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The number of significant digits in 0.1km is 1 and we know that 1 km = 1000m.
This then implies that the 0.1 km is equal to 100.0 m, but the rule says the change in units doesn't affect the number of significant digits
So how come the value in m has more significant digits?

I am really confused by it and the error I can find is that the value in m should be 100 m, then the number of significant digits would be same but then the question arises that can we remove decimal like this.

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    $\begingroup$ 1km is equal to 100000cm, not 1000 cm. $\endgroup$
    – 5xum
    Sep 18, 2018 at 13:39
  • $\begingroup$ you meant 1 km=1000 m may be...? $\endgroup$
    – pooja somani
    Sep 18, 2018 at 13:42
  • $\begingroup$ Sorry for the mathematical error I did $\endgroup$
    – Nick
    Sep 18, 2018 at 14:54

3 Answers 3

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1 km is 1000m, so 100 000 cm, therefore you'll get 0.1km = 10 000cm. Both 0.1km and 10 000cm have the same number of significant digits.

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    $\begingroup$ I would argue that notation 100.0 m implies that we have 4 significant digits. So 0.1km = 100m, but 100.0m = 0.1000km $\endgroup$
    – Vasily Mitch
    Sep 18, 2018 at 13:48
  • $\begingroup$ I would agree, but I would also argue quite plainly, that no where in my answer have I argued that 100.0m has the same number of significant digits as 0.1km. $\endgroup$
    – MRobinson
    Sep 18, 2018 at 13:52
  • $\begingroup$ I strongly disagree that 0.1km has the same number of significant digits as 10000cm. The first one has one significant digit; the second one has 5. If you want to express this length in centimeters with one significant digit, you should write $1 \times 10^4\mathrm{cm}$. (In keeping with this then it is neater to write the kilometer quantity as $1 \times 10^{-1}$km, but that goes beyond the scope of the question.) $\endgroup$
    – Mees de Vries
    Sep 18, 2018 at 13:57
  • $\begingroup$ @MeesdeVries Whilst I would also rather see something written as $1 \times 10^4$, the two are equivalent. The zeroes merely show the magnitude, nothing to do with the accuracy of the data, which is why more or less significant digits may be chosen. $\endgroup$
    – MRobinson
    Sep 18, 2018 at 13:59
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$0.1$km with one significant digit means the value you are interested in is between $0.05$km and $0.15$km. That puts it between $50$m and $150$m. In scientific notation that's $1 \times 10^2$m. which preserves the single significant digit precision. When you write $100$m you have (implicitly) three significant digits - the zeroes might be significant.

When you take significance seriously you should always use scientific notation, writing (for example) $$ a.bcd \times 10^n $$ for four significant digits. Then the unit conversions (in the metric system) affect just the exponent.

$0.1$km ($1.0 \times 10^{-1}$km) has two significant digits. $100.0$m ($1.000 \times 10^3$m) has four.

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    $\begingroup$ I wouldn't be so affirmative regarding the number of significant digits of $100$, because the zeroes must be there anyway. But $100.0$ (as in the question) clearly has four significant digits. $\endgroup$
    – user65203
    Sep 18, 2018 at 13:57
  • $\begingroup$ @YvesDaoust, as the answer shows in an example, the zeroes needn't "be there anyway": you can write $1 \times 10^2$ instead of $100$. $\endgroup$
    – Mees de Vries
    Sep 18, 2018 at 13:59
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    $\begingroup$ @MeesdeVries: you can, but it is not mandated. There is some ambiguity in the notation $100$. $\endgroup$
    – user65203
    Sep 18, 2018 at 14:00
  • $\begingroup$ @YvesDaoust $100$m is indeed ambiguous. See my edit. $\endgroup$
    – Ethan Bolker
    Sep 18, 2018 at 14:05
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You are right, writing $100.0\,m$ usually implies that the value is accurate up to the first decimal, hence with an accuracy of $1\,dm$ and $4$ significant digits, whereas unless otherwise stated $0.1\,km$ is accurate to a tenth of a kilometer and has a single significant digit.

So it would be more correct to say

$$0.1\,km=100\,m,$$ though some doubt can remain about the number of significant digits in the RHS.

After re-reading the question, I seem to understand that keeping one digit after the decimal point is your choice. This is not the way it works, you have to keep the same number of significant figures, not the same number of decimals.

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