of course there is an underlying reason for this semantical "unaccuracy", namely the following:

Consider the three digit number $999$. Here one unit of the last digit amounts to a relative error of about $0.001$. But for the three digit number $101$ one unit of the last digit amounts to a relative error of about $0.01$, ten times larger.

Christian Blatter
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I am Italian, and I almost always consider accuracy to $n$ significant figures, not to decimal places.

Anyways, if you round $T$ to $3$ decimal places you get $T=3.141$ which has the three decimal places correct

Lorenzo B.
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