Mathematical Derivation of Significant Figure Rules

[jrtayloriv]

Mathematical Derivation of Significant Figure "Rules"

Let's say I have a scale which can accurately read weights out to ten-thousandths of a gram so it might read 1.0005 g or 0.0005 grams ... why is it that the first reading has 5 significant figures and the second has only 1? Same instrument ... so how is it less precise just because the item weighs less? If I want to add two measurements together from the same scale ... say 0.0056 and 1.2345 --- why do I have to make it a two digit number? Why does that cause me to lose precision?

Please do not tell me how to apply the "rules" for significant figures -- I can read the tables in my chemistry/physics books just fine. I am asking how the rule is derived ... everywhere I've asked, I've had people saying "well leading zeroes aren't significant" ... I know this, and like a good monkey can apply the rules without a problem -- but I want to know why they aren't "significant".

I want an explanation centered around arithmetic of numbers in the decimal representation system -- something explaining why precision is lost because of leading zeros ... this is why I put it into the math section ... I figure this is more of a number theoretic question than anything else...


Thanks.

 

[mathman]

Typically precision is thought about in terms of fractional error. A number such as .05 could be off by 10%, while a number like 1.05 has a fractional error of 0.1%. That is the reasoning behind significant digits.

 

[prasannaworld]

Well here is a thought...

Errors are usually (like Mathman said) linked with percentage error. So in you example we have (in Absolute Error):

0.0005 +/- .00001 g = 0.0005 +/- 20%
1.0005 +/- .00001 g = 1.0005 +/- .001% (wlthough you might ceil the value to 1%)

Hence in practice, most of the time one would try to use higher masses or time to more oscillations (e.g. 10) etc...

I believe the answer to your question lies in the way we mathematically evaluate the number of significant figures in a number. However, in Physics (like MM said), it is better to use the Percentage error or sometimes absolute error when quoting precision/accuracy.

 

[jrtayloriv]

Got it -- that's the first time someone has been able to explain it to me clearly. Thanks a bunch!

 

[Borek]

Note that significant numbers are faulty by design. Take a look at 1 and 9 - one siginificant digit in each case. In each case that means that the number is known with +/- 0.5 accuracy. That in turn means 50% accuracy for 1 and around 6% accuracy for 9.

That's why they are not used in real science. Some even claim that they are ONLY taught to poor HS students for no apparent reasons, as they are not used anytime later.

 

[HallsofIvy]

It is much better to express the number in "scientific notation". 1.005 g has clearly been read on a scale where you CAN read to the nearest 0.0005 g. 0.005 is ambiguous. If you were to write it instead as 5 x 10^-3 you can see that there is, in fact, just the the one significant figure.