Gifted Issues Discussion homepage Forums General Discussion Semi-wonkish question about significant figures

No. There is no ambiguity. There is no additional information to be had.

Look the whole notion of significant figures is just a muddleheaded oversimplification of something that should be done properly with random variables (and you're back to mathematics again). And that's too advanced a topic for an elementary school class on basic arithmetic in the decimal system.

The confusion here is coming from the unfortunate fact that scientists and mathematicians use the same notation to mean two fundamentally different things.

In mathematics - and therefore, in mathematics classes in school - "3.14160" (I'm putting it in quotes to indicate that it's complete in itself - there's no hidden information I'm eliding) is a number. It's the same number as 314160/100000, which by the time children understand fractions they should know is the same number as 31416/10000 and hence (even if their understanding of decimals is wobbly enough that they have to go through this process) the same number as 3.1416.

In science, when people are being careful, they write "3.14160 to 6sf". Our entire problem arises from the fact that, being almost as lazy as mathematicians, scientists sometimes like to omit the last bit, assuming that any string of digits is "string of digits to n sf" where the value of n can be worked out from the number of digits given. For clarity of thought we need to keep this in, though. Now, "3.14160 to 6sf" is not a number. It is a representation of a quantity with uncertainty. Exactly what that means may depend on context (it may, as 22B suggests, need to be thought of as a random variable) but for simplicity let's use the interpretation where it's an interval. Saying that a quantity is "3.14160 to 6sf" means that it is somewhere in the real interval [3.141595,3.141605) (note closed end on the left, open end on the right) where those figures are actually numbers.

Now we can define addition on intervals: for example, we can write

[3.141595,3.141605) + [2.718275,2.718285) = [5.859870,5.859890)

Note that we've just defined + in that sum, by decreeing that [a,b) + [c,d) = [a+c,b+d) where the the + on the right hand side represent the + of real numbers [which, therefore, someone learning about this stuff needs to already understand]. In the case of addition this is easy to justify in terms of where the truth must lie, but we have to be similarly careful in redefining all the other arithmetic operations.

Now, the interval we got as an answer isn't one of those that can be represnted as "string of digits to n sf". So at this point, scientists say that the answer should be a conveniently expressible interval that contains this interval; this is justified because we shall consider that the important thing is that we know the true answer is in there somewhere, and we will be less bothered by the presence of numbers in the interval that couldn't possibly be the right answer. So instead of giving [5.859870,5.859890) as the answer to that sum, we give the bigger interval [5.85985,5.85995) which we can notate as "5.8599 to 5sf". This introduces extra complication into the definition of addition, and even more when you get to, say, exponentiation, and is why scientists end up saying things like


Even if we did carefully define the arithmetic operations so that there was a unique correct answer to every sum, or at least a well-defined understanding of whether an answer was correct or not, I really don't envy the primary school teacher who has to make sure the children understand what's going on. (And you wouldn't be proposing teaching them some rules by rote, would you?) Notice that addition now has no identity and no inverses (because uncertainty always increases when you do an arithmetic operation, so you never get back to exactly where you started).

Also notice that even in science, not all numbers are measurements, so you still have to deal with ordinary numbers sometimes. For example, you don't want people considering mc^1.5 and mc^2.5 when working out E...

This is all valuable stuff to understand. But the idea that it's appropriate to introduce it at the same time as basic arithmetic on decimals is the kind of crazy idea only the parent of an HG+ child could have...

Thanks ColinsMum. While I understand sig figs, I don't think it should be the focus for eight year old kids learning math. If they start to write out decimal division problems to a ridiculous number of digits, the teacher might make a brief mention of sig figs, but the teacher should leave it at that.

While engineers should stick to sig figs, I am an engineer and don't worry about significant figures - civil engineers typically round and approximate at random and no one cares. Surveyors care, but civil engineers don't.

This English major is proud that she actually understood (most of) this conversation.

My take on it is that it might be worthwhile to discuss this as a "By the way, you might want to know that...but in fact, you should do your problems as you have been taught...", but perhaps only at an older age or with math-gifted kids.

Interestingly, one concept my normally math-savvy DD has had trouble with is rounding.

Which, going all the way down the rabbit hole, makes this the PERFECT time to discuss the meaning of "degrees of freedom" and their relative loss, doesn't it??

I agree that the whole reveal is not a good idea.

I think that where Val and I are coming from is that there's no compelling reason to teach students something that they will have to unlearn later on in almost every other discipline they enter.

This is a serious problem for high school and undergraduate students-- and many of them never quite manage to "unlearn" that business of "just add a trailing zero" which leads to errors even in college-educated STEM individuals, frankly. Their errors on this point have some pretty significant consequences (if you'll pardon the pun).

I don't see any compelling REASON to add zeros to 2.5 just so that I may add the value 1.689 to that value.

Simply align the decimals correctly and add them, YK?

Maybe most 3rd-5th grade students really can't manage to keep this straight without the zeros-- but I guess I can't really wrap my head around that particular idea. Maybe that is my problem; I'm assuming ability that many students don't have, perhaps.

Aligning without zeros....graph paper or turn notebook paper sideways ...there you go!

I agree that a notion of these "fuzzy numbers" could be captured by random variables or by intervals or in various other ways too.

And surely everyone should agree that, pedagogically, it's best to try to understand non-fuzzy numbers (i.e. numbers) first, even those who believe that there's really no such thing as non-fuzzy numbers.

The discussion about intervals and rounding reminds me. DS7 was wondering how are you (officially) supposed to round off -3.5?

Ha I never would have thought of that and just rounded to -4 without thinking. But now that he asks, it is a good mathematical question.

Random sampling of the mathematicians around me this evening produces nothing better than a bored "it's policy-dependent", but I think I was taught that even negative halves round up, so -3. This policy has the advantage that 0 isn't unfairly disadvantaged as a target for rounding - makes me slightly queasy to think about the artefacts the other policy might produce, especially if something is added after this rounding stage.

I just googled "rounding negative numbers" and the mathematicians' answer generally is, you can do what you want, just list the various possibilities for rounding conventions, and give them different names.