Significant figures and rounding

[sp3sp2sp]

Homework Statement

This isn't a specific HW problem, its just something I keep tripping over for some reason so I guess it qualifies as HW. OK so, for example the number 1.5 has 2 sig fig . But if you compare that to another number with 3 sigfigs, like 1.58, then arent they actually equal? What is confusing me, is what exactly does 1.5 imply about tenths and the the hundredths places? They are insignificant, but how do you determine if 1.58 is greater or lessor than 1.5?

Homework Equations

The Attempt at a Solution

I think the numbers are equal but I am not seeing the reasoning. thanks

 

[QuantumQuest]

OK so, for example the number 1.5 has 2 sig fig . But if you compare that to another number with 3 sigfigs, like 1.58, then arent they actually equal? What is confusing me, is what exactly does 1.5 imply about tenths and the the hundredths places? They are insignificant, but how do you determine if 1.58 is greater or lessor than 1.5?

In order to compare $1.5$ and $1.58$ you have to write $1.5$ with three significant digits i.e. as $1.50$. Obviously $1.58$ is bigger. If you are just interested in two significant digits e.g. for the needs of a problem, then you may clip $1.58$ to $1.5$ but then there is no point to compare it to a three significant digits number.

 

[sp3sp2sp]

thanks ..so is it wrong to say that for two significant figures, it implies that the 3rd digit , fourth digit etc are all zero? What I mean is, if it means that we don't know about the 3rd digit, and that's why we're not including it, then couldn't the third digit in a 2 digit number be a 9 just as easily as it could be a 1?

 

[QuantumQuest]

... so is it wrong to say that for two significant figures, it implies that the 3rd digit , fourth digit etc are all zero? What I mean is, if it means that we don't know about the 3rd digit, and that's why we're not including it, then couldn't the third digit in a 2 digit number be a 9 just as easily as it could be a 1?

Yes, if you have a result of a calculation for instance and you are only interested in, say, two significant figures then you chop off the rest digits - the ones after the second significant digit and taking care of the rounding that takes place, and you have a two significant figures number. From this point on, we don't talk about a three, four etc. significant digits number anymore. On the other hand, if you are given a number with, say, two significant figures and there is a need to compare it to a three significant digits number for example (with the first two significant digits the same as in the first number) then you have to add a zero in the third significant digit place of the first number and compare it to the second number.

 

[jbriggs444]

But if you compare that to another number with 3 sigfigs, like 1.58, then arent they actually equal?

Given measured values of 1.5 (with an implicit uncertainty of plus or minus 0.05) and 1.58 (with an implicit uncertainty of plus or minus 0.005), the true value for the first could be greater, less than or equal to the true value for the second. There is no way to tell from the information at hand.

 

[Ray Vickson]

Homework Statement

This isn't a specific HW problem, its just something I keep tripping over for some reason so I guess it qualifies as HW. OK so, for example the number 1.5 has 2 sig fig . But if you compare that to another number with 3 sigfigs, like 1.58, then arent they actually equal? What is confusing me, is what exactly does 1.5 imply about tenths and the the hundredths places? They are insignificant, but how do you determine if 1.58 is greater or lessor than 1.5?

Homework Equations

The Attempt at a Solution

I think the numbers are equal but I am not seeing the reasoning. thanks

To two significant figures you would round 1.51, 1.52, 1.53 and 1.54 down to 1.5, and you would round up 1.56, 1.57, 1.58 and 1.59 up to 1.6. The case 1.550000... presents issues, and different methods have been suggested for dealing with such a case.

However, during computations, always keep more digits than the significant figure count that you want, and just round off the final answer. For example, if you are solving a pair of linear equations such as
$$\begin{array}{rrcl}
1.51 x & +\; 2.263 y &=& 1743\\
-2.82 x &+\;1.74 y &=& 763.2
\end{array}
$$
do NOT round off to 3 significant figures during the computational work. If you are doing it by hand, keep all intermediate quantities to full calculator accuracy. If you are doing it using a computer package, that package will handle all the accuracy issues for you. You can always round off the final solution.

 

[haruspex]

the true value for the first could be greater, less than or equal to the true value for the second.

Not in this case. 1.5+0.05 is still less than 1.58-0.005.
1.5 versus 1.53 would be a more interesting example. If these represent measurements then all you could say is that the true value of the second is probably greater than the true value of the first.