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Chemistry, 1.05 (V) Measurement and Significant Figures Part I

In this lecture, I'll be defining measurement and discussing the importance of significant figures, units, and unit conversions in measurements. I'll also be reviewing scientific notation. In the last lecture, we talked about how observations are used in the scientific method. Many observations are actually measurements. So the next logical question is, what is a measurement? Well, a measurement is a quantitative observation. It's a comparison to an agreed upon standard. How long is something, how heavy is it, how large is the volume, all of those things are measurements. Every measurement must have two things. It must have a number and a unit. It's important to have both the number and the unit whenever you report a measurement. Let's begin our discussion of measurements by talking about the number part of the measurement. The precision of that number can be communicated by using a different numbers of significant figures. Now, some numbers are accurate but not very precise. In other words, we shouldn't be using very many significant figures to express our measurement of that number. This diagram on the left shows what we mean by a number being accurate. In other words, if we averaged all of those points, we would get something that's about right but not very precise. Meaning, each time we make the measurement, there's a large variation in the number that we obtain. Ideally, we'd like to have measurements that are both accurate and precise. Because that way, we have a very careful measurement with a large number of significant figures. Probably the most dangerous thing is to have measurements that are precise. Meaning each time we take the measurement, we get the same value, but not very accurate because oftentimes those are misleading. Those can happen if you're measuring with an instrument that isn't measuring exactly the right length. Perhaps you don't have a good calibration or maybe you have, or a meter stick that is missing a couple of centimeters. That would give you always the same value. But those values would not be accurate and that's dangerous. So, in significant figures, we're writing different number of digits to express the level of precision of the measurement. Using different numbers of significant figures is a way to write numbers to reflect precision. Let's do some examples to show how that works. Any non-zero number that we write down is significant. But what about zeros? For example, this number, 0.0165. How many significant figures would we say that has? Well, zeros before non-zero numbers are not significant, so we would say that this number only has three significant figures. Let's do another example. What about a zero that is at the end of a number? That's a lot trickier. For example, what about this number 85.950? Zeros after non-zero numbers that are also to the right of the decimal point are significant. So all five of these digits are significant, and they are all relevant to our measurement. [SOUND] Zeros that are between non-zero numbers are also considered to be significant. So this number, 12.061, also has five significant figures. The most difficult situation is when you have a zero that is to the right of the non-zero numbers but is to the left of the decimal point. Let's look at one like that. For example, what about the plain old number 10? How many significant figures does that have? Well, I would say it only has one but other people would say, well, I really can't tell. So in order to be able to tell if I have one significant figure or two significant figures, one thing that I can do is to write the number in scientific notation, which I'm sure you learned about in middle school math. For example, I could write 1.0 times 10 to the first power, or I write 1 times 10 to the first power. 1.0 times 10 to the first power is showing two significant figures. And that indicates that I had a measuring device that allowed me to discriminate between, for example, a measurement of 9 and a measurement of 10 and number of, measurement of 11. A measurement of only 1 times 10 to the first power indicates there's only one significant figure. In this case, the number could range from anywhere from 6 to 14, and we would still round it to 10. Maybe we don't have a very good measuring device in that case. Another way that some people sometimes show the number of significant figures is to put a decimal place after zeros that are to the right of non-zero significant figure numbers. Let me show you what I mean by that because that sounded kind of complicated, but it's really not. I could write the number 10 as 1, 0 and then the decimal place point very firmly shown. That decimal point shows that there are two significant figures in this number. I think it's best to actually always use scientific notation when there's ambiguity like that because sometimes it's hard to tell if your wrote the decimal point or not. One of the tricky things that comes up with significant figures is when we have different numbers of significant figures for numbers that we use in calculations. The one that students seem to have the most trouble with, the example that is often the most difficult, is having different numbers of significant figures in measurements that I need to add or subtract. In this case, the number of significant figures is determined by the number of decimal places of the measurement that's being neither added or subtracted that has the least number of decimal places going in. For example, let's take the number 52.369, which has five significant figures, and add to that the number 69.2, 69.2 which has three significant figures. If we just added up all those digits, we'd get an answer of 121.569. Now, the way I've written that answer has six significant figures. But, really, I should draw a line to the right of the least significant digit, which here is this 2 of the 69.2, and that's how many decimal places the answer should have. So the answer here should be to the tenth place and it should have only one decimal place. In other words, this answer should have four significant figures. Let's do another example. Let's take the number 0.2597 and subtract from that 0.034. In this case, I have a number with four significant figures and I'm subtracting from it a number that has only two significant figures. If I just did that with the digits, I would get an answer of 0.2257. But, remember, I have to draw a line to show the most significant decimal place going in, which here is the thousandths place. So, the answer should be expressed only to the thousandths place. And I'm going to round it up to 0.226, so I have three decimal places. Let's look back at the first example. In the first example, I was adding a number that had five significant figures to a number that had three significant figures. But, the answer that I wrote actually had four significant figures. And sometimes that happens with addition and subtraction. You can gain or lose significant figures. Here's another example. Let's look at the next one. I had a number that had four significant figures added to a number that had three, two significant figures, excuse me. And the answer there, I said, had three significant figures because it's to three decimal places. So when you add or subtract, you can't just count up the number of significant figures in the quantity that you're using in the calculation. You actually need to look at the decimal place to determine the significance of the answer to determine the correct number of significant figures for the answer. Let's do some more examples. This time, let's look at multiplication and division. And that's actually quite a bit easier to do. Because in the case of multiplication and division, the number of significant figures in the answer should be equal to the least number of significant figures of any of the quantities you use in the calculation. So you just look for the least number of significant figures of the numbers that you're using when your doing the multiplication and division, and that is the number of significant figures in the answer. Here's an example, 12.3 times 5.0. Well, that would equal 61.5 if I plot it into the calculator. But this number with the least number of significance going in is 5.0, which has two significant figures. So my answer also must have two significant figures. So I'm going to round my answer to 62. The same is true is for division. I can take the number 16.7 and divide it by 9.004 and my calculator might actually give me eight decimal places. But if I just wrote down a few, I would say 1.8547. But I see that there, in this case, the quantity going in that had the least number of significant figures had three significant figures, so my answer should also have three significant figures. It doesn't matter how many quantities you're using. If you're only doing multiplication and division, you just look for the quantity going in that has the least number of significant figures. And that's the number of significant figures in the answers, quite a bit simpler than the addition and subtraction case. Here's another example. Here I have all multiplication and division as my operations. So, in this case, I look for the quantity going in that has the least number of significant figures, which here is 3. I have three significant figures in each of those and four in the other number. So, my answer should have three significant figures. It should be expressed to three significant figures. Why am I doing that again? I'm doing that to show the level of precision, okay? We don't want to show a level of precision than, that is higher than what we actually had in any of our individual measurements. Now remember when you do math, you have to perform certain mathematical operations in the correct order. So you do the operation that's inside the parentheses first. Then, you do multiplications and divisions, and then you do additions and subtractions. You have to be careful here about what happens to the number of significant figures when you do each operation. I like to keep track of the significant figures by underlining the last digit that's significant in any intermediate answer that I have. Remember in addition, you might gain a significant figure. And in subtraction, you might lose a significant figure. So let's do an example of that. In this problem, I have 2.86 minus 2.43. That's inside the parentheses, so I would do that first. The number that results from that, does it have three significant figures, even though both of those numbers do have three significant figures? The number that results from that is 0.43, and it only have two significant figures. Okay. So just from looking at what's on this side of the parentheses, the answer from that, remember, can only be to the hundredths place because I'm using the decimal places when I do subtraction. That has two significant figures. I, then, take the number with two significant figures and divide it by 10.57 to give me an answer of 0.04068, which I'm going to round to two significant figures, 0.041. Let's do another example, 3.58 plus 6.73. In this case, I have two quantities that each have three significant figures in the measurement. But both of those measurements were precise to the one hundredth place. So when I add them together inside the parentheses, I get an answer of 10.31. That has four significant figures. Therefore, when I take that four significant figure number and divide it by a five significant figure number, my answer needs to be expressed to four significant figures. Does that make sense? Now it looks like, when I'm doing these operations, that I might actually be rounding during an intermediate operation step. And you need to be really careful about that, this. So, I'm going to give you a quick word about rounding errors. One of the things that can happen, if you have a calculation where, let's say you have five or six different operations that you need to do in the calculation, is that if you round an intermediate calculation, you can end up with lots of errors. So what I recommend that you do, and I'll show you an example so you see what I mean is that you always carry at least one extra non-significant digit through the multistep calculation, and then round at the end. And I actually like to carry a couple of extra digits the whole way through and use a line to underline the last significant digit to help me keep track. So here's an example. I can take the number 12.456, multiply it by 12 and then divide it by 3.697. The result of those operations is the answer 40.43. Now there's two significant figures. In the quantity going in that had the least number of significant figures. So my answer should also be expressed as two significant figures. And I can do that again by either showing that decimal point or using scientific notation. So I could say 4.0 times 10 to the first power. If I do the calculation inside the parenthesis and round it and then do the second calculation, I can introduce quite a bit of error. In other words, if I did it in two steps, if I took 12.456, multiplied it by 12, and got 149.47, but then I rounded it to two significant figures. Okay. And then I take that quantity, which is 150 now, and divide that by 3.697, now my answer becomes 41 if I round it to two significant figures. And that's quite a bit different than 40. How much error have I introduced? I've introduced a 2.5% error. That can be a really large error. The difference between 2.5% concentration of something in your bloodstream, for example, could be toxic. Missing by 2.5% if you're shooting at the moon could mean you just fly off into interstellar space. So 2.5% can be a really big mistake. So it's important to not do intermediate rounding. What I could have done when I did this two-step calculation, is instead of rounding it to 1.5 times 10 to the 2, what I could've done is written down my intermediate answer and underlined the 4, just to remind myself that I only had two significant figures. That's one way that I'd do it. Something you might want to try.



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In this lecture, I'll be defining measurement and discussing the importance of significant figures, units, and unit conversions in measurements. I'll also be reviewing scientific notation. In the last lecture, we talked about how observations are used in the scientific method. Many observations are actually measurements. So the next logical question is, what is a measurement? Well, a measurement is a quantitative observation. It's a comparison to an agreed upon standard. How long is something, how heavy is it, how large is the volume, all of those things are measurements. Every measurement must have two things. It must have a number and a unit. It's important to have both the number and the unit whenever you report a measurement. Let's begin our discussion of measurements by talking about the number part of the measurement. The precision of that number can be communicated by using a different numbers of significant figures. Now, some numbers are accurate but not very precise. In other words, we shouldn't be using very many significant figures to express our measurement of that number. This diagram on the left shows what we mean by a number being accurate. In other words, if we averaged all of those points, we would get something that's about right but not very precise. Meaning, each time we make the measurement, there's a large variation in the number that we obtain. Ideally, we'd like to have measurements that are both accurate and precise. Because that way, we have a very careful measurement with a large number of significant figures. Probably the most dangerous thing is to have measurements that are precise. Meaning each time we take the measurement, we get the same value, but not very accurate because oftentimes those are misleading. Those can happen if you're measuring with an instrument that isn't measuring exactly the right length. Perhaps you don't have a good calibration or maybe you have, or a meter stick that is missing a couple of centimeters. That would give you always the same value. But those values would not be accurate and that's dangerous. So, in significant figures, we're writing different number of digits to express the level of precision of the measurement. Using different numbers of significant figures is a way to write numbers to reflect precision. Let's do some examples to show how that works. Any non-zero number that we write down is significant. But what about zeros? For example, this number, 0.0165. How many significant figures would we say that has? Well, zeros before non-zero numbers are not significant, so we would say that this number only has three significant figures. Let's do another example. What about a zero that is at the end of a number? That's a lot trickier. For example, what about this number 85.950? Zeros after non-zero numbers that are also to the right of the decimal point are significant. So all five of these digits are significant, and they are all relevant to our measurement. [SOUND] Zeros that are between non-zero numbers are also considered to be significant. So this number, 12.061, also has five significant figures. The most difficult situation is when you have a zero that is to the right of the non-zero numbers but is to the left of the decimal point. Let's look at one like that. For example, what about the plain old number 10? How many significant figures does that have? Well, I would say it only has one but other people would say, well, I really can't tell. So in order to be able to tell if I have one significant figure or two significant figures, one thing that I can do is to write the number in scientific notation, which I'm sure you learned about in middle school math. For example, I could write 1.0 times 10 to the first power, or I write 1 times 10 to the first power. 1.0 times 10 to the first power is showing two significant figures. And that indicates that I had a measuring device that allowed me to discriminate between, for example, a measurement of 9 and a measurement of 10 and number of, measurement of 11. A measurement of only 1 times 10 to the first power indicates there's only one significant figure. In this case, the number could range from anywhere from 6 to 14, and we would still round it to 10. Maybe we don't have a very good measuring device in that case. Another way that some people sometimes show the number of significant figures is to put a decimal place after zeros that are to the right of non-zero significant figure numbers. Let me show you what I mean by that because that sounded kind of complicated, but it's really not. I could write the number 10 as 1, 0 and then the decimal place point very firmly shown. That decimal point shows that there are two significant figures in this number. I think it's best to actually always use scientific notation when there's ambiguity like that because sometimes it's hard to tell if your wrote the decimal point or not. One of the tricky things that comes up with significant figures is when we have different numbers of significant figures for numbers that we use in calculations. The one that students seem to have the most trouble with, the example that is often the most difficult, is having different numbers of significant figures in measurements that I need to add or subtract. In this case, the number of significant figures is determined by the number of decimal places of the measurement that's being neither added or subtracted that has the least number of decimal places going in. For example, let's take the number 52.369, which has five significant figures, and add to that the number 69.2, 69.2 which has three significant figures. If we just added up all those digits, we'd get an answer of 121.569. Now, the way I've written that answer has six significant figures. But, really, I should draw a line to the right of the least significant digit, which here is this 2 of the 69.2, and that's how many decimal places the answer should have. So the answer here should be to the tenth place and it should have only one decimal place. In other words, this answer should have four significant figures. Let's do another example. Let's take the number 0.2597 and subtract from that 0.034. In this case, I have a number with four significant figures and I'm subtracting from it a number that has only two significant figures. If I just did that with the digits, I would get an answer of 0.2257. But, remember, I have to draw a line to show the most significant decimal place going in, which here is the thousandths place. So, the answer should be expressed only to the thousandths place. And I'm going to round it up to 0.226, so I have three decimal places. Let's look back at the first example. In the first example, I was adding a number that had five significant figures to a number that had three significant figures. But, the answer that I wrote actually had four significant figures. And sometimes that happens with addition and subtraction. You can gain or lose significant figures. Here's another example. Let's look at the next one. I had a number that had four significant figures added to a number that had three, two significant figures, excuse me. And the answer there, I said, had three significant figures because it's to three decimal places. So when you add or subtract, you can't just count up the number of significant figures in the quantity that you're using in the calculation. You actually need to look at the decimal place to determine the significance of the answer to determine the correct number of significant figures for the answer. Let's do some more examples. This time, let's look at multiplication and division. And that's actually quite a bit easier to do. Because in the case of multiplication and division, the number of significant figures in the answer should be equal to the least number of significant figures of any of the quantities you use in the calculation. So you just look for the least number of significant figures of the numbers that you're using when your doing the multiplication and division, and that is the number of significant figures in the answer. Here's an example, 12.3 times 5.0. Well, that would equal 61.5 if I plot it into the calculator. But this number with the least number of significance going in is 5.0, which has two significant figures. So my answer also must have two significant figures. So I'm going to round my answer to 62. The same is true is for division. I can take the number 16.7 and divide it by 9.004 and my calculator might actually give me eight decimal places. But if I just wrote down a few, I would say 1.8547. But I see that there, in this case, the quantity going in that had the least number of significant figures had three significant figures, so my answer should also have three significant figures. It doesn't matter how many quantities you're using. If you're only doing multiplication and division, you just look for the quantity going in that has the least number of significant figures. And that's the number of significant figures in the answers, quite a bit simpler than the addition and subtraction case. Here's another example. Here I have all multiplication and division as my operations. So, in this case, I look for the quantity going in that has the least number of significant figures, which here is 3. I have three significant figures in each of those and four in the other number. So, my answer should have three significant figures. It should be expressed to three significant figures. Why am I doing that again? I'm doing that to show the level of precision, okay? We don't want to show a level of precision than, that is higher than what we actually had in any of our individual measurements. Now remember when you do math, you have to perform certain mathematical operations in the correct order. So you do the operation that's inside the parentheses first. Then, you do multiplications and divisions, and then you do additions and subtractions. You have to be careful here about what happens to the number of significant figures when you do each operation. I like to keep track of the significant figures by underlining the last digit that's significant in any intermediate answer that I have. Remember in addition, you might gain a significant figure. And in subtraction, you might lose a significant figure. So let's do an example of that. In this problem, I have 2.86 minus 2.43. That's inside the parentheses, so I would do that first. The number that results from that, does it have three significant figures, even though both of those numbers do have three significant figures? The number that results from that is 0.43, and it only have two significant figures. Okay. So just from looking at what's on this side of the parentheses, the answer from that, remember, can only be to the hundredths place because I'm using the decimal places when I do subtraction. That has two significant figures. I, then, take the number with two significant figures and divide it by 10.57 to give me an answer of 0.04068, which I'm going to round to two significant figures, 0.041. Let's do another example, 3.58 plus 6.73. In this case, I have two quantities that each have three significant figures in the measurement. But both of those measurements were precise to the one hundredth place. So when I add them together inside the parentheses, I get an answer of 10.31. That has four significant figures. Therefore, when I take that four significant figure number and divide it by a five significant figure number, my answer needs to be expressed to four significant figures. Does that make sense? Now it looks like, when I'm doing these operations, that I might actually be rounding during an intermediate operation step. And you need to be really careful about that, this. So, I'm going to give you a quick word about rounding errors. One of the things that can happen, if you have a calculation where, let's say you have five or six different operations that you need to do in the calculation, is that if you round an intermediate calculation, you can end up with lots of errors. So what I recommend that you do, and I'll show you an example so you see what I mean is that you always carry at least one extra non-significant digit through the multistep calculation, and then round at the end. And I actually like to carry a couple of extra digits the whole way through and use a line to underline the last significant digit to help me keep track. So here's an example. I can take the number 12.456, multiply it by 12 and then divide it by 3.697. The result of those operations is the answer 40.43. Now there's two significant figures. In the quantity going in that had the least number of significant figures. So my answer should also be expressed as two significant figures. And I can do that again by either showing that decimal point or using scientific notation. So I could say 4.0 times 10 to the first power. If I do the calculation inside the parenthesis and round it and then do the second calculation, I can introduce quite a bit of error. In other words, if I did it in two steps, if I took 12.456, multiplied it by 12, and got 149.47, but then I rounded it to two significant figures. Okay. And then I take that quantity, which is 150 now, and divide that by 3.697, now my answer becomes 41 if I round it to two significant figures. And that's quite a bit different than 40. How much error have I introduced? I've introduced a 2.5% error. That can be a really large error. The difference between 2.5% concentration of something in your bloodstream, for example, could be toxic. Missing by 2.5% if you're shooting at the moon could mean you just fly off into interstellar space. So 2.5% can be a really big mistake. So it's important to not do intermediate rounding. What I could have done when I did this two-step calculation, is instead of rounding it to 1.5 times 10 to the 2, what I could've done is written down my intermediate answer and underlined the 4, just to remind myself that I only had two significant figures. That's one way that I'd do it. Something you might want to try.


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