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Technology
Significant Digits
I feel like we've talked about this before, but I can't find anything on it.I remember one time in Stanley's class (7th hour Chemistry, Jr Year) and a few of us railed against an aspect of significant digits. I can't remember specifically what our argument was, but it was good, and I think by the end of it even Stanley conceded we we're correct.
Take that chemists!
Edit: A cursory glance at the link above makes me think our problem was with #3:
3. All zeros appearing to the right of an understood decimal point and non-zero digits are significant. Example: '12.2300' has six significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant).
Since in order to get the thousandths you pretty much HAD to measure the tenth and hundredth spots, they just happened to be zero, so those not counting as significant seemed stupid to us. Maybe that wasn't what it was about, but either way, that is stupid.
**This originally appeared in the PCW thread, but it was made its own topic.
View External Link [en.wikipedia.org]
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 | Carlos44ec - Since 1980! 08/09/2007 @ 02:07:54 PM |
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nerd Edit: And by Nerd I mean, I agree. |
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| Carlos44ec messed with this at 08/09/2007 2:58:31 pm | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/27/2007 @ 11:54:39 PM. Reason: Off Topic | ||
 | Matt - 3941 Posts 08/09/2007 @ 09:01:00 PM |
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Jeremy Wrote - 08/09/2007 @ 01:48:16 PM A cursory glance at the link above makes me think our problem was with #3: 3. All zeros appearing to the right of an understood decimal point and non-zero digits are significant. Example: '12.2300' has six significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant). Since in order to get the thousandths you pretty much HAD to measure the tenth and hundredth spots, they just happened to be zero, so those not counting as significant seemed stupid to us. Maybe that wasn't what it was about, but either way, that is stupid. But then you could just choose how many significant figures you wanted something to be and change the unit to fit that. For example if you measured something as 12 mm, that's 2 significant figures. But you could also express that as 0.012 m or 0.000012 km. It wouldn't make sense to then let those be 3 or 6 significant figures respectively. |
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| Matt screwed with this at 08/09/2007 9:02:00 pm | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/27/2007 @ 09:53:52 PM. Reason: Off Topic | ||
 | Alex - But let history remember, that as free men, we chose to make it so! 08/09/2007 @ 09:42:28 PM |
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Matt Wrote - 08/09/2007 @ 09:01:00 PM Jeremy Wrote - 08/09/2007 @ 01:48:16 PM A cursory glance at the link above makes me think our problem was with #3: 3. All zeros appearing to the right of an understood decimal point and non-zero digits are significant. Example: '12.2300' has six significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant). Since in order to get the thousandths you pretty much HAD to measure the tenth and hundredth spots, they just happened to be zero, so those not counting as significant seemed stupid to us. Maybe that wasn't what it was about, but either way, that is stupid. But then you could just choose how many significant figures you wanted something to be and change the unit to fit that. For example if you measured something as 12 mm, that's 2 significant figures. But you could also express that as 0.012 m or 0.000012 km. It wouldn't make sense to then let those be 3 or 6 significant figures respectively. Word. |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/27/2007 @ 09:57:06 PM. Reason: Off Topic | ||
 | Jeremy - I hate our freedoms 08/09/2007 @ 11:35:05 PM |
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Sure, it breaks down in converting the original measurement, but lets say you had a very precise scale that measured in grams and something was .0000000002 grams. That's a very precise measurement, but now no matter what math you use that number in you are only "allowed" to keep one digit even though you did infact MEASURE 10, it just happened to not weigh enough to trigger those. It's a 0, but it's still a zero you measured. Why is weighing something that happened to be .000003 on the same scale as something that weighs .001003 less precise because nothing happened to trigger up the 0's? It's just an arbitrary rule to get around situations like you described "breaking" the system so, like energy, significant digits can't be lost or created during conversion. I would propose some sort of notation to denote how many digits were in fact measured (from the decimal on, you all know what I mean, don't point out how my logic could imply infinite sig digits), rather than base anything off of zeros. |
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| Jeremy perfected this 4 times, last at 08/09/2007 11:52:10 pm | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/27/2007 @ 11:53:31 PM. Reason: Off Topic | ||
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| Carlos44ec - 2079 Posts 08/10/2007 @ 06:58:03 AM |
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| this has nothing to do with PCW! Go away with your .000000002 km! |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/27/2007 @ 11:56:07 PM. Reason: Off Topic | ||
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| Matt - 3941 Posts 08/10/2007 @ 09:44:08 AM |
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Jeremy Wrote - 08/09/2007 @ 11:35:05 PM Why is weighing something that happened to be .000003 on the same scale as something that weighs .001003 less precise because nothing happened to trigger up the 0's? If it's on the same scale, then neither one is more or less precise than the other. As I recall, precision is to what place the scale/balance/whatever can measure to, not how many sig figs the measurement has. Jeremy Wrote - 08/09/2007 @ 11:35:05 PM I would propose some sort of notation to denote how many digits were in fact measured (from the decimal on, you all know what I mean, don't point out how my logic could imply infinite sig digits), rather than base anything off of zeros. Someone beat you to it. It's called scientific notation. Which leads me into my defense of the rule. It's not that all those zeros aren't "measured", it's just that they aren't significant, because they only really affect what "place" the non-zero numbers show up in (or in scientific notation what exponent you use). For example if you take 0.0003 x 0.1003 you get 3.009 x 10-5 = 3 x 10-5 using sig figs. Now, you can put any number of zeros you want in front of the 15 or the 3, but you'll still get 3.009 x 10n = 3 x 10n. So then you really just need to worry about the 3 and the 1003 as the digits that can affect the accuracy of your answer, and since the last digit is subject to rounding for both, you can only say for sure that your answer is some where between 2.5 x 10-5 and 3.5 x 10-5, which when rounded would give you 3 x 10-5. |
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| Matt edited this 2 times, last at 08/10/2007 11:28:05 am | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:02:11 AM. Reason: Off Topic | ||
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| Jeremy - 9543 Posts 08/10/2007 @ 10:07:51 AM |
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The whole point of the significant digit system is that your final answer can only be as precise as your least precise measurement. Therefor, it's fair to argue a very precise measurement deserves to be treated as such, regardless of how many numbers ahead of it just happened to be a zero. To separate the argument of precision from significant digits gets away from why it was made and makes it that much more arbitrary. You didn't measure .100003 and .000003 to any different levels of precision. Why should calculations using the former get to be 6 times more "precise" than those using the latter just because nothing happened to toggle closer to the decimal? You certainly could finagle scientific notation, I suppose, but it still isn't really what I'm talking about. Something like if you measured .003 grams you could say 3 x 10-6 (~3) kg to denote that even though it's been converted, you have 3 sig figs at your disposal. Incidentally has any thread ever been more dramatically derailed? |
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| Jeremy perfected this 2 times, last at 08/10/2007 10:10:45 am | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/27/2007 @ 11:52:49 PM. Reason: Off Topic | ||
 | Matt - Nutcan.com's MBL 08/10/2007 @ 01:12:58 PM |
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Jeremy Wrote - 08/10/2007 @ 10:07:51 AM Incidentally has any thread ever been more dramatically derailed? Yes Jeremy Wrote - 08/10/2007 @ 10:07:51 AM You didn't measure .100003 and .000003 to any different levels of precision. Why should calculations using the former get to be 6 times more "precise" than those using the latter just because nothing happened to toggle closer to the decimal? Calculations using 0.100003 are not "6 times more "precise"" (whatever that means anyway) than those using 0.000003 From my view, they are either equally "precise" (ex. multiply by 2 [assuming 2 is an exact number] = 0.200006 and 0.000006 respectively.) or at worst you lose one "place" (ex. multiply by 6 [assuming 2 is an exact number] = 0.600018 and 0.00002). Anyway, it seems to me that you're equating number of sig figs with how precise a number is and that isn't the case. How I understand it is that sig figs are just a way to round numbers without representing a level of accuracy/precision that you don't have. Going off of what I said before, if you take 1003 x 3 you don't really have 3009. What you really have is something between 1002.5 x 2.5 = 2506.25 and 1003.5 x 3.5 = 3512.25. Using your way though, would give 3 the same number of sig figs as 1003 and would give the answer as 3009 and thus represent that you know for sure that the answer is between 3008.5 and 3009.5 when all you really know for sure is that the answer is about 3000 give or take a few hundred. |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:18:35 AM. Reason: Off Topic | ||
 | Alex - You've got to trust your instinct, and let go of regret 08/10/2007 @ 01:26:13 PM |
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I almost choked on my lunch over this.  |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:19:20 AM. Reason: Off Topic | ||
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| Jeremy - 9543 Posts 08/10/2007 @ 01:29:36 PM |
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I don't see where I gave this 3 of yours any more significant figures than the one it has. Matt Wrote - 08/10/2007 @ 01:12:58 PM Anyway, it seems to me that you're equating number of sig figs with how precise a number is ... sig figs are just a way to round numbers without representing a level of accuracy/precision that you don't have. Does not compute. You HAVE the precision, there was just nothing there. I think we've gone in circles enough on this. |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:20:21 AM. Reason: Off Topic | ||
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| Matt - 3941 Posts 08/10/2007 @ 01:31:52 PM |
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Jeremy Wrote - 08/10/2007 @ 10:07:51 AM Incidentally has any thread ever been more dramatically derailed? {channeling Scott} Speaking of significant figures, I don't think there is a more significant figure for the Brewers' pennant chances than a healthy Ben Sheets (who has 12 quality starts so far this year). {/channeling Scott}  |
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| Matt perfected this at 08/10/2007 1:32:49 pm | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:21:08 AM. Reason: Off Topic | ||
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| Jeremy - 9543 Posts 08/10/2007 @ 01:32:58 PM |
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For Merits in Outstanding MBLness |
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| Jeremy edited this 2 times, last at 08/10/2007 1:35:05 pm | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:22:45 AM. Reason: Off Topic | ||
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| Jeremy - 9543 Posts 08/10/2007 @ 01:54:21 PM |
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Though I've declared the end of the debate I just thought I'd add that I understand the rules for significant figures as they stand. What I am proposing is a "separate" rule, if you will, for denoting things you actually measured. One that supersedes the other rules. I understand 5 grams shouldn't get 3 significant figures because you convert it to .005 kg. However, something you actually measured at .005 grams shouldn't have to "lose" digits just because there happened to be nothing in the tenth and hundreds spots that you did, in fact, measure. |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:23:28 AM. Reason: Off Topic | ||
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| Matt - Washington Bureau Chief 08/10/2007 @ 01:59:57 PM |
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Jeremy Wrote - 08/10/2007 @ 01:29:36 PM I don't see where I gave this 3 of yours any more significant figures than the one it has. If you say that 0.003 grams should count as 3 significant figures, then why wouldn't 3 mg have the same? After all it really is the same as 003 mg. In my example I first used 0.1003 and 0.0003, which, again, by your arguments I assume you would give the same number of significant figures, so 1003 and 3 (or 0003) would have the same number as well. Jeremy Wrote - 08/10/2007 @ 01:29:36 PM Matt Wrote - 08/10/2007 @ 01:12:58 PM Anyway, it seems to me that you're equating number of sig figs with how precise a number is ... sig figs are just a way to round numbers without representing a level of accuracy/precision that you don't have. Does not compute. You HAVE the precision, there was just nothing there. I think we've gone in circles enough on this. 1003 x 3 = 3009 implies a "precision" of something like 1003.00 and 3.00 (or 0.1003 x 0.0003 = 0.00003009 implies 0.100300 and 0.000300 if you prefer), which you wouldn't have. |
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| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:24:11 AM. Reason: Off Topic | ||
 | Jeremy - 9543 Posts 08/10/2007 @ 02:09:51 PM |
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I guess what confused me is that you just started saying "3" without denoting you "converted" anything and then keep saying, and using examples, of 3 sig figs when I am proposing it has 4 You're also still trying to cram it back into rectifying with the old rules when my whole point is that they are flawed. Forget about the stupid rules for a second man! Assuming we are talking about a number you measured: What is fundamentally different than the last 2 zeros than the first 3 in .000300? Everyone of them was measured and just happened to be zero. Isn't it the same thing, in principal? "Rule 3" is just there because it worked out that way by definition of the other rules, where everything is based of of zero counting, so no digits are "created or lost." It's not there because it's some core truth to mathematics/numerology. |
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| Jeremy screwed with this 2 times, last at 08/10/2007 2:22:12 pm | ||
| This comment originally appeared in The Official PCW Homepage it was moved on 08/28/2007 @ 12:25:29 AM. Reason: Off Topic | ||
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| Alex - Ignorance is bliss to those uneducated 08/28/2007 @ 12:39:47 AM |
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The last 2 zeros were measured because there was something to be measured at that level. You can't measure that which doesn't exist, therefore the zeros to the left are meaningless. Put another way, .030 means you measured thirty thousandths, but there was nothing to measure in terms of tenths. You could say you measured .3 tenths, but in that case there is no 0 anymore, which is the whole point. I think.  |
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| Jeremy - Cube Phenomenoligist 08/28/2007 @ 08:40:18 AM |
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I'm going to go ahead and "measure" the number of full grown African Elephants running around my office at zero. I guess since that's impossible to quantify I better watch out, because for all I know the number is really 127. |
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 | Jon - many posts 08/28/2007 @ 08:42:44 AM |
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| I think this thread should be renamed "zero" because it's insignificant. |
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| Matt - Washington Bureau Chief 08/28/2007 @ 11:25:56 AM |
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Jon Wrote - 08/28/2007 @ 08:42:44 AM I think this thread should be renamed "zero" because it's insignificant. You're insignificant. |
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| Jeremy - 9543 Posts 08/28/2007 @ 11:31:58 AM |
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| For a little bit there I made his thing say "Jon - I'm Insignificant - 08/28/2007" for his tagline. Then he changed it to something about me eating poo, then he changed to his current "Nutcan.com's kitten expert" |
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| Alex - 3619 Posts 08/28/2007 @ 01:13:37 PM |
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Jeremy Wrote - 08/28/2007 @ 08:40:18 AM I'm going to go ahead and "measure" the number of full grown African Elephants running around my office at zero. I guess since that's impossible to quantify I better watch out, because for all I know the number is really 127. Wikipedia says:, "A number with all zero digits (e.g. '0.000') has no significant digits, because the uncertainty is larger than the actual measurement." |
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| Jeremy - I hate our freedoms 08/28/2007 @ 01:26:49 PM |
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What the hell does that mean? "The uncertainty is larger than the actual measurement" sounds like dumbass, "let's try and sound smart," reasoning. Zero is a number, just as countable/quantifiable as any other. I get why a measurement of ALL zeros shouldn't have any amount of significant digits. However, if one of your measurements comes up all zeros significant figures is more likely than not going to be irrelevant, or be the least of your problems. Now perhaps when measuring something on a scale that say only has the granularity to read out in tenths of grams and you put one 1/100 of a gram and it doesn't register then you can't say "The weight is 0.0 with one significant digit" because you know that's inaccurate. Besides, all of that is just more of an argument for the zeros to the right of the 3 in .00300 to be MORE meaningless than the ones to the left. There MIGHT BE something to the right if you had a sensitive enough machine, you KNOW there isn't anything to the left. (Though you still measured the spots immediately to the right, I'm just saying...) |
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| Jeremy screwed with this at 08/28/2007 1:52:45 pm | ||
 | PackOne - 1528 Posts 08/28/2007 @ 02:26:14 PM |
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| I choose to give this thread 0 nuts. |
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| Matt - Washington Bureau Chief 08/28/2007 @ 05:07:11 PM |
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Jeremy Wrote - 08/28/2007 @ 01:26:49 PM Besides, all of that is just more of an argument for the zeros to the right of the 3 in .00300 to be MORE meaningless than the ones to the left. There MIGHT BE something to the right if you had a sensitive enough machine, you KNOW there isn't anything to the left. (Though you still measured the spots immediately to the right, I'm just saying...) The zeros to the right are far from meaningless. Since the last number of the measurement is subject to rounding, 0.003g really means some number in the range of 0.0025g to 0.0035g. Add the two zeros to the right and it becomes some number between 0.002995g and 0.003005g (or in other words, you go from a potential range of 1/1000 of a gram to one of 1/100,000 of a gram). That's a big difference. |
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| Matt - Nutcan.com's MBL 08/28/2007 @ 05:22:27 PM |
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Jeremy Wrote - 08/28/2007 @ 08:40:18 AM I'm going to go ahead and "measure" the number of full grown African Elephants running around my office at zero. I guess since that's impossible to quantify I better watch out, because for all I know the number is really 127. I'm not exactly sure what you're getting at here, but I'll just add that, you're not really measuring there, you're counting. Therefore, it can be assumed that whatever count you get has an infinite number of significant figures. So, if you counted zero elephants, you can feel safe that you won't be trampled in your office by one (unless you missed one, of course). |
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| Jeremy - 9543 Posts 08/28/2007 @ 05:22:33 PM |
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| I never said they were meaningless. Just that there's more "uncertainty" to the right of the 3 than the left, regarding Alex's Wiki blurb. |
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