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This (foolish) doubt on significant figures is giving me sleepless nights.

Say, A given number is 20000 I would say it has 1 significant digit right?

But Now, Say The value given is 20000 m (metres) , Now how many significant digits does it have?? I actually am in a dilemma because the textbook that I have says that trailing zeros (without a decimal) are considered Significant if there is a "unit" included.. (so according to the textbook, 20000m has 5 sig. digits) But, I find this rule nowhere online!

Can someone help me here? Also if you feel it changes/doesn't change, WHY is it so? (Give a reason too please)

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    $\begingroup$ "Say, A given number is 20000 I would say it has 1 significant digit right?": False statement.... from there on it's all downhill for you. $\endgroup$
    – PcMan
    May 26, 2021 at 17:40

3 Answers 3

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You cannot tell the amount of significant digits if you are simply presented an expression like 20000. Without further information two equally valid assumptions are possible, you can assume it has 5 significant digits or you can assume that it has only 1 significant digit. That is why scientific notation is a much clearer notation when it comes to significant digits. There you would write $2\cdot 10^4$ if you had only 1 significant digit and $20000 \cdot 10^0$ if all digits were significant. Without this notation you can't tell the difference unless you are given more information or context.

Note that this is not a problem of units. Units would only change the exponents and numerical values but wouldn't affect the number of significant digits.

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To be honest, a lot of the "significant figures" rules presented in textbooks are overly pedantic and confusing. Almost always, in a corner case like this with a lot of significant zeros, the true meaning is explicitly stated or else clear from context, and sensible people don't rely on arbitrary conventions to convey this information. There is a huge difference in the level of precision between obtaining a measurement with 1 and 5 significant digits. It should be clear from context whether:

(a) you are being given a rough estimate like "the distance between these towns is approximately $2 \times 10^5\ {\rm m}$ or $20\ {\rm km}$"

or

(b) you are being given a precision measurement of something that is actually 20 km long to 1 meter precision. For example, maybe you are reading a document mapping race course and the specification says "We have determined the length of the course to be $2 \times 10^{5}\ {\rm m}$ to within the precision of $1\ {\rm m}$". I would expect such a document to be very clear about the measurement and not rely on a convention about significant figures to convey the information. Actually I would expect the information to be either presented with an error bar: $20000 \pm 1 {\ \rm m}$ or as a difference... $|L-20000\ {\rm m}|<1\ {\rm m}$ where $L$ is the length of the race track.

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In addition to the scientific notation approach mentioned by @Hans Wurst, where you would write $2 \ 10^4$ for 1 significant digit or $2.0000 \ 10^4$ for 5 significant digits, it is also important to remember that in the scientific community and in publications significant figures are typically not even relevant. This is because, instead of just allowing the number as written to give information about the uncertainty, most scientific measurements will explicitly report the uncertainty. For example, you might see $20000 \pm 5000$ for a "1 significant figure" measurement or $20000 \pm 0.5$ for a "5 significant figure" measurement.

Using an explicit statement of the error allows much more clear and explicit communication of the uncertainty. Also, it allows for a more refined expression of uncertainty. For example both $20000 \pm 0.1$ and $20000 \pm 0.2$ would map to $2.0000 \ 10^4$ even though the first measurement is twice as precise as the second.

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