Question after 1 day of Chemistry class: Can anyone help me grok significant figures?

Factual Questions
1

That is, I can and have memorized the rules and give my chemistry teacher the answers she wants, but I’d like to understand what I’m doing a bit better than I do now.

Two points in particular I’m not “getting”:

  1. Why is a zero to the right of the decimal but the left of any digits not significant? That is, .0012 has two significant figures, whereas .1112 has four.

  2. Why, in a number written with no decimal point, are the zeroes to the right of the digits not significant, but if there is a decimal point, the zeroes to the right of the digit ARE significant? That is: 12000 - 2 s.f.; 12000. - 5 s.f.

Here’s what I think I understand about significant figures as a concept: a measurement can only be as accurate as the tool and the eye of the measurer. So that we all know how precise we’re being, we write down all the numbers that the tool gives us for certain, and then we wing the last one. So if a ruler is marked in millimeters, we can read that a bit of string is at the 12 cm mark, but we don’t really know how much over it the edge is, so we estimate, and say that it’s 12.3 cm. That three is a guess, but it’s significant in our work because it lets us know that 12 was measured to the limits of the tool, and the string is definitely less than 13cm.

So, back to my question #1 - why aren’t those zeros “significant”? Don’t they tell us that the string is definitely less than 1/100 of the unit being measured? Or am I conflating “significant” meaning “important” with “significant” in “significant figure”?

As for question #2: I thought that a decimal could always be assumed if a number was written as a whole number. Is this wrong in chemistry simply because a decimal indicates that no rounding took place and without a decimal, we have to assume that the number’s been rounded? That’s my best guess.

Any Chem 101 teachers out there? Use small words, please! :smiley:

2

Significant figures are a short cut for the error of a measurement. If I have 12 pounds of something. This tends to mean 12 pounds + or - 1/2 pound. So if I have 12 pounds of something and .003 pounds of something else saying I have 12.003 pounds total implies a precision of measurement which I just don’t really have.

Significant figures for addition and subtraction. The right most no zero digit is the most significant digit. When you add numbers you round to the left most significant digit of all the numbers.

.0012 + .103 = .104
When multiplying or dividing things the significant digits are the total number of digits from the first non zero number to the last. You take the lowest number of significant digits of all the factors.

so say 10.7 amps of current at 3.2 volts is 34.2 Watts.

So .0012 + .1112 is .1124 but .0012 * .1112 is .00013

3

  1. It makes much more sense if you consider the numbers in scientific notation. In scientific notation 0.0012 is 1.2E-3 whereas 0.1112 is 1.112E-1.

  2. For this example, significant figures can actually be flawed if you don’t use scientific notation. The decimal tells you that the zeroes are precise, but what you want four significant figures in 120000? It’s better to write it as 1.200E5, that way you are sure of the number of significant figures.

4

Imagine that you measure something using a kilometer stick (yes, it requires imagination) and come up with something like 25.7 Km. If you then convert this to mm, you get 25700000 mm. This still has three significant figures because that is the accuracy to which you made the original measurement. The point of the rules with the zeros is so that you don’t artificially imply more precision when doing unit conversions. If you actually DID use a meter stick with 1 mm resolutions, and came up with 25700000. mm, then you add the decimal point to indicate that, even though it came out to a nice even number, you did actually measure that many millimeters.

5

Basically, the number of significant figures should stay the same even if the unit of measurement changes. Say .0012 is a measurement in meters. In millimeters you would write that as 1.2 mm. Dropping the zeros between the decimal and the first digit means that the number of significant digits is the same even if you change units.

Something similar is going on in the second case. Say you measure a distance that 1.2 kilometers – two significant digits. Convert to meters and you get 12000 meters. The zeros have to be there, but the number isn’t any more precise than it was when it was written as kilometers. If you did measure down to the meter you need a way to show it. That’s what the the trailing decimal indicates.

6

I made a mistake here. That should be 34 Watts because volts is only 2 significant digits.

7

That is certainly true, and I agree that when written 1.2 X 10[sup]-3[/sup] it’s easy to see why it has 2 s.f. But I’m still not clear WHY, when written 0.0012 it still only has 2 s.f. This is where I get tripped up on this idea that significant figures help us determine the precision of measurement, and if a number is precise (plus one number after the precise ones), it’s significant. Essentially, the number “0” (after a decimal and before a digit) can be precise and yet not significant, unlike all the other numbers, is that it?

Maybe the answer is simply “convention: memorize it and stop thinking so hard”? :smiley:

Ooh! Ooh! I think I get it! So if you’re indicating precision, use the decimal point, and then we all know it’s been measured all the way to the ones column. (Points to Pachacco, too, who said the same thing, I think.) If the decimal isn’t there, we can’t assume that it’s really that precise. So I was on the right track only I wasn’t thinking about converting from one unit to another.

8

I don’t teach chemistry, but I am a biochemist, so I’ll try to help you out.

Zero’s to the right of the decimal point that come before other digits (such as in 0.0012) are not significant because they are place holders.
I find it easiest to use kindergarten counting and money as an example… if you have $10.00 you could have 10 $1 bills… so the zero in the “ones” place is significant, we’ve counted all the way through the “ones” so we needed to move up to the “tens”. So the zero in the "ones place is significant.
If you have $0.01 you couldn’t have 10 dimes… the zero in the “tenths” place is not significant, we’ve not counted up to the “tenths” yet, so the zero is just holding ground to get you to where the information starts.
Note: This is a tricky concept, and it may be easier to just go with memorizing it as a “rule”.

9

I had a rough time when significant figures was introduced to me. It’s already been explained better than I could by the previous posters, but I’ll try to address this part. You’re right in thinking that insignificant doesn’t mean not important, it just implies the degree of precision (as opposed to accuracy, you’ll probably touch on that next week).

As you progress in your course, one thing you’ll learn about science in general, and chemistry specifically, is that nomenclature is very important. Things are written in a very specific way, so that I can show it to you, a dude from Europe, or a gal from Asia, and we’ll all understand what it means.

ETA: To expand on precision, check out the famous dart board example.

10

The key thing to remember is that the precision of a measurement is independent of how it is written.

1.23 meters = .00123 kilometers = 1230 millimeters

The rules you learned insure that you treat each of these representations as having the same precision: 3.

11

The number of zeros to the right of the decimal and to the left of the digits is arbitrary.

This is partially convention, I guess, but it’s also related to scientific notation–mentioned earlier–and to the metric system. In the metric system one can talk about an inch(not metric system) being 25.4 millimeters, or 2.54 centimeters, or .0254 meters or .0000254 kilometers. All of these measurements have the same number of significant figures.

12

I actually taught college-level general chemistry for five years, so I’m your man.

In the example of 0.0012 (two sig figs), the zeros before the “1” are not significant because they are just placeholders. They are only there because if we eliminated one or both of them, the number changes (i.e. 0.012 or 0.12).

So 0.0012 has two significant figures. In scientific notation, it would be written 1.2 x 10[sup]-3[/sup].

The number 0.00120, on the other hand, has three significant figures. The trailing zero is not necessary for the quantity, so it must be significant. We are indicating a precision to the hundred-thousandths place, and the figure happens to be a zero. In scientific notation, it would be written 1.20 x 10[sup]-3[/sup].

Similarly, 1,200 has two significant figures. The trailing zeros are only there as placeholders. In scientific notation, it would be written 1.2 x 10[sup]3[/sup].

The number 1,200.0 has five significant figures (scientific notation 1.2000 x 10[sup]3[/sup]). The zero after the decimal point is not necessary for the quantity, so it must be significant.

By convention, if we want to write the same number (1,200) with four significant figures, we write it as “1,200.” The decimal point after the trailing zero indicates that the preceding figure is significant. In scientific notation, it would be written 1.200 x 10[sup]3[/sup].

So how do we write the same number (1,200) with three significant figures (i.e. the tens place is significant, but the ones place is not)? One convention is to write a horizontal line over the tens place, but the only unambiguous way is to write it in scientific notation (1.20 x 10[sup]3[/sup]).

Hope this helps.

13

Ah, that helps. That helps a lot.

Thank you **all **so much for helping me out here! As you might guess from my username, I’m the kind of learner who always wants to know why (or why not), so this is all very valuable for me. I’m smart enough just to memorize by rote and pass the class, but that would leave me bored and frustrated! Although, of course, I know that this is just the beginning, so I’m willing, to a frustratingly limited degree, to just go with it and trust that it will either become clear or important or both. And that’s why I’m not (yet) pressing my teacher for further explanations - plus, there’s a bit of a language barrier. She’s great with the “what”, but I’m not sure she’ll be great with the “why”. But, again, only one day of class, so I should give her a chance! But it really helps to hear it explained different ways by different people.

Another good way to look at it!

14

I have no idea why chemistry has this baggage of teaching sig figs. It is fundamental to all of science. Possibly it is because chemistry has the wides variety of measurement tools. By the time I got to grad school, I basically had no need of sig figs.

Actually, WhyNot, it can get quite a bit more complicated when you start genuinely calculating uncertainty in measurements. You can have a measurement of 1.32 +/- 0.14, and it makes perfect sense as long as you reference how you calculate the uncertainty. You will soon get into the difference between precision and accuracy, and those are very important concepts to understand so pay attention. Once again, why the hell chemists have to teach this stuff is beyond me.

15

I think the examples given about thinking of them as scientific notation is really the key here. The fact that there are more zeroes in 0.0012 than in 1.2E-3 is an artifact of the notation. As far as the rules, try thinking about them this way.

Leading zeroes: One key point that makes significant digits important is that it’s the right most digit that determines the precision because it is the least significant digit. This is why leading zeroes never count, because you can have an indefinite number of leading zeroes, but that will have no bearing on the least significant digit’s value

Trailing zeroes: This one is trickier, but only because of the context of the rule regarding leading zeroes. We’re use to thinking that 1.000~ is the same as 1, but because the right most digit determines precision, the first number is more precise, even if it is conceptually the same value. I think the whole “only if there’s a decimal” rule is what confuses it, because then we look at 10, 10., 10.0, 1E1, 1.0E1, 1.00E1, etc. as the same, but they’re really varying degrees of precision. To save yourself the trouble, just try to always use scientific notation.

Another way to look at the second part would be with an example. When the last digit isn’t zero, it’s intuitively obvious what the precision is; that is if I say points A and B are about 112’ apart, you’ll assume +/- 6". If I say points A and B are about 100’ apart, without any other measurements for context, what are you going to assume is the error? If the context of other measurements is 200’, 400’, and 1300’, you’d assume +/- 50’; if the other measurements were 130’, 170’, 80’, you’d assume +/- 5’; and, again, if they were 134’, 87’, 103’, you’d assum +/- 6". That standardized notation allows you to easily denote the precision without going “btw, the precision on this measurement is 4 digits”.

What really stumped me when I was learning it was an odd example on a worksheet where we were given numbers and told to change the number of significant digits. For instance, say I had 112 and they wanted 2 significant digits, the answer was 110, but one of the examples was something like 1030 and they wanted 2 significant digits, but if you round it you get 1000 which is only 1; the only non-ambiguous way to denote the precision of that number to 2 digits is to 1.0E3.

Really, I think it’s silly to teach significant digits outside the context of scientific notation, because you’re bound to run into a counter-intuitive problem like that eventually; so really, I think you should just try to keep it in that context until you’ve really grasped the idea conceptually.
*For those of you wondering, E is my lazy shorthand for “x10^” that I learned from using graphing calculators because I don’t feel like playing with the superscript function.

16

When I taught at a military prep school, I always led into this example with an actual target drawn on the board. Then I pulled out my “dayglo-orange single action, five-round suction cup pistol,” and picked one of the Marines in the class to let off five rounds as fast as they could.

Always livened up the first week of class… :smiley:

17

Yep, we got that worksheet yesterday in class. I had a pretty good grasp of the tricky ones like you mention (and we had to round to a specific number of s.f., as well, which gave me a moment’s pause before I could figure out how to round 10063.32 to three significant figures (1.01 X 10[sup]4[/sup], right?).

But the one that caused the whole class to screech to a halt, riots to break out and dogs to lie with cats wasn’t even related to rounding or changing sig figs. It was way “easier” than that:

How many significant figures are in each of the following numbers?
a) 0.00072
b) 72000

Well, a) we all got immediately, no problems: 2; b) made people’s heads start spinning and all hell broke loose because of the decimal/no decimal thing. She hadn’t mentioned it in her lecture (on purpose), but we were all SO SURE that we should treat it like we treated zeroes to the right of the decimal, assuming that there was a decimal there. Nope, nuh-uh! When we all flipped righteously back to our notes, we were astounded to find out that she never had said all zeros to the right of a digit were s.f., only that zeroes between or after digits after a *decimal *were s.f…

We got our ASSumptions handed back to us on a plate of crow. Lesson learned. I bet none of us will ever forget that zeros to the right of digits don’t count if there’s no decimal!

18

I’m having trouble wording this correctly, but hopefully this will make sense, and maybe even help you understand…!

Say I have a measurement of 0.0012km.

We say the zeros aren’t significant, but it’s a little bit hard to understand why. We know, though, that this is really the same as 1.2m, and it’s easy to see that only the last 2 numbers are significant in that case.

So, going back to 0.0012… it really is just a case of having conducted the measurement using the wrong tool.

A kilometre stick doesn’t make sense for something that small; we should use a metre stick if we can. Had we used a metre stick, we might have been able to be even more precise, and say 1.264m. But when a kilometre stick is all we have, we are forced to make do.

In the “real world” there isn’t always a better tool to be used, so we are stuck with measurements in units that are inappropriate for the object in question. That leads us to have some error in the result; we can only say that something is “true” up to a point, and then things like the accuracy of the pipette, the volumetric flasks etc kick in.

Does that make any sense?

19

It has nothing to do with the tool that is used. 0.0012 Km is exactly the same as 1.2 meters. The units you use to report your measurement are whatever is most appropriate for your presentation, and it has nothing to do with the instrument used to measure it.

20

Oooooooooooh!

So, it’s not significant because it’s really just mucking about with units, not numbers. After measuring, we could just keep sticking zeros in there and changing our units each time. It doesn’t change the essence of the number or the size of the thing or how precisely we measured it.
0.0000012 megameters or
0.0012 km or
1.2 m or
120 cm or
1200 mm

It’s all the same (assuming I have my units right.) Nothing is more or less precise than any of the others; therefore they ALL have 2 s.f.; none of them is more accurate or “significant” than any other.

Am I getting it? I feel like I’m getting it! (At least until I learn about this “precision” vs. “accuracy” thing. Right now, I think y’all are nuts! :wink: )

Christopher, while I see what you’re saying in response to mnemosyne, I think her (his) way of phrasing it helped me understand what **Pochacco **and **Eureka **said before; I wasn’t understanding the relevance when they said: