 Greetings and welcome to Math Help for Science Courses. In this video, we are going to be talking about how and why we go about rounding numbers. So we're going to look a little bit about why we do that, and then talk about how we go about rounding numbers when we are doing calculations. So let's get started here. First of all, why do we round numbers? Well, we can do it for a couple of different reasons. We can approximate a number. We may just need a rough idea of a number and not need to know the exact value, and maybe we don't have a calculator available, we want to do something in our head. So we can make the calculation simpler by rounding the numbers. For example, if we were to multiply 1,987 by 195, we could do that. You could write it out and do long multiplication, you could plug it into a calculator, and find out the answer is 387,465. However, if you just need to do a quick approximation, we know that 1,987 is approximately 2,000. We know that 195 is approximately 200, and we can multiply those together and find out that the answer is about 400,000. If we don't need an exact value but just a rough calculation, we can see that 400,000 is not all that far off from 387,465. Now, we also round numbers to present it in the correct number of significant figures. So we might go through a calculation, and we might end up with our calculator giving us all sorts of decimal places that we know are not significant. So if we go through, for example, a calculation, and we get an answer of 34.6978, and we find that that is only supposed to have three significant figures, then we can use rounding and that we only want to cut it off here at three significant figures, but because the figure to the right of that where we're rounding is five or greater, we would then round that up. So we would round this up to 34.7. So we can do it in those cases, and we can also look at things like measurement inaccuracies. We can only measure so accurately, so we may just round our numbers based on that. So we may be only able to measure something, say, to the nearest millimeter with a ruler. We might not be able to get it to a tenth of a millimeter. It'd be very difficult to measure it accurately to a tenth of our millimeter. So therefore we round the numbers as well. So let's look at the rules for rounding numbers, what we need to do here, and first of all we have to determine what is the rounding digit. So where are we rounding to? Then we look to the digit to the right of the rounding digit. So if we were going to round 387 to the tens place, the eight would be the rounding digit, and the number to the right of the rounding digit would be a seven. If that number is zero, one, two, three, or four, we just drop all numbers to the right, put in placeholder zeros as needed, and that would be the end of it. However in this case it is a five, six, seven, eight, or nine, and in those cases we increased the rounding digit by one, so we would increase the eight to a nine, and we would need to add in a place holding zero to not change the number, because we certainly don't want to change 387 to 39. There is a big difference there. When rounding it it would be 390. So it just depends on what the number is to the right of your rounding digit. If it's four, three, two, one, or zero, then you just drop the numbers, add in a zero placeholder if needed, and you're done. If it's a five, six, seven, eight, or nine, you increase your rounding digit by one, and then drop the remaining numbers to the right. Now if your rounding digit is to the right of a decimal point, then you just drop the numbers completely. So if we were rounding 0.3862, and we wanted to round that to the hundredths place here, again we look at our digit, we would say we would increase that by one, which would make it 0.39, and we're done. We don't have to add any zeros, and in fact adding zeros would be a bad thing, because we would be counting them as significant figures, because they are to the right of the decimal point. If your rounding digit is to the left of a decimal point, then you need to replace the drop numbers with a zero as a placeholder, as we did in the example here. And we're going to do a couple more examples to show you this. So let's look at a few examples here. And our first example, we want to round 3521 to the tens place. So a 2 is our rounding digit. The number to the right of that is a 1, and since that is in the small range, then we just drop it, and this would become 352, but we need to put the zero in, because the numbers are to the left of the decimal place. So we have to have that placeholder zero, so we do not change the actual number. So if we rounded 3521 to the tens place, we would get 3520. What if we were to round 0.3521 to the hundredths place? Well, that would be rounding it to here, and that would mean we would then round it. We look at the number to the right, which is a 2, which means we can just drop the numbers, because this is all to the right of the decimal. We just drop the numbers and get 0.35. So we don't add any zeros when it is to the right of the decimal point. Now let's look at the next couple. The next one is we want to round 1,582,804 to the thousands place, which is right here. We look at the number to the right, that is an 8, because that is greater than 4, then we need to up this number a little bit. So we need to do 1, 5, 8. We up the 2 by 1 to a 3, and then because these are to the left of the decimal point, we have to add in placeholder zeros for each of these numbers. Meaning that if we round 1,582,804 to the thousands place, we get 1,583,000. Let's look at a couple more examples here. If we have 0.89578, and we want to round it to the tens place, again we look at the number to the right, which is a 9, because that is a 5 or greater than we up this number by 1, so that becomes 0.9, and because we're to the right of the decimal place, again we drop everything. So we can just drop the rest of those numbers, we've rounded it up, and our answer to the tens place would be 0.9. And our last example here, 25.9854, and we want to round it to the ones place. Again look at the number right to the right, that's a 9, since that is 5 or greater, then we up this by 1, that becomes 26, and now that we are still all to the left of, all of our digits are to the left of the decimal point, we are done, we just drop all of those decimals, and our answer rounded, rounding 25.9854 to the ones place would be just 26. So let's finish up with our summary here, and what we find is that numbers can be rounded for various reasons, because of approximation, to simplify calculations, or perhaps to present the number in the correct number of significant figures. There are specific rules that we went over to determine how to go about rounding a number. You always look at the number, you determine your rounding number, then look at the number to the right of it. If it is 5 or greater, then you increase the rounding digit by 1. If it is 4 or less, then you simply ignore the numbers to the right of the rounding digit, and you always have to add placeholders if the number is to the right of the decimal point, I'm sorry, to the left of the decimal point. If it is to the left of the decimal point, you add placeholders, if it is to the right of the decimal point, you do not. So it is the key is always looking at the number to the right of that rounding digit. So that concludes our lecture on rounding numbers. We'll be back again next time for another math topic for science courses. So until then, have a great day everyone, and I will see you in class.