### Significant Figures

I've found I don't like the idea of significant figures, from a mathematical perspective. If you look up the wiki on it, in the section called Ineffectiveness, you'll see that, apparently, the subject receives more attention in high-school and college chemistry courses than it does in real-world laboratories. Labs use a different method of notating uncertainty.

For the following discussion, it might be helpful to review the rules for adding and subtracting using correct significant figures:

1. For multiplication and division, the result should have as many significant figures as the measured number with the smallest number of significant figures.

2. For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.

So here's why I don't like significant figures. There are at least two mathematical problems I see with them.

1. Multiplication is not associative. Here's an example:

(1.11 x 5.79) x 6.34 = 6.43 x 6.34 = 40.8

1.11 x (5.79 x 6.34) = 1.11 x 36.7 = 40.7

These answers are clearly not equal in the least significant digit. Which answer is correct? I'm hanged if I know.

2. Multiplication does not necessarily give the same result as the equivalent addition problem. Here's an example: suppose we want to multiply 55.55 by the exact number 3. No problem. Integers, by definition, have an infinite number of significant figures. So we go 3 x 55.55 = 166.7. Ah, but now suppose we add instead: 55.55 + 55.55 + 55.55 = 111.10 + 55.55 = 166.65. The rounding rules for addition seem a bit more favorable towards retaining significant figures than multiplication does. And we see that we get a different answer here depending on which way we do things.

So I don't like 'em. I'm not entirely sure what alternatives there are. I suppose sig figs do have the pedagogic value of instilling in students the idea that you can't gain certainty by multiplication, at least (though you can by addition!).

For the following discussion, it might be helpful to review the rules for adding and subtracting using correct significant figures:

1. For multiplication and division, the result should have as many significant figures as the measured number with the smallest number of significant figures.

2. For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.

So here's why I don't like significant figures. There are at least two mathematical problems I see with them.

1. Multiplication is not associative. Here's an example:

(1.11 x 5.79) x 6.34 = 6.43 x 6.34 = 40.8

1.11 x (5.79 x 6.34) = 1.11 x 36.7 = 40.7

These answers are clearly not equal in the least significant digit. Which answer is correct? I'm hanged if I know.

2. Multiplication does not necessarily give the same result as the equivalent addition problem. Here's an example: suppose we want to multiply 55.55 by the exact number 3. No problem. Integers, by definition, have an infinite number of significant figures. So we go 3 x 55.55 = 166.7. Ah, but now suppose we add instead: 55.55 + 55.55 + 55.55 = 111.10 + 55.55 = 166.65. The rounding rules for addition seem a bit more favorable towards retaining significant figures than multiplication does. And we see that we get a different answer here depending on which way we do things.

So I don't like 'em. I'm not entirely sure what alternatives there are. I suppose sig figs do have the pedagogic value of instilling in students the idea that you can't gain certainty by multiplication, at least (though you can by addition!).