# Significant digits

I'm reading my physics book and I found something confunsing. In one place my book says that 10,3 km have 5 significant digits (I concluded that is because 10,3 km is 10 300 m ) and in one place it says that 576 km/h has also 3 significant digits. Is it because km/h? How many significant digits has 576 km/s? (3 or 6 (576 000 m/s))? Thx for your replay.

Last edited:

Borek
Mentor
Converting to other units doesn't change number of significant digits. 576 is three. 576 000 m/s is 576x103 ms/s - that is, still three. Unless you were told 576.00 - that would mean 5 significant digits.

What I am going to write is quite not pedagogical, but don't pay too much attention to sig figs, they are taught, but rarely used outside of school.

so 10,3 km has 3 significant digits, not 5 as book says?

Borek
Mentor
Previously you wrote 10.3 has 3 SD? And that's the correct answer.

Note: if there are zeros listed at the end they are meaningful. But if you add them as effect of the conversion to other units, they don't mean anything.

Previously you wrote 10.3 has 3 SD? And that's the correct answer.

Note: if there are zeros listed at the end they are meaningful. But if you add them as effect of the conversion to other units, they don't mean anything.

sorry, my mistake. Book says that 10.3 km has 5 SD, obviously that is wrong.

Borek
Mentor
Yep. 10.300 would have 5 SD, but 10.3 has 3.

10300 m still has 3 SD. If the measurement was written as 10300. m or as Borek said, the SD would be 5.

Borek
Mentor
10300 m still has 3 SD.

No, it has 5. Doesn't matter where you put the decimal point. If you report distance as 10300m you mean 5 SD, if you report it as 10.3 km or 1.03*104m (or 10.3*103m) you mean 3 SD.

No, it has 5. Doesn't matter where you put the decimal point. If you report distance as 10300m you mean 5 SD, if you report it as 10.3 km or 1.03*104m (or 10.3*103m) you mean 3 SD.

Trailing zeros are not considered significant unless they are after a decimal point. In order to detail that trailing zeros are significant, you must put a decimal at the end.

After checking wikipedia, I think I found why we disagree on this issue.

From Wikipedia:
The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:

My textbooks and professors use the convention of putting a decimal point at the end to show that the trailing zero are significant. If a decimal point is omitted, the trailing zeros are assumed to not be significant.

Borek
Mentor
Significant digits are ambiguous and in most cases wrong, there are much better ways of expressing uncertainty. As I wrote before, this is one of these things that are taught and used in schools - and that's the end of their applicability.

But you are right it is a matter on convention. For me - if they are written, it means they are significant, doesn't matter where the decimal point is. If they are not significant, use scientific notation so that you don't have to write them.

HallsofIvy
That's one reason why, when you want to be clear about "significant digits", you should use "scientific notation". $1.0300\times 10^4$ km clearly has 5 significant digits. "10300" is ambiguous. It could mean "about 10300 km, rounded to the nearest 100 km" or "10300 km rounded to the nearest km".