Thread: Significant Figures

While the main purpose of using significant figures is to reduce false precision, in certain calculations it seems like it can also reduce precision and even introduce false precision.

Loss of Precision

According to the rules of addition & subtraction the result should be equally accurate to the least accurate number entering the calculation. For example in the following calculation: even though .007 contains a single significant figure the answer contains two significant figures.

.007 + .007 = .014

However, in the rules for multiplication & division the result should have an equal number of significant figures as the number with the least number of significant figures. For example in the following calculation: both 2 and .007 contain a single significant figure, thus limiting the result to one significant digit.

2 x .007 = .01

The above multiplication operation results in 29% error.


False Precision

Significant figures can also introduce false precision (i.e. mathematically increase precision)

10. x .0010 = .010

In the above operation a comparatively crude measurement is multiplied by a number precise to 1/10,000^th to achieve a result presumed to be more precise (1/1000th) than the first measurement.

Am I overlooking something? It seems like as long as you are dealing with numbers of the same units you should be able to retain digits as you would with addition/subtraction. Also significant figures alone would do a poor job of handling uncertainty of different top and bottom ranges (e.g. 5.5 + 3 / -0) , so are significant figures eliminated when uncertainty is able to be expressed +/- (i.e. only use significant figures if uncertainty is unknown)?

Originally Posted by Runmc

While the main purpose of using significant figures is to reduce false precision, in certain calculations it seems like it can also reduce precision and even introduce false precision.

Loss of Precision

According to the rules of addition & subtraction the result should be equally accurate to the least accurate number entering the calculation. For example in the following calculation: even though .007 contains a single significant figure the answer contains two significant figures.

.007 + .007 = .014

However, in the rules for multiplication & division the result should have an equal number of significant figures as the number with the least number of significant figures. For example in the following calculation: both 2 and .007 contain a single significant figure, thus limiting the result to one significant digit.

2 x .007 = .01

The above multiplication operation results in 29% error.


False Precision

Significant figures can also introduce false precision (i.e. mathematically increase precision)

10. x .0010 = .00010

In the above operation a comparatively crude measurement is multiplied by a number precise to 1/10,000^th to achieve a result presumed to be even more precise, to 1/100,000^th.


Am I overlooking something? It seems like as long as you are dealing with numbers of the same units you should be able to retain digits as you would with addition/subtraction. Also significant figures alone would do a poor job of handling uncertainty of different top and bottom ranges (e.g. 5.5 + 3 / -0) , so are significant figures eliminated when uncertainty is able to be expressed +/- (i.e. only use significant figures if uncertainty is unknown)?

Loss of precision is unavoidable (and much more complicated than the way you describe it). To understand how computers minimize (but cannot cancel) the effects, you would need to read a book on Computer Arithmetic. I could recommend a few, if you are interested. Here is a good start.

Howard,

Thanks for the quick response. So basically computers can use an alternative floating point format called "extended precision" to handle round-off error?

Still when dealing with a physical measurement, the best method would have to be defining the uncertainty. Ultimately the limit of precision is the capability of the instrument. Therefore wouldn't the uncertainty of the instrument supersede any result mathematically extrapolated? As is the case with addition/subtraction.

Originally Posted by Runmc

Howard,

Thanks for the quick response. So basically computers can use an alternative floating point format called "extended precision" to handle round-off error?

The computers use both "rounding" and "truncation".

Therefore wouldn't the uncertainty of the instrument supersede any result mathematically extrapolated?

That would make a very bad instrument for the experiment.

Originally Posted by Runmc

While the main purpose of using significant figures is to reduce false precision, in certain calculations it seems like it can also reduce precision and even introduce false precision.

Loss of Precision

According to the rules of addition & subtraction the result should be equally accurate to the least accurate number entering the calculation. For example in the following calculation: even though .007 contains a single significant figure the answer contains two significant figures.

.007 + .007 = .014

However, in the rules for multiplication & division the result should have an equal number of significant figures as the number with the least number of significant figures. For example in the following calculation: both 2 and .007 contain a single significant figure, thus limiting the result to one significant digit.

2 x .007 = .01

The above multiplication operation results in 29% error.


False Precision

Significant figures can also introduce false precision (i.e. mathematically increase precision)

10. x .0010 = .00010

In the above operation a comparatively crude measurement is multiplied by a number precise to 1/10,000^th to achieve a result presumed to be even more precise, to 1/100,000^th.


Am I overlooking something? It seems like as long as you are dealing with numbers of the same units you should be able to retain digits as you would with addition/subtraction. Also significant figures alone would do a poor job of handling uncertainty of different top and bottom ranges (e.g. 5.5 + 3 / -0) , so are significant figures eliminated when uncertainty is able to be expressed +/- (i.e. only use significant figures if uncertainty is unknown)?

The number .007 contains 3 significant figures, not 1, the numbers to the left or right of the decimal point.
Significant means measured within the precision of the method used, such as 1mm using a caliper.
If x=.007 ± .0005 (margin of error), then 2x = .014 ± .001, 7% for the components and the total. Multiplication is repeated addition, so there is no difference, with this example.
Your second example is incorrect.
10 x .0010 = .010
The precision of the 1^st is ± .5, or 5%, and the 2^nd is ± .0005, or 5%.
The product precision is .010 ± .0006, or 6%.
You cannot get more precision out than you put in!

Originally Posted by Howard Roark

That would make a very bad instrument for the experiment.

It would. But what is stopping a chemistry student from performing an experiment with instruments well within the necessary precision, then on a final calculation magically have a result of increased precision and simply citing significant figures? My point is that significant figure rules for 10. x .0010 = .010 disregard propagation of error, if the instrument was more precise it would have been communicated through proper notation.


phyti,

Thanks I edited my post from "10. x .0010 = .00010" to "10. x .0010 = .010"
However, in rules of significant figures your statement is wrong, .007 has only 1 significant figure, but otherwise what you are saying is exactly my point.