Explanation of Significant Figures
Date: 09/11/2007 at 17:55:10 From: Forest Subject: Significant figures My Physical Sciences teacher introduced our Honors class to the concept of "Significant Figures". At first it wasn't a problem. She made us take notes and include the book-given definition. But then we got a graded worksheet back, and almost everyone got a certain question on significant figures wrong. So, for the past two days, there has been a heated debate going on over significant figures. Ultimately, here's my question: How can 1,000,000 have only one significant figure, but 1,000,000. have 7? I understand that it's the decimal point, but saying that 1,000,000 has only one significant figure is coming across to me the same as saying that 1 = 1,000,000 because none of the zeros are important or "significant". Also, my teacher asked two other teachers for information, and one mentioned the Atlantic-Pacific rule which "works most of the time", and the other said it works "only some of the time". What's going on here? And what's up with the whole 1,000,000 thing? And how many significant figures does .0549 have, and how about 0.0549? I read someone else's question on this site, and it was answered that 390 has only 2 SF, and 390. has 3. However, 3900 has three also, but 3900. has four. What?????? Why??? To me, saying that 1,000,000 has only 1 SF, but 1,000,000. has 7 SF seems to me to mean the same as 1 = 1,000,000; but this doesn't make sense to me. Having $1,000,000 is a lot different than having one dollar. Is there a decent, constant rule about significant figures that I can remember? And what makes "placeholders" unimportant, and what makes a number a "placeholder"? 0.0008 is a lot different than 0.08. Again, how are these zeros unimportant??? I'm very confused on a lot of this, and right now I'm so confused that I can't even explain everything that I'm confused about! I understand this: 1) That 0.435 has only three significant figures. 2) That 435 has three significant figures as well. 3) That 4.035 has four significant figures because the zero has a digit of a value of one or higher on each side. I'm not sure about the rest of the significant figures thing...If you have time, in your answer, would it be out of your way to touch on all "combinations" at least a little? (By this I mean: 0.03 has ___ amount of SF because______, and 3.03 has... and 303 has... and try to get all of those? Thanks!)
Date: 09/11/2007 at 22:50:13 From: Doctor Peterson Subject: Re: Significant figures Hi, Forest. Have you looked at all we've said about significant digits? Some comments are found here: Selected Answers: Significant Digits http://mathforum.org/library/drmath/sets/select/dm_sig_digits.html I can't tell for sure which one(s) you are referring to. I think most of what you're asking is in there somewhere. I'll also reply to your questions. >...what's up with the whole 1,000,000 thing? And how many >significant figures does .0549 have, and how about 0.0549? I read >someone else's question on this site, and it was answered that 390 >has only 2 SF, and 390. has 3. However, 3900 has three also, but >3900. has four. What?????? Why??? A significant digit is one that we can properly assume to have been actually measured. We make such an assumption based on a set of conventions about how to write measurements: we only write digits that we have actually measured, unless the digits are zeros that we just have to write in order to write the number at all. For example, if I use a ruler that shows millimeters (tenths of a centimeter), then if I wrote 12.34 cm I would be lying, because I can't really see whether that 4 (tenths of a millimeter) is correct. If I wrote 12.30 cm, I would still be lying, because I would be claiming that I know that last digit is exactly zero, and I don't. I have to write it as 12.3 cm. All three of these digits are significant--they really represent digits I measured. They are still significant if I write it as 0.123 m (just changing the units to meters), or as 0.000123 km (using kilometers this time). Here, the zeros I added are there ONLY to make it mean millionths; they are placeholders. It isn't really quite true, as far as I am concerned, that 1,000,000 has one significant digit. It MIGHT have only one, or it might have seven. The problem is that you can't be sure; the rule of thumb that is being used doesn't allow you to indicate that a number like this has 2, or 3, or 6 significant digits. Maybe I know that it's between 95,000 and 1,005,000, so that the first three digits are all correct, but the rules don't let me show that just from the way I write it. I'd have to explicitly tell you, the value is 1,000,000 to three significant digits. Significant digits are really best used only with scientific notation, which avoids this problem. We can write 1,000,000 with any number of significant digits: 1: 1 * 10^6 2: 1.0 * 10^6 3: 1.00 * 10^6 4: 1.000 * 10^6 5: 1.0000 * 10^6 6: 1.00000 * 10^6 7: 1.000000 * 10^6 The trick is that we don't need any "placeholders" in this notation; every digit written is significant. >Is there a decent, constant rule about significant figures >that I can remember? And what makes "placeholders" unimportant, and >what makes a number a "placeholder"? 0.0008 is a lot different than >0.08. Again, how are these zeros unimportant??? I'm very confused >on a lot of this, and right now I'm so confused that I can't even >explain everything that I'm confused about! The main idea of significant digits is as a rough representation of the RELATIVE precision of the number; that is, it is related to the percentage error that is possible. For example, 0.08 indicates that the measurement might be anywhere between 0.075 and 0.085; a more sophisticated way to indicate this is to write it as 0.08 +- 0.005, meaning that there might be an error of 0.005 in either direction. This error is 0.005/0.08 = 0.0625 = 6.25% of the nominal value. If you do the same thing with 0.0008, you get the same relative error; although the value itself is very different, the error is proportional to the value. Both have one significant digit, which likewise indicates the relative error, NOT the actual size of the error. >I understand this: > 1) That 0.435 has only three significant figures. > 2) That 435 has three significant figures as well. > 3) That 4.035 has four significant figures because the zero has a >digit of a value of one or higher one each side. > >I'm not sure about the rest of the significant figures thing...If >you have time, in your answer, would it be out of your way to touch >on all "combinations" at least a little? (By this I mean: 0.03 has >___ amount of SF because______, and 3.03 has... and 303 has... and >try to get all of those? Thanks!) The significant digits of a number are all the digits starting at the leftmost non-zero digit, through the rightmost digit, if there is a decimal point. If there is no decimal point, then you count only through the rightmost nonzero digit, because zeros beyond that MIGHT be there only to give other digits the correct place value. (Zeros to the left of the first nonzero digit serve a similar purpose, but aren't counted anyway.) Here are the cases I can think of: Decimal point to the left of all nonzero digits: Count from the leftmost nonzero digit all the way to the end. .103 0.103 0.00103 0.0010300 \_/ \_/ \_/ \___/ 3 3 3 5 Decimal point in the middle: Count from the leftmost nonzero digit all the way to the end. 10.305 00103.0500 \____/ \______/ 5 7 Decimal point on the right: Count from the leftmost nonzero digit all the way to the end. 10305. 1030500. \___/ \_____/ 5 7 No decimal point: Count from the leftmost nonzero digit to the rightmost nonzero digit. 10305 1030500 [here we don't know whether the last 2 zeros are \___/ \___/.. measured or just placeholders] 5 5? If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 09/12/2007 at 08:22:00 From: Forest Subject: Thank you (Significant figures) Dr. Peterson, Thank you so much for the help on Significant Figures! I think I understand it now completely, and I liked the way you described everything...especially the examples. Oh, and sorry for making you write all of those out...that probably took up a lot of your time. But thanks again for the help! The Dr. Math service is amazing and I always come out understanding whatever I had a problem with completely. Great job! Forest
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