What is the proof of the rules of significant figures?

[fxdung]

Please prove the rules of significant figures. I do not know why when multiplying and dividing we have to retain the same number of significant figures as in the number with the least of them.

 

[Orodruin]

The rules of significant figures is a poor man’s version of error analysis and propagation. It is more rules of thumb about how far you can trust your precision than actual rules.

 

[RPinPA]

Let's say for instance you have a measured value which you write as $x = 2.3$. What that is supposed to mean is that you're confident of the 2 and the 3, but not of any further figures. You don't mean that the real value is 2.3000. You mean it's approximately 2.3. You mean that it could be 2.32. It could be 2.27. You can't distinguish between those possibilities with your measuring equipment. It's something that rounds to 2.3, so it could range from 2.25 to just under 2.35. We could write this as 2.30$\pm$0.05. Let's call this $x + dx$ where $-0.05 \leq dx \lt 0.05$ (technically it can't equal +0.5 exactly as that would round up, not down).

Now let's say we have another measured value $y = 4.12$. On this one, I'm confident of the three digits, but no more. So actually it's 4.12$\pm$0.005 or $y + dy$ with $-0.5 \leq dy \lt 0.5$.

What will we say about a calculated value $z = xy$? Well since the correct values of $x$ and $y$ are actually a range of values, we have a range of possible values for $z$.
$z + dz = (x + dx)(y + dy) = xy + y\;dx + x\;dy + dx\,dy$
So the error part that we add to $xy$ is $dz = y dx + x dy + dx dy$.
That means that $dz/z = (y\;dx)/z + (x\;dy)/z + dx\,dy = (y\;dx)/(xy) + (x\;dy)/(xy) + dx\,dy/(xy)$ = $(dx/x) + (dy/y) + (dx/x)(dy/y)$.
In the example we have $dx/x = 0.05/2.3 = 0.022$ a $2.2\%$ error and $dy/y = 0.005/4.12 = 0.0012$, a $0.12\%$ error. So the relative error in $z$ is going to be at least $2.2\% + 0.12\%$, and that last term is even smaller, a tiny fraction of a percent. So we typically ignore it.

We know $y$ to about 0.1%, but we only know $x$ to about $2\%$, and that causes us to only know $z$ to about $2\%$. It is the error in your least precise term that dominates in your overall error.

 

[DaveC426913]

I do not know why when multiplying and dividing we have to retain the same number of significant figures as in the number with the least of them.

This isn't a "proof", rather an intuit:

You're trying to measure the area of your rectangular table to make a pattern on it out of beads. You need to go to the bead store with a value for the area to get the right amount of beads.

You use a millimetre tape measure to measure the length of your table and get a value of 1,000 millimetres (1.000 m).
Your tape measure gets busted and now you're left with only a metre stick with all the markings worn off.
You measure the width of your table and get a value of 1m. Without any markings, you have no way of knowing whether the precise width of your table is 0.5m or 1.5m or anything in between.

When you go to the bead store, what can you tell them about the area of your table?

1.000 x 1(rounded up or down) is not 1.000. You simply don't know its area to within a millimetre. All you know is that your table's area is somewhere between 0.5 and 1.5 m².

In other words the only meaningful thing you can say, without artificially adding digits, is that your table is 1 metre square.