Arithmetic (all content)
Rules of significant figures
This video teaches significant figures rules, crucial for measurements and calculations. It covers identifying significant digits, including non-zero digits, zeros in between, leading zeros, and trailing zeros. Additionally, it introduces scientific notation for clearer representation of significant figures. Created by Sal Khan.
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- What would you use significant figures for?(10 votes)
- In engineering significant figures has to do with how accurately you are able to measure something. According to Google, Mt. Everest is 29,029 feet tall. Well, I suspect it is not EXACTLY 29,029 feet tall... to the inch? the thousandth of an inch? how accurately is it measured?
If the height were given as 29,000 feet, that would still be a pretty good measurement, but it implies the number is really only 'good' to the thousands place (plus or minus 500 feet). 20,029 implies that measurement is 'good' to the individual foot (plus or minus 6 inches)... a very different impression of accuracy.
The intent (as I understand it) is to avoid giving the impression of accuracy where it does not exist.(52 votes)
- What is the difference between 5.60, 5.600, and 5.60000? We all know that the zero is a placeholder for a non-existing value when placed after a decimal and after a non-zero digit. I understand that that would show that the measurer took the time and had an appropriate measuring device to arrive at a more and more accurate measurement, and that it is essentially superfluous to write additional zeros after the first trailing zero. What is a little ambiguous to me, is the difference between 5.0, 5.00 and 5.000... (All trailing zeros that reflect additional accuracy). I guess my real question is, what are the circumstances for counting additional trailing zeros as significant?? ...is it simply the amount of significant figures that the measuring device reads that decides this scenario?(3 votes)
- Mathematically there is no difference between the exact, precise numbers 5.0 and 5.00 and 5.000.
However for measurements in the real world (for science, engineering etc.) there is always a limit to the precision of a measure. Someone wanting to make use of a measurement needs to know how precise it is because they need to know how precise accurate their final result is.
A measurement of 5.0 (2 SF) could be anywhere between 5.04 or 4.95
A measurement of 5.00 (3 SF) could be in reality anywhere between 5.004 and 4.995
A measurement of 5.000 (4 SF) could be in reality anywhere between 5.0004 and 4.9995.
Measuring the carpet to cover a floor as 5.0 Meters (2SF) by 6.0 Meters (2SF) means the area could be as little as 4.95x5.95, 29.4525M^2. However, at that precision the area could be 5.04x6.04, 30.4416M^2. That is a difference of 1 M^2
When measuring accurately to 4 SF means a difference of 2 thousandths of a meter squared.
Knowing accuracy and knowing possible error for future calculations is the difference between 5.60, 5.600 and 5.6000.
It is the precision of the measuring device that decides the significant figures.(20 votes)
- How do we remember trailing 0s, leading, whatever? It's really confusing.(0 votes)
- Remember it like a group of three people walking on the road. The one in the front is leading the others. the one in the back is trailing them. So, the leading zeroes are the ones in front (like 0.052; the first two zeroes are leading) and the ones in the back are trailing (like in 56.00, the last two are trailing). Hope this helps!(8 votes)
- Why are the zeroes significant when there is a unit?(3 votes)
- Whether or not a value has a unit does not affect whether a zero is significant or not.(2 votes)
- The 0,00700 is considered 3 significant figure for the 700 part, however, the decimal is quite further off, and we are ignoring the 0.00. Thats' conflicting. Can anyone explain?(2 votes)
- The first three zeros are just placeholders, not significant.(3 votes)
- I have several conflicting sources regarding leading zeros. I have always been told that if the number is less than one, then a single zero before the decimal is considered significant. The reason for this is because it shows a purposeful inclusion of the place in the accuracy of the measurement, rather than just leaving the place blank, and therefore ambiguous. An example would be 0.345. Can you verify this? I DO understand that leading zeros, such as the zeros in the famous ( or infamous, depending on your opinion) 007, would not be considered significant. In the Rules of significant figures video, at1:18Sal simply states that all leading zeros are not significant. Are there exceptions to this? Thank you.(3 votes)
- how many significant figures do percentages have? Do they have one? (5%) or 3? (.005)(3 votes)
- then it would just be 1 sig fig because of 0.005(1 vote)
- So like the problem shown what if the 37,000 was actually a 37,500
Would you say they had 2 significant figures or 3 significant figures by counting the 3,5, and 7? or would you have to count the zeros too?(1 vote)
- If you have 37500 with no other notations (bars over zeros, decimal points, etc.) then the number as written has THREE sig figs. Not four. I have no idea how Kyle came up with four.
37500 with a bar over the first zero would be four sig figs
37500 with a bar over both zeros would be five sig figs
37500. (with a decimal at the end) would be five sig figs
Writing them in scientific notation avoids confusion because the zeros cannot be "mistaken" as only placeholder zeros. Hope that helps...(4 votes)
- How many significant figures are in 870.050(2 votes)
870.050has 6 significant figures.
If it were
870.05, it would have 5. However, the trailing zero above is included, so there are six.(1 vote)
- How many significant figures are there in:-
a) 25,000 kg
b)5.67 x 10^3 m/s
and why?(2 votes)
- To figure out the number of significant digits, turn the number into scientific notation and then cross out the 10^. The numbers you end up with are the significant digits.
25,000 in scientific notation is 2.5 x 10^4.
Get rid of the 10^ and it's just 2.5, which is a two digit number. So there are two significant digits in that number. Your second example is already in scientific notation, so 5.67 are the sig figs. This is because numbers that can be replaced by 10^() or 10^-() are not significant. For example:
2 x 10 = 20
0.5 x 10^-1= 0.05
So the 0s are not significant.
0.55 x 10= 5.5
This is, however, not the only rule. Zeroes used for "cosmetic" purposes (or used only to show where other numbers are; e.g. the 0s in 0.05) are not significant.(1 vote)
Based on the examples in the last video, let's see if we can come up with some rules of thumb for figuring out how many significant figures or how many significant digits there are in a number or a measurement. So the first thing that is pretty obvious is that any non-zero digit and any of the zero digits in between are significant. Clearly, the 7 and the 5 here are significant. And the 0 in between them, it's also going to be significant. So let's write this over here. So any non-zero digits and zeros in between are going to be significant. That's pretty straightforward. Now, the zeros that are not in between nonzero digits, these become a little bit more confusing. So let's just make sure we can rule out some of them. So you can always rule out when you're thinking about significant figures, the leading 0's. And when I'm talking about leading zeros, I'm talking about the zeros that come before your non-zero digit. So these are leading zeroes here. These are leading zeroes. There is no leading zeroes here. No leading zeroes in this one, this one, and this one. But in any situation, the leading zeros are not significant. So leading zeros not significant, I'll write it over here. Leading zeros not significant. And so the last question, all you have left, I mean you only have non-zero digits and zeros in between. You could have some leading zeros, which you've already said are not significant. And so the only thing left that you have to figure out is what do you do with the trailing zeroes, the zeroes behind the last non-zero, or to the right of the last non-zero digit. So these trailing zeroes here. There's actually two trailing zeroes over here. And then there's three trailing zeroes over here. So let me make a little-- so trailing zeroes. Trailing zeroes, what do we do with them? So the easy way to think about is if you have a decimal, if there's a decimal anywhere in your number, count them. If you have a decimal, count them. Count them as significant. They are significant, count them as significant. If there's no decimal anywhere in the number, then it's kind of ambiguous. You're kind of not sure and this is a situation. So clearly over here, there's a decimal in the number, so you count the trailing zeroes. These are adding to the precision. Over here there's a decimal, so you count the trailing zero. There's a decimal here, so you count the trailing zeroes. There are no trailing zeroes here. And over here-- well, the way I later put a decimal here. Here you would count it. So if you have the decimal there, you would count all five. If you didn't have the decimal, if you just had 37,000 like that, it's ambiguous. And if someone doesn't give you more information, your best assumption is probably that they just measured to the nearest thousand. That they didn't measure exactly the one and just happened to get exactly on 37,000. So if there's no decimal, let me write it this way-- it's ambiguous, which means that you're not sure what it means, it's not clear what it means. And you're probably safer assuming to not count it. If someone really does measure, if you were to really measure something to the exact one, then you should put a decimal at the end like that. And there is a notation for specifying. Let's say you do measure-- and let me do a different number. Let's say you do measure 56,000. And there is a notation for specifying that 6 definitely is the last significant digit. And sometimes you'll see a bar put over the 6, sometimes you'll see a bar put under the 6. And that could be useful because maybe your last significant digit is this zero over here. Maybe you were able to measure to the hundreds with a reasonable level of precision. And so then you would write something like 56,000, but then you would put the bar above that zero, or the bar below that zero to say that that was the last significant digit. So if you saw something like this, you would say three significant digits. This isn't used so frequently. A better way to show that you've measured to three significant digits would be to write it in scientific notation. There's a whole video on that. But to write this in scientific notation, you could write this as 5.60 times 10 to the fourth power. Because if you multiply this times to the fourth, you would move this decimal over four spaces and get us to 56,000. So 5.60 times 10 to the fourth. And if this confuses you, watch the video on scientific notation. It will hopefully clarify things a little bit. But when you write a number in scientific notation, it makes it very clear about your precision and how many significant digits you're dealing with. So instead of doing this notation that's a little bit outdated-- I haven't seen it used much with these bars below or above the high significant digit, instead you could represent it with a decimal in scientific notation. And there it's very clear that you have three significant digits. So hopefully that helps you out. In the next couple of videos, we'll explore a little bit more why significant digits are important, especially when you do calculations with multiple measurements.