 Greetings and welcome to Math Help for Science Courses. In this class lecture, we are going to be discussing significant figures and how those are used in calculations. And what essentially it means is that when you put numbers into your calculator and do a calculation, your calculator assumes everything is exact and will give you this tremendous number of figures, most of which are really meaningless. That does not really mean anything because they are only as accurate as the numbers that went into the calculation. So let's get started and look at a few things here. First of all, what do we mean by significant figures? And some numbers are exact. So if we are right accounting the number of eggs in a dozen, we should, if ten different people do it, they should all get exactly twelve eggs. So that is an example of an exact number. Two wheels on a bicycle, there are always exactly two. So that if we account them, everybody is going to get exactly the same number. So these are numbers that have as many significant figures as we need. They are exact and if you wanted to write them out, you could write twelve point and you could put all the zeros you want there because there is no doubt as to how many eggs there are in a dozen or how many wheels there are on a bicycle. So these have unlimited significant figures because they are exact. However when we are measuring things as we often do in science, well we get different measurements. For example, you might measure a piece of paper. And if we have everybody measure it and do it with different instruments, we might find one student gets 220 millimeters, which is two significant figures. Someone with a slightly more accurate device might get 218 millimeters, which is three significant figures. And finally, someone with an even more accurate device may be able to measure it to the tenth of a millimeter and get 217.6 and that is four significant figures. Note that that does not mean that one of the answers is right and the other is wrong. These are all the same. And this is what it means when we make measurements. Sometimes we cannot measure things as accurately, so we have less significance in it. So there are only two figures. This might be 220, but it could also be, if measured more accurately, it could be 218, or maybe something else measured would be 222. It means we simply do not know other than that it is approximately 220. So it makes a difference when we are doing our calculations how many figures we will leave in that answer. So let's look at the rules that we use for determining significant figures. The first is that any non-zero digit is always significant. It does not matter what it is. One, two, three, four, five, six, seven, eight, or nine. If it is not zero, it is always a significant digit. The other rule tells us that leading zeros are never significant. So if you have the number 12.5, I can put zeros before that all I want, but they are never significant. They do not count for significant figures. Also, where this really matters is in decimals. Let's write 0.003. That is only one significant figure. These leading zeros are needed in this case as placeholders because there is a big difference between this and just writing 3, or just writing .3. There is a big difference in those numbers. So we need the placeholders here, but they are not significant. So those leading zeros, when you start from the left to the right, until you find a non-zero number, those leading zeros are never significant, no matter where they appear. So we have non-zero digits. We looked at leading zeros. We can also have embedded zeros. Any zeros in between two non-zero numbers, that is called an embedded zero, that is always significant because it is in between. And the other possibility, we can have leading zeros, we can have them in between, and we can have trailing zeros. They are only significant if the decimal point is specified. So 250, the zero is not significant. However, 250, with the decimal point specified, this one has two significant figures. This one has three significant figures. If we specify the decimal point, the trailing zeros are significant. And we're going to look at a couple of examples on this on the next page. But finally, we want to look at numbers in scientific notation. In often cases, we use numbers written in scientific notation. And how do we tell what's significant there? Well, if we write 2.53 times 10 to the eighth, this is essentially a placeholder for a bunch of zeros. All we look at is the number out here, and it should be all the digits to the left should always be significant if it's written in standard scientific notation form. So in this case, it would have 1, 2, 3 significant figures. This is needed. You can't get rid of the 10 to the eighth power. However, it is not used for calculating the number of significant figures. So let's look at a couple of examples of this. And here we have a few that we can look at. First of all, 300. 300, these last two are trailing zeros, so they are not significant. So this first one has one significant figure, only the three. In the second example, we still have 300, but now it's 300.0. So now we have that decimal point in there, and that makes, remember our rule that says if the decimal point is specified, then all of those trailing zeros become significant. So this one would then have four significant figures. Now, the next one is a very tiny number, 0.00052. These are all leading zeros, and recall leading zeros are never significant, so this would then have two significant figures. The next number here, we have a number of zeros and a number of non-zero numbers. The zeros are all embedded in between non-zero numbers, and embedded zeros are always significant, so that would mean that this one actually has seven significant figures. Here we have a kind of a combination. We have a leading zero here, and we have some trailing zeros. There is a decimal point specified, meaning that the trailing zeros are significant, but the leading zeros are never significant. So this would then have four significant figures. That would have the two and the three trailing zeros. Finally, looking at an example in scientific notation, again, we only look at the part in front. This is just placeholders for zeros, so 6.58, none of those are zeros, and that would be three significant figures. So that's some of the examples as to how you can figure out how many significant figures, and you go back and use those rules that I gave you on the previous slide that will allow you to tell exactly how many significant figures there are in a number. So how does this apply to calculations that we might be doing? Well, we have to look at the rules for calculations in significant figures. When we are adding or subtracting, the last digit to be saved is the input number that is the most estimated. Now, that's a little difficult to say, means it ends at the highest place value. So if you were adding 2.5 plus 17, say, you would then not look at the number of significant figures in each, but you would add these together to get 19.5 first, but then you would say that this ends at the tenth's place, but this one ends at the one's place. You cannot go beyond the one that is most estimated here, so your answer has to stop at the one's place, and therefore 19.5 becomes 20 with a decimal point to specify that it is two significant figures. If you just write 20, then that's looking like it's only one significant figure, so it has to be 20 with the decimal point in this case. So when you're adding or subtracting, you look at the last place that you go to. When you're multiplying or dividing, then you look at the number of significant digits in each input number and count them, and your answer has to go with the least, whatever has the least. So if you had one input number with two significant figures and one with three, then your answer has to have three significant figures. So let's look at a couple more examples of these, and what we find is if we multiply, for example, this first one, if we multiply these two numbers together and you put those in your calculator, you would get 1.622124 times 10 to the sixth power. Now, to figure out, since we're multiplying, we would then use the number of significant figures in each. This one has four significant figures, this one has three, so we go with the least one of three, and we have to round this to just three significant figures. So we count from here 1, 2, 3. We would keep this portion of it. This part then gets dropped or rounded, and your answer would be 1.62 times 10 to the sixth power. Now, you cannot get rid of the 10 to the sixth power. That is still needed as a placeholder. Now, this could also have been written. Just to look at it slightly different, you could write it as, without writing it in scientific notation. So if we write it out as 1,622,124, we can do the same thing. We still need three significant figures in the answer, which would be these first three. However, we can't just write 162, because there is a big difference between 162 and 1,620,000, which we'd be rounding it. The other number itself is well over a million. We can't just round that down to 162. We still need to use placeholder zeros for these, so we'd put the four zeros there, and that would then give us 1,620,000. We do not put a decimal point at the end, because that would then specify that these zeros are significant when they are not. All that matters is that when you multiply these two numbers, this would be the answer, either of these, in the correct number of significant figures. Now let's look at a second example. If you put all of this into your calculator and write out your answer of 3.476190476 times 10 to the fifth, then you again have to look at these for the number of significant figures. I will put one aside as a warning here. If you try this in your calculator and you do not get this answer, make sure you look at the video which explains how to use a scientific calculator and how to enter these numbers correctly. If you enter this as a multiplication, using the multiplication key, that is wrong, and your calculator will see it as two separate numbers, and it will then divide and then multiply by the exponent. It will see it as two separate numbers. This is really one number, so make sure you're using the E or the double E or the EXP key to enter numbers in scientific notation. So let's count how many significant figures. We have three here, two here. How many do we have in two-thirds? Well, you could look at it as each of these having one, but in reality, two-thirds is an exact number. It's .6666666 going on forever. So fractions like this would have as many significant figures as we need. So this has an infinite number of significant figures. We don't need to use it. We just need to look that this one has two, this one has three. We then have to round this to two significant figures, which would be three-point. We would round this up to 3.5 because this is greater, a larger number here, and it would be 3.5 times 10 to the fifth would be our answer in scientific notation. Now, let's look at a couple examples of adding and subtracting. If for this first one we add these three numbers given, we would get 737.6598. Now, when we're adding and subtracting, we do not look at the number of significant figures in each number. That does not matter. We look at the places where the number ends. So this first number goes down to the ten thousandths place. This one stops at the ones place, and this one stops at the tenths place. So our answer has to stop at the least of these, which would be the one most rounded would be the ones place. So we have to round this to the nearest one, which would mean we'd round this, we'd get rid of this, round this up to 8, and our answer would be 738 in the correct number of significant figures. So we are looking at the places where things end in this case, and not the number of significant figures in each as we did with multiplication and division. Now, one more we can do here, if we add the add and subtract these three numbers, we would find we get negative 0.09. So where are we going to do this one? Remember, we look at the places where they stop, this one is at the tenths, this one is at the tenths, and this one is at the hundredths. So we need to round this to the tenths place. And if we round this to the tenths place, that would be negative 0.1. So that would be rounded to the correct number of significant figures. And going anything more than that would be, while sometimes students feel like that makes it things more inaccurate, in reality when you're starting to add more and more to that, you are making the numbers inaccurate. We don't know it accurately enough. We do not know the numbers that went into these accurately enough to say these exact answers are correct. If we knew things were precise, if this was precisely 25.1, 41.5, and 16.31, then we could say this about the answer. However, if there is measurement uncertainty in the last digit, we don't know what the hundredths place was in any of these. Could 25.1 really have been 25.06, or was it 25.14? Either of those would round to 25.1, and we would not know which is correct. So we could get various answers for the actual answer, but when we use the correct number of significant figures, we know that is as accurately as we possibly know the answer. So let's finish up with our summary here. We talked about exact numbers and some that are measured. There is a big difference. Exact numbers have as many significant figures as you need. The measured numbers have a limited number of significant figures. We talked about the specific rules for determining how many significant digits there are in a number, and the specific rules for determining the number of significant digits in an answer after a calculation has been done, and we call that it was different if you were multiplying or dividing, and or adding and subtracting. So that concludes our lecture on significant figures. We'll be back again next time for another Math Help video. So until then, have a great day, everyone, and I will see you in class.