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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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  • leafers ultimate style avatar for user NinaNYC123
    What would you use significant figures for?
    (13 votes)
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    • blobby green style avatar for user DeWain Molter
      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.
      (55 votes)
  • leafers tree style avatar for user zhawkins42
    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?
    (4 votes)
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    • leaf green style avatar for user PeterO
      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.
      (22 votes)
  • duskpin ultimate style avatar for user Lorenzo Miguel P Delos Santos
    Why are the zeroes significant when there is a unit?
    (4 votes)
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  • aqualine seedling style avatar for user Brooke
    How do we remember trailing 0s, leading, whatever? It's really confusing.
    (0 votes)
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    • piceratops tree style avatar for user A Very Helpful Guy
      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)
  • hopper cool style avatar for user Adarsh
    So if I had o choose in between having to count the zeros as shown in the last example on , I should count only the non-zero digits, but if there is a decimal point, it automatically means I should count the zeros as significant digits as well?
    (3 votes)
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  • female robot grace style avatar for user Milpals
    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)
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  • primosaur sapling style avatar for user Jill Anne Murphy Frampton
    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, at Sal simply states that all leading zeros are not significant. Are there exceptions to this? Thank you.
    (3 votes)
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  • duskpin seedling style avatar for user Aaron Shey
    how many significant figures do percentages have? Do they have one? (5%) or 3? (.005)
    (3 votes)
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  • aqualine ultimate style avatar for user Caleb EJ
    Is this the syllabus for 2023 to 2024 according to CBSE?
    (3 votes)
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  • duskpin tree style avatar for user Kianeliz596
    How many significant figures are in 870.050
    (2 votes)
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Video transcript

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.