#### Raschedian

If you read any introductory physics or chemistry course, in the very first chapter, you'll come across measurement, uncertainty, etc. There, you learn some rules about rounding off numbers, etc.
One of the rules discussed there is that if you add/subtract two numbers with different decimal places, the result should always be rounded off to the number of decimal places that the number with the least number of decimal places has.
Could you please tell me why this rule makes sense? If possible, please use some practical example like measuring the length of two objects and then adding them together for example. Thank you

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#### Wrichik Basu

Gold Member
2018 Award
Let's take two simple examples: Instrument A measures a certain length as 1.235m, while instrument B measures a different length as 2.35m.

Note that A can measure accurately only upto three decimal places, while B can measure upto two decimals.

What you have to know is, the third decimal place for B is inaccurate. So, when you sum the two measurements as 3.585m, you are not actually sure of the third decimal place, because one instrument (B) gives an error in that place.

That is why in such cases, what you have said is done. Remember that this is a convention and not a law. There is no hard and fast rule regarding this; it's a matter of convention.

Often, we add, subtract, multiply and divide numbers with different decimal places, and round off only at the end to say four or five decimal places, regardless of decimal places of the numbers that we used. Rounding off at the end is always better.

#### James Pelezo

Gold Member
"There is no hard and fast rule regarding this; it's a matter of convention. "

Forgive, I may not understand your point here, but the 'rule' is the final answer/result is to be rounded to an accuracy relating to the 'least accurate measuring device in ones experiment'. Rounding, Standard Deviations, Confidence Intervals, Significant Figures, etc. are 'experimental/laboratory' practices giving results reasonable values with respect to the instruments and tools being used in the data collection process. One does not arbitrarily assign an accuracy to a lab results at the end of calculated results, the tools of measurement do. To be fair, you did compare data from instruments having different accuracy factors but there is a rule of practice. For the original question, the rounding to the least number of decimal places reflects the instrument that is least accurate in the data collection process.

#### Dale

Mentor
One of the rules discussed there is that if you add/subtract two numbers with different decimal places, the result should always be rounded off to the number of decimal places that the number with the least number of decimal places has.
Could you please tell me why this rule makes sense?
The definitive reference for measurements and uncertainty is https://www.nist.gov/sites/default/files/documents/2017/05/09/tn1297s.pdf

The best way to handle measurement uncertainty is to calculate and report the standard error. Significant figures are just a lazy shortcut. If you measure a length of 1.00 m what you mean is that the length could be anywhere between 0.995 m and 1.005 m, which according to the NIST document (see 4.6) would give a standard error of 0.003, so you would write the measurement as $1.00\pm0.003$. If you had a second measurement of 0.100, then that second one could be from a length of anywhere between 0.0995 and 0.1005 m. So the standard uncertainty would be 0.0003 and the full measurement would be written $0.100\pm0.0003$

Now, if you were to add these two measurements together, the estimates just add, for an estimate of 1.100 m, but the standard errors add in quadrature for a total standard error of $\sqrt{0.003^2+0.0003^2}\approx 0.003$. So the summed measurement would be written $1.100\pm0.003$. So if we translate that “official” method back into the significant figures shortcut, we drop all of the digits at or beyond the place where the uncertainty enters in and we get 1.10 m, which is the significant figures from the lowest accuracy measurement

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#### Borek

Mentor
Simplest example: you measure 1.0 L of water, then you add 0.1 mL of water. What is the final volume? Does it make sense to say you have 1.0001 L?

#### Raschedian

Thank you Dale with the pdf document. I think I understand what you're saying but I'll have to read the pdf document and moreover think a little bit about the good example that you gave here.
If I have any further questions, I'll post them here and your kind assistance would be much appreciated. Thank you very much sir!

#### Raschedian

Hi Borek. I think I get your point but would you explain a little bit more please. Thank you!

#### DaveE

I think the key to this is to recognize that measurement all have uncertainty. Whenever you write a number, the digits imply an amount of uncertainty. So, if I write x = 1.00 what I am really saying is 0.995 ≤ x < 1.005, I am also saying that there isn't enough precision to write x = 1.000. If y = 1.0 (i.e. 0.95 ≤ y < 1.05), it would be wrong to say that x + y = 2.00 (i.e. 1.995 ≤ x + y < 2.005) because x + y = 1.004 + 1.04 = 2.044 is a possibility.

#### Raschedian

Thank you DaveE. Your explanation clarifies things for me a lot more.

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