Yes. Well done.
Nitpick: It would be 120 ± 10 cm and 1200 ± 100 mm.
Wheee! Sometimes I think intellectual break-throughs are more fun than orgasms.
You’ll all be here on Thursday after my second class, right? Oh, and I have Bio on Saturdays, so I expect you all in here bright and early on Sunday as well! 
If it’s to help you have a little fun, WhyNot?
I think you’re close, but at the same time, I think the hang up between precision and accuracy is sort of what’s hurting you here. Try these definitions from dictionary.com:
I think the confusion comes because we treat them as synonyms in vernacular, but they’re different concepts in science and mathematics. Of course, I like to think is slightly less vague language:
Accuracy: How consistent your instrument is or how close your measurement is to the actual value. For an instrument, if you have two scales that will show down to 0.01lbs, and you try weighing the same 1lb weight a few times on each, and the first one comes up 1.12, 0.84, 1.04 and the second 1.03, 0.98, 0.99, the second scale is more accurate. For a measurement, if one person says a football field is 99 yards from endzone to endzone and another person says it is 115 yards, the first measurement is more accurate.
Percision: How fine your instrument or the degree of representation of your measurement. A ruler with 1/32" marks is more precise than one with only 1/2" marks. In the case of a representation, this is where significant digits come in.
Often, particularly in the sciences, precision and accuracy go hand in hand; for instance, 3.14 is both a less precise and less accurate measurement for pi than 3.14159. Real world measurements will almost never fall exactly on whatever arbitrary unit we use. To go back to the football example, it is possible for one person to say 99 yards and another to say 115.437 yards; the first is more accurate, the second is more precise.
Of course, don’t take my examples too seriously, because any instrument that isn’t accurate within its precision makes the extra precision worthless.
Man, I hope I didn’t just confuse matters more for you. :eek:
Don’t worry, I’m skimming your last post until next class when we cover those topics. 
I did say I was having trouble writing it! I think WhyNot understood what I was trying to say, as a step in the direction of clearing up confusion about those pesky zeros. Pretty much everyone else in this thread has explained things better than I ever could, though, so perhaps I’ll just bow out 
Congrats! I still remember the moment I *understood * the Fourier Transform. I’d been doing homework problems with it for a week, just working through things by rote. One night as I was lying in bed I suddenly had a flash and I could see exactly how it functioned. Very fun.
And soon the joys of figuring out how many significant figures remain after doing calculations with data of various precision!
Ooh! Ohh! Pick me! Pick me!
Multiplication and division: the result will have the least number of significant figures
2.5*12.6=31.5 on the calculator, but you only use 2 s.f., so the correct answer is 32
Addition and subtraction: the result will have the least number of sig. figs after the decimal point:
1.2+26.65 = 27.85 on the calculator, but you only use 1 s.f. after the decimal, so the answer is 27.9
However, if the calculation involves an Exact number (one gotten from counting whole items or from a rule or definition), then you ignore the number of s.f.'s in that number. So 3 tins of spam at 10.5 ounces yields 31.5 ounces - you don’t limit that to 1 s.f, because 3 is an Exact number.
How’d I do?
I didnt understand sigfigs at first either. In fact, I still dont entirely understand it, nor do I enjoy it. Basically I am chiming in just for the hell of it.
One question I had with sigfigs is: if you have 10 and 12, why does 10 have only one sigfig, but 12 gets to have two? This, to me, seems unfair. Sure, we can add a decimal point and it equals things up, but why does it have to come to that? Why are we discriminating against zero? Is it because it is fat?
My T.A., having exhausted all possible explanations (including all of those detailed above), eventually broke down and told me the truth: it doesnt really matter. Know the importance of accuracy, but be relieved to know it is based on a bunch of rules that are kind of weird and silly, but they have been being used for a long long time and chemists are not going to change the rules just because they dont make sense. Essentially, memorize it and move on.
My T.A. was really smart, and also cute, so he can be believed 100%.
But yeah. In other news, I totally agree with you - intellectual breakthroughs are awesome. And chemistry in general rocks! I am totally jealous you are taking bio - I am taking physics this year, and dont get to take bio until next year, and I while I have a serious crush on physics as a whole, I cant wait for bio.
I could go on and on about how I love science. But I wont.
Anyway, good luck in chem!
love
yams!!
One thing that has bugged me in labs is roundoff error. If you’re doing multiple steps of calculation, you don’t want to round off until you get the final result, right? So you should really keep a few extra digits until then. In most lab courses I’ve always gotten forms or worksheets that have spaces for a number of results, just one space for each number.
In reality, you should have two sets of the numbers: one for reporting values, and another with extra digits if you’re using them for further calculation. You’d mark which ones are significant with a bar. Am I totally wrong?
Oh, and yams: you can indicate the last significant digit by putting a bar over it. That tells you if, say, the number 1000 has more than one sig. fig. You could put a bar over any of the zeroes.
Fairly well. If it helps your thinking about it, imagine that an Exact number has an infinite number of significant figures. If there are three tins, then there are 3.0000000000… tins, because there are no fractional tins. So the other measurement – whatever it is – has the fewest significant figures.
Sig figs are important in engineering and machining disciplines too, because they define how much time a machinist spends checking and refining the part. Machinists are paid by the hour, and so this bears directly on the cost of a part. If you’re dropping fence posts in your yard and your post-hole digger is about 0.1m in diameter, that’s good enough to specify the diameter of the posts, too. You wouldn’t go in to the woodshop and turn each post on a lathe until it was 10.0cm diameter. If you’re mounting a steel axle in a bearing, though, you had better hope the parts are machined so that you can call them 100.0mm or better.
Great thread. I was always confused by this aspect of significant figures as well. The explanations provided in the thread now make me think I understand it at last.
From what’s been said in the thread, it sounds like the assumption is that, unless there’s a decimal point, numbers that end in 0 or multiple 0’s are assumed to have been rounded off to that value rather than to be exact. So yes, discrimination against 0 because it’s a nice round value.
I beg the more sig fig-knowledgeable denizens of the thread to correct me if I’m wrong.
I think that it’s more that you just don’t know whether or not its been rounded or to what place value the measurement was taken. If you have been exact, like, say, you counted 10 grapefruits, then you write 10. just so everyone knows you’ve measured by ones. If you write 10 then it’s possible you were exact and didn’t write it down, but it’s just as likely or more likely that you were not all that precise, or that you were counting/measuring in another unit.
Say we run a food pantry and someone donates crates of grapefruits that each hold 10. Someone delivers them with a few extra grapefruits balanced on top. I ask you to tell me how many crates of grapefruits we have. You look, and the best answer you can give me is “2”. There aren’t enough grapefruits to make another crate, and I didn’t ask you how many grapefruits we have, but how many crates. So you’re correct, we have 2 crates of grapefruits. Easy to see how this is 1 significant figure, right?
If I then wrote down how many grapefruits we have based on your information, I’d write down 20 because I know there are 10 grapefruits to a crate and you told me we have 2 crates. “20” is correct because it’s a conversion from another unit - we know there are at LEAST 20 grapefruits, or you couldn’t have answered 2 crates - but it’s only so-so correct; we say it’s correct to one significant figure. I can’t say it’s 2 significant figures, because it came from only 1 significant figure - I don’t get more significant with my pencil - the number of sig figs was determined when you measured by counting crates.
Then I decide to get a more perfect measurement, so I go in and count grapefruits, instead of counting crates, and I find that we have 23 grapefruits. “23” is *more *correct, because I used another significant figure, from a more precise way of measuring. 20 wasn’t *wrong *- we did have 20 grapefruits, only we had 3 more besides those 20!
If we had exactly 20 grapefruits, then you’d tell me we had 2 crates, and if I converted that into grapefruits, I’d again write 20 to indicate that it was a conversion. Then if I went and counted grapefruits, I’d write 20. to indicate that I counted and by golly there were 20 grapefruits and not a seed left over. The decimal tells us that, for sure, it was measured to the ones column, and not counted by tens or converted from another unit.
And yeah, I know because I used a counting example that I’m risking confusion on the Exact numbers have infinite significant figures rule. Mea culpa. But it’s the unit conversion thing I’m trying to illustrate, without using the same km → m thing we already used.
With your grapefruit example, if there were 9 grapefruits in one case and 8 in the other, I’d still tell you there were two crates. In other words, it works both for numbers a little bigger or a little smaller, hence why many tolerances are written as +/-.
I’ve been pondering this for a while, and I think this is a common misconception. On the otherhand, it is likely what WhyNot is being taught.
In my experience, the ± designation represents uncertainty in the measurement which may or may not be dependent on the precision of the instruments being used to measure it. For example, in my dissertation, I had to measure reaction rates. While all of the instrumentation involved gave me 3 significant figures, each time I did the experiment, the result would give me a result that varied by as much as 10%. That gave me figures that looked like 1.00E–4 ± 0.10E–4.
This is unfortunately, a very common error that you will find in many synthetic (inorganic and organic) chemistry papers. The fact is, synthetic chemists generally are not number crunchers. Often times a measurement like a reaction rate will be measured once, then they calculate the uncertainty in the curve they created and report that. This is completely wrong, because the experiment will naturally vary from trial to trial due to uncontrolled factors.
Synthetic chemistry, unfortunately, is notorious for “enhanced” results and missing details. This is partly due to the ignorance of the researchers, and partly due to the way funding is distributed. For the record, I find it appalling.
Oh my yes. One of the toughest courses I took in computer science grad school was on numerical methods and stability. We spent literally weeks on just significant digits, working through the formal proofs you have to use to insure that you maintain precision during lengthy computations. Believe me, WhyNot, it could be much, much worse.
Based on your user name, **29A **, I suspect you already know this … .