# How do I write orders of magnitude in scientific notation when margin of error is involved?

My TA has me confused about how to use scientific notation when margin of error and orders of magnitude are involved.

The number I wrote: 3.78 ± 0.02 * 10^-3

My TA told me that this means: 3.78 ± 0.00002

I thought that this meant: 0.00378 ± 0.00002

Which one is correct? I've been looking online to find something that explains the proper notation when writing out margin of error with orders of magnitude but I haven't been able to find anything (I'm at a loss of what terms to search for). If what I wrote isn't the correct way to say 0.00378 ± 0.00002, then what is? Writing it as 3.78 * 10^-3 ± 0.02 or 3.78 * 10^-3 ± 0.02 * 10^-3 seems kind of foolish.

• Maybe $(3.78\pm0.02)\cdot10^{-3}$? – Ivan Neretin Jan 19 '19 at 7:04
• – Loong Jan 19 '19 at 15:15
• Writing $3.78\pm0.02\cdot10^{-3}$ instead of $(3.78\pm0.02)\cdot10^{-3}$ is clearly wrong for the same reason as writing $3.78\pm0.02\ \mathrm g$ instead of $(3.78\pm0.02)\ \mathrm g$ is wrong. – Loong Jan 20 '19 at 10:29

Sometimes you need to go the extra mile to be entirely unambiguous. Your original formulation (3.78 ± 0.02 * 10^-3) could be interpreted in two ways, the way you intended and the way your TA suggested. The majority of people, I expect, would use context to interpret it correct as you intended, but some (as your TA demonstrates) will not.

There is no clear cut widely agreed standard that will resolve this. But you can apply standard mathematical notation (which will look superfluous to some) to make what you write entirely unambiguous. As Ivan suggested in a comment the simple addition of brackets giving (3.78 ± 0.02) * 10^-3 makes your statement unambiguous in its interpretation. This is more compact than 3.78 * 10^-3 ± 0.02 * 10^-3 but just as unambiguous.

If you were using properly typeset formulae, you might well be able to avoid some of the apparent redundancy of the last formulation, but that would take more effort. Better to accept that redundancy is sometimes required to make things clear.

• This is good to know, thank you! I wanted to make sure I hadn't missed some crucial part of how to use margin of error with scientific notation. I'm supposed to be learning how to write scientific papers that could be put in journals. I'll keep in mind that redundancy is sometimes okay. – Bradley H Jan 19 '19 at 18:57
• The answer Loong linked to the question mentions accepted standards of notation. It also notes that the general form discussed here should be avoided due to flawed interpretation. // Redundant is also the wrong word to use, because for unambiguity it is necessary to write it out, and therefore the exact opposite. – Martin - マーチン Jan 20 '19 at 12:06

I think your TA is wrong, based on the following argument.

When you write a number derived from calculation or experiment, and following standard protocols for the propagation of uncertainties, you regard the last reported integer in the number as uncertain. Therefore, it simply does not make sense to report an uncertainty of ±0.00002 for 3.78, since the uncertainty in 3.78 must be at least ±0.01. I would say that although there may be cases in which explicit notation should be followed, it is generally safe to write

$$3.78 ± 0.02 \times 10^{-3}$$

as meaning

$$0.00378 ± 0.00002$$

A safe alternative (consult a standard analytical chemistry textbook such as Skoog, West and Holler's) is to write

$$3.78 (±0.02) \times 10^{-3}$$

The use of parentheses emphasizes that the uncertainty defines a range of possible values.

Postscript: Another similar argument can be made for the placement of units, for instance should one write

$$3.78 ± 0.02 ~kcal/mol$$

or

$$3.78 ~kcal/mol ± 0.02 ~kcal/mol$$

Clearly the second case borders on the absurd. This is a recurrent problem during drafting of scientific text. Another similar situation is the following: should one write

A and B were determined to be 1.0 and 2.0 kcal/mol, respectively.

or

A and B were determined to be 1.0 kcal/mol and 2.0 kcal/mol, respectively.

or

We determined values of A=1.0 kcal/mol and B=2.0 kcal/mol.

Clearly the first case is acceptable, even if it could be construed as ambiguous, and is not necessarily the most concise. Scientific writing is subject to different rules than standard prose.

Clarity is very important, conciseness too, and there are, in general, recommended conventions. See this answer for instance, which suggests that using the "±" symbol is not recommended unless you mean to use the uncertainty to estimate the distribution of errors.

• Thanks! I was thinking it would be along these lines. I feel like my TA might have graded too many papers by the time she got to mine because there is no reason I can see that an uncertainty of 0.00002 could be meaningfully attached 3.78. – Bradley H Jan 19 '19 at 18:53
• @BradleyH to play TAs advocate here, they are trying to make sure you understand uncertainty and are grading accordingly. If they are interpreting what you wrote in a way where it couldn't "be meaningfully attached" to what you wrote, then that could mean (from their perspective) you don't understand uncertainties and so they mark it as incorrect – Tyberius Jan 20 '19 at 4:08
• I think as long as the question is how to write a number in the format yyy ± xxx, and not "what is the uncertainty in yyy", then my answer holds. – Buck Thorn Jan 20 '19 at 9:02
• @BradleyH There is no point in reporting an uncertainty, if you are sloppy with the notation. Your TA certainly is correct to point out your mistakes, after all that is how you learn. The is enough sloppy notation in well known textbooks, old and directed theories, it is important to teach to the new generation not to make the same mistakes again. // I think this answer also shows very bad style; a construct like 'the value is determined to be' should never be used in a scientific paper. Use active voice instead 'the value is'. – Martin - マーチン Jan 20 '19 at 11:53
• You're right @Martin-マーチン in general, but the literature is full of similar expressions, since humans write articles for other humans. – Buck Thorn Jan 20 '19 at 12:26