This topic has to be the most poorly taught topic in both high school and college. Can someone give me some guidelines especially when converting a large number into scientific notation?
Suppose I say, hypothetically, that there are 92,826,324 miles to the sun from point “X”. That’s 9.28 x 10^7 miles. Now, have I committed myself to only being accurate to two decimal places? Even if I multiply this by some factor, say 0.17865 mudpies/mile. And, I want to know how many mudpies from Point “X” to the sun?
Should I round-off this factor to 0.18? Or, is it best to carry all places and round off my answer to two decimal places? Sig figs…the rules are as clear as mud! I think there’s a lot of bad info and WAGs floating through our schools…
You don’t count decimal places when multiplying, you count signifigant figures. Sig figs are every digit except for 0s at the end or beginning of a number.
So, for example, 34,000 would have 2 sig figs. The last three zeros don’t count because they’re at the end.
.00000422 would have 3 sig figs. The first 0s don’t count because they’re at the beginning.
1.0000000001 would have 11 sig figs. The 0s are in the middle and so are counted.
9.28 x10^7 has three sig figs. The 9, 2, and 8 count, the 7 0s at the end don’t. 0.17865 has five. Since the smallest number of sig figs in the equation (9.28 x 10^7) x .17865 is three, then the answer will have three sig figs.
Also, note that it only works with non exact numbers. Exact numbers have an unlimited amount of sig figs. If you had .434323423 dozen of something and you wanted to know how many individual items you had, the answer would have nine sig figs because 1 dozen is exactly equal to 12.
The basic rule is, the final answer is only as precise as the least accurate data. If an someone says he ran 12 miles today, and you need to report that in a European newspaper, you multiply that by the conversion factor 1.60930 to get km. So can you report that he ran 19.3116km that day? Of course not, you should say 19km because you only know the distance to two significant digits. It doesn’t matter how accurately you know the conversion factor. And since you only need 2-digit accuracy, you could have done the calculation with a conversion factor of 1.6 instead of 1.6093, though this usually isn’t a good idea - if the calculation involves multiple steps, the errors will add up. Using one or two extra digits for those factors is usually good enough.
Counting digits really isn’t a precice method of dealing with uncertainties. If you’re dealing with experimental data, you should state the uncertainties explicitely, and propagate the errors properly for the calculations. If your measuring device has 1% accuracy you’d say, for example, 24.4 +/- 0.2 miles. To convert that into km you’d say 39.3 +/- 0.3 km. Counting digits is just a quick and dirty way of doing this.
You can have exactly 12000 of something. The fact that the digit is zero does not mean it is not significant. Most often exact numbers are not that round, but it is not necessarily the case that a zero on the right is not significant.
With decimals, in fact, the inclusion of zeros implies that they are significant. If I list the weight of something as 0.0100 grams, it implies that I have measured it to four significant figures, unless otherwise noted.
It’s not so much there are exceptions to the rule, it’s that there are more rules.
Think of it this way - exactly how precise was your measurement? That precision needs to remain with the number. The number 12,000 can be a rough approximation, like 12,000 years ago, or it can be an exact number, 12,000 feet between my house and mom’s house. In fact, that’s one reason sci notation was devised. Is that 1.2 x 10^4, or 1.2000 x 10^4? Depends on the precision of the measurement. If you rounded within a dozen feet, then you’d want 1.20 x 10^4.
scr4, while it is true you need to track uncertainties for some calculations (like experimental data, statistics, etc), it is not always important in calculations. However, sig digits is important to keep you from manufacturing precision.
Jinx, in your example, you rounded your answer to 9.28 x 10^7 miles when you wrote it down. If you wanted to retain more sig digits, you would write those out. As far as calculations, it is best to carry as many sig digits on each number as you have, and then do all rounding at the end of your calculations. This keeps error from creeping in due to sequential roundings.
I disagree. The inclusion of zeros implies they are significant only if they are not necessary to express the quantity. In other words, the trailing zeros in 0.0100 are significant, but those before the “1” are not. (The number 0.01 expresses the same quantity to a different degree of precision, and only has one significant figure.) Similarly, those zeros trailing the 2 in 12,000 would generally be considered NOT to be significant, but writing 12,000.0 would imply that ALL of the zeros are significant.
Before I go on, a caveat: As stated above, the use of significant figures is only an approximate method. The use of +/- an amount or percentage is the proper method. Nevertheless, there are accepted methods of using significant figures, and having taught 7 years of general chemistry in the past, I believe I have the gist in how they are commonly used.
Anyway, the number of significant figures in the number 12,000 would generally be considered to be two. If you wanted to indicate three or more significant figures, one method is to place a horizontal line over the last significant figure, but this method is not widely used. A method to indicate that all of the figures are significant is to place a decimal point after the number (12,000.), but this is also not always understood. The only definitive way to indicate three to five significant figures in this number is to use scientific notation: (1.20 x 10[sup]4[/sup], 1.200 x 10[sup]4[/sup], 1.2000 x 10[sup]4[/sup]) To indicate six significant figures, scientific notation may also be used, but one could also write: 12,000.0
The measurement 0.0100 g, while precise to the ten-thousandth of a gram, has three significant figures, not four. The measurement 0.1100 g is also precise to the ten-thousandth of a gram, and does have four significant figures. The number 125.0100 g is also precise to the nearest ten-thousandth of a gram, and has seven significant figures.
The product of multiplication (and division) should have the number of significant figures equal to that of the least number of significant figures that went into the calculation.
For addition (and subtraction), significant figures are not used. The result of addition (and subtraction) should have the number of decimal places equal to that of the least number of decimal places that went into the calculation.
Many conversion factors are exact, and have an infinite number of significant figures. Some are: 60 seconds = 1 minute; 100 cm = 1 meter, and (by definition) 2.54 cm = 1 inch.
This is true, but incomplete. The U.S. survey foot and all fundamental survey units (e.g., rods, chains, statute miles, acres, sections, and townships) depend on the definition of 1 meter equaling exactly 39.37 inches. The difference is about two parts per million, but it’s there.
Doesn’t this just mean that a survey foot is not exactly the same as a standard foot? This isn’t really a significant figure issue, but rather a units issue.
All this quibbling over what significant digits are can be resolved simply by expressing every single number (that’s 1.0 * 10^0 or 1.0E0) in scientific notation. If we all did this than it would be easy to see how many significant figures we had in our numbers. A number like 1200000 is confusing when it is written out like that, but as 1.200 * 10^6, it is clear that I have four (that’s 4.0 * 10^0, count 'em 1.0E0,2.0E0,3.0E0,4.0E0) significant figures. Likewise with such numbers as 0.0100 which is much better written as 1.00E-2
Sailor is right. If you are only accurate to five sig figs, then the meter equals 3.9370E1 inches. However, the meter is also equal to 3.937007874015E1 inches.
When precision matters, trust the inch to mm conversion quoted above.
In 1959, the directors of the National Bureau of Standards and the United States Coast and Geodetic Survey agreed on a redefinition of the inch-centimeter relationship. This redefinition defined 1 inch as equal to 2.54 centimeters, exactly, or 1 foot as equal to 0.3048 meters, exactly. However, their agreement stipulated that the older value for 1 meter equaling 39.37 inches, exactly, be retained for identifying the U.S. survey foot. One of the reasons for this retention was that the State Plane Coordinate Systems, which are derived from the national geodetic control network, are based on the relationship of 1 meter equaling 39.37 inches, exactly. The difference between these two values for the foot is very small, two parts per million, which is hardly measurable but not trivial when computational consistency is desired. Fundamental survey units, such as rods, chains, statute miles, acres, sections, and townships, all depend on the relationship of 1 meter equaling 39.37 inches, exactly.
The only other thing that catches people on sig figs is when to apply the limiting factor when a series of calculations are done; that is, when the output of an addition step is multiplied by some other measured value, and so on. I have never heard a specific rule for what to do in these situations, and I have always rounded at the end of the entire calcuation.
Nope. One inch is 2.54 cm exactly although there’s the “survey inch” defined so that 39.37 inches is exactly one metre. But an “inch” is 25.4 mm exactly.
Well I think we’re splitting hairs here, but you’re technically correct, and perhaps Nametag misspoke. His post implied that, for land survey measurements, the meter was “redefined.” In fact, the meter is still a meter; it’s the inch that was redefined.
BTW: All this talk about “how many significant digits does 0.0100 have” and “how do you carry significant digits” is totally ridiculous. It has no meaning in the real world. Any teacher or professor who spends any time spewing such nonsense should be tar and feathered. The only worthwhile statement made so far is that you should “record and retain all significant digits, through all calculations, until the end.” True enough.
A little background… I spent 3 years as a metrology engineer at a Department of Energy facility, and have spent the last 6 years running a university’s calibration control facility (among other things). During this time I have worked with dozens of other metrologists, including NIST staff members. I’m saying this not to brag, but to back up my position with mild authority.
In a nutshell, teachers (who should know better) have taught that the no. of significant somehow implies the uncertainty of the value. If you ever do this in a professional setting, you will be sent to work in the mailroom.
In the World of Measurements, all measured values are assigned an uncertainty. This quantity usually has an absolute component and relative component. How you determine an instrument’s uncertainty, and how you combine uncertainties of system components are the real and only concerns. (Metrologists have spent decades trying to figure out how to do this.) Sure, there’s a correlation between “significant digits” and uncertainty, but you should never even guess what the latter is based on the former. In fact, it is much better to simply say “I have no idea what the uncertainty is” than “based on the number of significant digits the uncertainty is 0.001 grams.” Utter the latter, and to the mailroom you go.
Crafter, while sig figs say nothing about the uncertainty of the value, they do say something about how one should report one’s answers when in doing calculations.
Sometimes one wishes to do back of the envelope calculations to get ballpark ideas of quantities. Many times it is completely misleading and wrong for the student to report all the sig figs the calculator gives when in doing this calculation even if the student knows and has dutifully entered the Newton’s gravitational constant to such precision. The problem is that calculators today give students the impression that they have more precision than they actually do. This whole mess wasn’t a problem 30 years ago when people didn’t carry enormous computational power along with them in their front pockets. Today’s scientific calculators have succeeded in making a horde of people who don’t know when or why to stop writing down what the calculator says. The calculator will give you an answer, but it’s up to the calculating person to determine what the answer means. Thus the inevitable appeal to significant figures.
Reporting a value with five sig figs is intellectually dishonest if you have precision in certain values of only two sig figs. I don’t care if during the calculations one hundred fifty sig figs are kept, the reported answer better not have that many or off to the mailroom you go.
The way to do calculations then is to enter every value in as precisely as you can and then think before writing the answer down. Most of the time the portion of the calculation that has the most error will determine the manner in which you report your answer, but in practice, it makes more sense to report the number of sig figs you are confident are useful and then tack on your uncertainty measurements afterwards. Unfortunately, many students don’t yet have the ability to think this well, so the rule-of-thumb techniques of keeping only thus-and-such number of sig figs is given as a starting point. I don’t know if I have been around only competant people or what, but I have yet to come across the incompetancy you describe where someone assigns uncertainty based on sig figs. Such hooha will get its just desserts in due time, I imagine. Nevertheless, I can understand the motivation behind appealing to uncertainty when in trying to explain to the dense student why he shouldn’t fill his page up with meaningless digits beyond the decimal point. He isn’t after all, yet working for NIST and perhaps doesn’t yet have any way to get a handle on what the errors in his calculations are.
If you think you can explain it better, go out and teach it!