Understanding and Mastering Rounding Numbers

Hi, welcome to Math Antics. In this video, we're going to learn about an important math concept called rounding. To help you understand what rounding is, let's think about how numbers are usually used. Most of the time, numbers are used to represent amounts of things, like how many miles it is to the supermarket, or how many days until your birthday, or how many students went to your high school. How many students went to my high school? Oh, about 2,000. Okay, but it wasn't exactly 2,000, was it? Well, no. It was more like 1,900. Ah, but it probably wasn't exactly 1,900 either, was it? Well, no. It was more like 1,860. Alright, fine. 1,863. See what I did there? At first, the number used to represent the students at high school was a round number. It was a good estimate of how many students there were, but it wasn't exact. The next two numbers were a little closer to the truth, but they were still estimates. Only the final number represented the exact amount of students at the school. All three of the estimates are rounded versions of the exact count, but they have different levels of precision. 1,860 was the most precise estimate and 2,000 was the least precise estimate. So rounding a number basically means making a less precise version of it. And as you can see, there's usually multiple ways to round a number depending on the level of precision that you need. A really good way to understand what's going on when you round a number is to look at a number line. Here's 1,863. If we want to round it to the nearest 10, we need to decide if it goes up to 1,870 or down to 1,860. But if we want to round it to the nearest 100, we need to decide if it goes up to 1,900 or down to 1,800. And if we want to round it to the nearest 1,000, we need to decide if it goes up to 2,000 or down to 1,000. And in each case, the decision was based on which round number was closer to the original exact number. But you might be wondering… Why would we ever want to make a number less precise in the first place? What is rounding good for? Well, rounding numbers can often make them a lot easier to do calculations with. Like, it would be a lot easier to quickly add 300 and 500 than it would be to add 312 and 498. Or sometimes you just don't need very much precision. Like, you might not need to know that your dog weighs 55.83297 kilograms, 55.8 kilograms might be precise enough. And some numbers, like repeating decimals or irrational numbers, have to be rounded off, because we can't just keep writing decimal digits forever. OK, now that you know what rounding is and why we do it, for the rest of this video we're going to focus on learning the procedure we follow to round off a number. Do you remember how our number system is based on digits and number places? Each digit of a number occupies a particular number place, and each number place is named according to the amount it represents, or counts. And it's important to know those names whenever you're rounding a number, because you'll usually be asked to round to a specific number place. For example, you may be asked to round a number to the nearest 10, or the nearest 100. Or you might be asked to round a number off to the nearest tenth, or hundredth. You may even be asked to round to the nearest whole number. which is another way of asking you to round to the ones place. OK, so when you're asked to round a number, the first step is to pay very close attention to which number place you need to round to. That number place is important because it represents the smallest unit of counting that you're going to keep in your rounded version of the number. In fact, that number place and the digit inside it is so important that I'm going to give it a special name just for this video. Let's call it the Target. As I mentioned, rounding a number means making a new, less precise version of it. In that new number, any digits that are in number places smaller than the target will automatically get replaced with zeros. And in most cases, any digits that are in number places larger than the target will automatically be kept the same in the new rounded version. There are some exceptions, as we'll see later in this video. So that seems pretty simple. All the bigger digits you keep, and all the smaller digits you zero. But what about that target digit itself? What do we do with that? Well, we're going to do one of two things. We're either going to keep that digit the same, or we're going to increase it by one. If we keep that target digit the same, that's called rounding down. Which might seem strange at first. I mean, how can leaving the digit the same be rounding down? But remember, we're going to automatically replace all of the smaller places with 0. And doing that makes a rounded number smaller, even if the target digit stays the same. On the other hand, increasing the target digit by 1 is called rounding up, since the new rounded number will be larger than the original number. Alright, but how do we decide which to do? How do we know if we keep the target digit the same or increase it by one? The key is to look at the digit in the next smaller number place, the digit that's just to the right of the target digit. If that digit is less than 5, in other words, if it's a 0, 1, 2, 3 or 4, then we'll leave the target digit the same in the rounded version. But if the digit is a 5 or greater, 5, 6, 7, 8 or 9, then we'll increase the target digit by 1. OK, so now that you know the basic procedure for rounding numbers, let's try a few specific examples. Here's the first one. Round 24,623 to the nearest hundred. Since we need to round to the nearest hundred, we first need to identify the digit in the hundreds place. That digit is a 6, so that's our target. And we know that any digits to the right of the target will be replaced with zeros in our rounded version. Next let's decide what to do with the target digit. We either keep it the same or we increase it by one. To decide, we look at the value of the next digit to the right. Since that digit is only a 2, which is less than 5, we round down, which means that we'll keep the target digit the same in the rounded number. Last, we just keep all the digits in bigger number places the same in the rounded version. There! We've rounded the original number to the nearest hundred and the answer is 24,600. Let's try another problem. This one has some decimal digits, 32.725, and we're asked to round it to the nearest whole number. That means our target digit is in the ones place. We need to round it to the nearest one. So any digits to the right of the ones place will just be replaced with zeros in the rounded version. Now, to decide what to do with the target digit, We look at the next digit to the right. Since that digit is a 7, we'll round up this time. That means we'll increase our target digit by 1. And finally, we keep any digits to the left of the target digit the same. In this case, that's just the 3. So we've rounded this number off to 33.000. Or just 33, since we don't really need those extra zeros after the decimal point. Ready for one more? Let's round 65.7991. to the nearest hundredth. The first step is to identify the hundredth's place as our target. That place contains the digit 9. All the digits in smaller number places will just be replaced with 0 in the rounded version. Next, we need to decide if we leave the target digit the same or increase it by 1. So we look at the digit to the right of the target. It's a 9 also, so we'll definitely be rounding up. But since the target digit is already a 9, raising it by 1 is a little more complicated. When you add 1 to a digit that's already 9, you need to change it to 0 and increase the digit in the next bigger number place by 1. So that means that our target digit will become 0 and we need to increase the digit in the next bigger number place. That digit is a 7, so we'll increase it to an 8. The rest of the digits in the original number will be kept the same in the rounded version. So our rounded version will be 65.80. As you can see, In some cases, rounding can actually change the digits to the left of the target digit also. It's sort of a domino effect that can happen when rounding numbers. If you have a lot of 9s, rounding can bump them all up like a chain reaction. Like, what if you need to round 1,999,999 to the nearest 10? The 9 in the ones place tells us that we need to round our target digit up by 1. But it's already a 9, so we need to zero it and increase the next number place. But that's already a 9, so we need to zero it and increase the next number place. But that's already a 9, and so the pattern continues until we end up with 2,000,000 as our final rounded number. So sometimes rounding a number is pretty simple, and other times it's a little more involved. The key is to remember the rule that if the digit to the right of the target is less than 5, we leave the target digit the same. But if it's 5 or more, we increase the target digit by 1, even if that causes a chain reaction with the bigger number places. Alright, so now you know a lot about rounding numbers. You know why we round numbers, and you've seen the basic procedure in action. But just watching a video about rounding isn't enough to get really good at it. The only way to do that is to practice. So be sure to try rounding some numbers on your own. In fact, rounding is such an important math skill that you should probably practice it a lot. until you've really got it mastered. As always, thanks for watching Math Antics and I'll see you next time. Learn more at www.mathantics.com

How many students went to my high school? Oh, about 2,000. Okay, but it wasn't exactly 2,000, was it? Well, no. It was more like 1,900.

Ah, but it probably wasn't exactly 1,900 either, was it? Well, no. It was more like 1,860. Alright, fine. 1,863.

See what I did there? At first, the number used to represent the students at high school was a round number. It was a good estimate of how many students there were, but it wasn't exact. The next two numbers were a little closer to the truth, but they were still estimates.

Only the final number represented the exact amount of students at the school. All three of the estimates are rounded versions of the exact count, but they have different levels of precision. 1,860 was the most precise estimate and 2,000 was the least precise estimate. So rounding a number basically means making a less precise version of it. And as you can see, there's usually multiple ways to round a number depending on the level of precision that you need.

A really good way to understand what's going on when you round a number is to look at a number line. Here's 1,863. If we want to round it to the nearest 10, we need to decide if it goes up to 1,870 or down to 1,860.

But if we want to round it to the nearest 100, we need to decide if it goes up to 1,900 or down to 1,800. And if we want to round it to the nearest 1,000, we need to decide if it goes up to 2,000 or down to 1,000. And in each case, the decision was based on which round number was closer to the original exact number. But you might be wondering… Why would we ever want to make a number less precise in the first place? What is rounding good for?

Well, rounding numbers can often make them a lot easier to do calculations with. Like, it would be a lot easier to quickly add 300 and 500 than it would be to add 312 and 498. Or sometimes you just don't need very much precision. Like, you might not need to know that your dog weighs 55.83297 kilograms, 55.8 kilograms might be precise enough.

And some numbers, like repeating decimals or irrational numbers, have to be rounded off, because we can't just keep writing decimal digits forever. OK, now that you know what rounding is and why we do it, for the rest of this video we're going to focus on learning the procedure we follow to round off a number. Do you remember how our number system is based on digits and number places?

Each digit of a number occupies a particular number place, and each number place is named according to the amount it represents, or counts. And it's important to know those names whenever you're rounding a number, because you'll usually be asked to round to a specific number place. For example, you may be asked to round a number to the nearest 10, or the nearest 100. Or you might be asked to round a number off to the nearest tenth, or hundredth.

You may even be asked to round to the nearest whole number. which is another way of asking you to round to the ones place. OK, so when you're asked to round a number, the first step is to pay very close attention to which number place you need to round to.

That number place is important because it represents the smallest unit of counting that you're going to keep in your rounded version of the number. In fact, that number place and the digit inside it is so important that I'm going to give it a special name just for this video. Let's call it the Target.

As I mentioned, rounding a number means making a new, less precise version of it. In that new number, any digits that are in number places smaller than the target will automatically get replaced with zeros. And in most cases, any digits that are in number places larger than the target will automatically be kept the same in the new rounded version. There are some exceptions, as we'll see later in this video.

So that seems pretty simple. All the bigger digits you keep, and all the smaller digits you zero. But what about that target digit itself? What do we do with that?

Well, we're going to do one of two things. We're either going to keep that digit the same, or we're going to increase it by one. If we keep that target digit the same, that's called rounding down.

Which might seem strange at first. I mean, how can leaving the digit the same be rounding down? But remember, we're going to automatically replace all of the smaller places with 0. And doing that makes a rounded number smaller, even if the target digit stays the same. On the other hand, increasing the target digit by 1 is called rounding up, since the new rounded number will be larger than the original number. Alright, but how do we decide which to do?

How do we know if we keep the target digit the same or increase it by one? The key is to look at the digit in the next smaller number place, the digit that's just to the right of the target digit. If that digit is less than 5, in other words, if it's a 0, 1, 2, 3 or 4, then we'll leave the target digit the same in the rounded version.

But if the digit is a 5 or greater, 5, 6, 7, 8 or 9, then we'll increase the target digit by 1. OK, so now that you know the basic procedure for rounding numbers, let's try a few specific examples. Here's the first one. Round 24,623 to the nearest hundred. Since we need to round to the nearest hundred, we first need to identify the digit in the hundreds place.

That digit is a 6, so that's our target. And we know that any digits to the right of the target will be replaced with zeros in our rounded version. Next let's decide what to do with the target digit. We either keep it the same or we increase it by one.

To decide, we look at the value of the next digit to the right. Since that digit is only a 2, which is less than 5, we round down, which means that we'll keep the target digit the same in the rounded number. Last, we just keep all the digits in bigger number places the same in the rounded version.

There! We've rounded the original number to the nearest hundred and the answer is 24,600. Let's try another problem. This one has some decimal digits, 32.725, and we're asked to round it to the nearest whole number.

That means our target digit is in the ones place. We need to round it to the nearest one. So any digits to the right of the ones place will just be replaced with zeros in the rounded version. Now, to decide what to do with the target digit, We look at the next digit to the right.

Since that digit is a 7, we'll round up this time. That means we'll increase our target digit by 1. And finally, we keep any digits to the left of the target digit the same. In this case, that's just the 3. So we've rounded this number off to 33.000. Or just 33, since we don't really need those extra zeros after the decimal point.

Ready for one more? Let's round 65.7991. to the nearest hundredth. The first step is to identify the hundredth's place as our target. That place contains the digit 9. All the digits in smaller number places will just be replaced with 0 in the rounded version.

Next, we need to decide if we leave the target digit the same or increase it by 1. So we look at the digit to the right of the target. It's a 9 also, so we'll definitely be rounding up. But since the target digit is already a 9, raising it by 1 is a little more complicated.

When you add 1 to a digit that's already 9, you need to change it to 0 and increase the digit in the next bigger number place by 1. So that means that our target digit will become 0 and we need to increase the digit in the next bigger number place. That digit is a 7, so we'll increase it to an 8. The rest of the digits in the original number will be kept the same in the rounded version. So our rounded version will be 65.80.

As you can see, In some cases, rounding can actually change the digits to the left of the target digit also. It's sort of a domino effect that can happen when rounding numbers. If you have a lot of 9s, rounding can bump them all up like a chain reaction.

Like, what if you need to round 1,999,999 to the nearest 10? The 9 in the ones place tells us that we need to round our target digit up by 1. But it's already a 9, so we need to zero it and increase the next number place. But that's already a 9, so we need to zero it and increase the next number place.

But that's already a 9, and so the pattern continues until we end up with 2,000,000 as our final rounded number. So sometimes rounding a number is pretty simple, and other times it's a little more involved. The key is to remember the rule that if the digit to the right of the target is less than 5, we leave the target digit the same. But if it's 5 or more, we increase the target digit by 1, even if that causes a chain reaction with the bigger number places. Alright, so now you know a lot about rounding numbers.

You know why we round numbers, and you've seen the basic procedure in action. But just watching a video about rounding isn't enough to get really good at it. The only way to do that is to practice. So be sure to try rounding some numbers on your own. In fact, rounding is such an important math skill that you should probably practice it a lot.

until you've really got it mastered. As always, thanks for watching Math Antics and I'll see you next time. Learn more at www.mathantics.com

Transcript for:Understanding and Mastering Rounding Numbers

Transcript for:
Understanding and Mastering Rounding Numbers