# What do final zeros to the right of decimal point indicate?

Final zeros to the right of the decimal point are considered significant. What do those zeros indicate and why are they significant? For example, in 2.000 there are four significant figures.

• They are treated just the same as any other numbers; the uncertainty is in the next place and assumed to be $\pm 0.0005$. – porphyrin May 15 '17 at 7:32
• $10, 1000, 12340$ etc. are integers so are absolute numbers. $0.001$ can be written as $10^{-3}$ or $1/1000$ so the zeros are necessary to indicate the magnitude of the number. – porphyrin May 15 '17 at 8:04
• it means that this is a real number not an integer and that its values lies somewhere between $1.9995$ and $2.0005$ but that we don't know exactly where between these two numbers and as a result of our ignorance we write the best guess which is $2.000$ . – porphyrin May 15 '17 at 8:17
• Try not to answer a question in the comment session guys. – Jeppe Nielsen May 15 '17 at 8:26
• Related question on Mathematics.SE. – Apoorv Potnis May 15 '17 at 11:25

"$2.000$" does not mean $2.000 \pm 0.0005$.

"$2.000$" does not mean the interval [1.9995,2.0005].

"2.000" means that there is an unspecified amount of uncertainty in at least the last, and possibly the last two digits.

Example 5

The correction for a weight is computed to be 285.41 mg and the uncertainty is 33.4875 mg. First, round the uncertainty to two significant figures, that is 33 mg. Then, round the correction to the same number of decimal places as the uncertainty statement, that is, 285 mg

So in other words, for people who follow the NIST standard of stating uncertainty to two significant digits:

2.000 can mean anything from 1.990-2.010 to 1.901-2.099

Others only express uncertainty using one significant digit, in which case:

2.000 can mean anything from 1.999-2.001 to 1.991-2.009

• This might be the irrelevant question but I have to ask. In above example 5 you rounded off the number 33.4875 to two significant figures as 33 but my teacher taught me that if you want round off a number to some desired sig fig you must do it in different steps like 33.4875 = 33.488 = 33.49 = 33.5 = 34. So which is correct one? – AksaK May 15 '17 at 11:38
• @AksaK I didn't write the example, I'm just quoting from NIST. NIST is correct. Teacher is wrong. "33.4875" is closer to "33" than "34" so you round to "33". – DavePhD May 15 '17 at 11:42
• Though I didn't understand NIST but I understand rest of your answer. – AksaK May 15 '17 at 11:49
• @AksaK In physics publications it's more common to see two uncertain digits reported. For high school or undergraduate science, it's more common to see just one. I was told in college to use one digit, unless the digit would be "1", in which case to use two digits. – DavePhD May 15 '17 at 11:54
• Please note that my answer is dealing with precision of measurement, not uncertainty! The two are different! In terms of precision 2.000 definitely means that the measurement is in the interval [1.9995,2.0005[ as you would otherwise state a different measurement value! The uncertainty of a measurement is only relevant when you do more than one measurement! – Jeppe Nielsen May 15 '17 at 13:30

The zeros to the right of the decimal point denotes the expected precision of a measurement.

Thus a value of 2.0 indicates that the the measurement falls in the interval [1.95,2.05[. The value 2.00 correspond to the interval [1.995,2.005[ and 2.000 corrospond to the interval [1.9995,2.0005[.

Therefore the amount of zeros are significant for indicating the precision of a measurement.

• Why is there uncertainty of +/- x.5 ? Why not x.1? – AksaK May 15 '17 at 9:49
• @AksaK: Standard rounding rules. <.5 is rounded down, >.5 is rounded up. – MSalters May 15 '17 at 12:20

I think it's worth noting that the number of 0s is not guaranteed to indicate significance/precision, as it's possible the author did not follow standard conventions. For instance, they might have reported the number of digits that some measurement device displays, which may or may not correlate with the overall measurement precision in the context of the particular experiment. When in doubt, look for a discussion of the context of the measurement and an analysis of possible errors; if that's not present, consider asking the author to confirm whether they did, in fact, mean for the number of digits to reflect the precision.