Math Antics - The Number Line

Hi, I'm Rob. Welcome to Math Antics. In this video, we're going to learn about an important tool for visualizing how different numbers relate to each other. That tool is called the number line. In the place value videos, you learned that our base 10 number system uses 10 different digits to express any number. But we didn't just use those digits randomly. We used them in a specific order for counting. We started with 0, which represents the idea of having nothing, and then the next digit we used was 1, which represented a single apple in our example. Then, if we got another apple, we used the digit 2, then another 3, then 4, 5, 6, 7, 8, and finally 9, which is the last of our 10 digits. Do you see how this pattern formed by the apples increases as it goes from left to right like going upstairs and decreases as it goes from right to left like going downstairs? Well, that's how the number line works. It consists of a long horizontal line with an arrow at the end, which I'll explain in a minute. For each digit, there's a tiny vertical mark showing where that amount belongs on the line. This particular version of the number line shows how the digits 0 through 9 relate to each other. It's set up so that as you go from left to right, each new number is bigger or greater than the number just before it, and as you go from right to left, each number is smaller or less than the number just before it. And do you notice that all the marks are the same distance apart? That's because each of these numbers differs from its immediate neighbors by the same amount. They're all 1 apart. Starting from 0, moving right to the first mark is an increase of 1. Likewise, moving from 1 to the next mark 2 is also an increase of 1. In other words, you just add 1 to get to the next number. And even though our current number line stops at 9, we could continue the process of adding 1 to make it longer if we wanted to. For example, 9 plus 1 equals 10, 10 plus 1 equals 11, 11 plus 1 equals 12, and so on. In fact, technically the number line goes on forever because there is no biggest number. Whatever number you decide to stop at, you could always just add one more to make the line longer. And that's what the arrow on the right side is for. It tells you that the numbers keep on going forever even though we could never write them all down. You'll probably see many different versions of the number line as you learn math. Some may stop at different numbers and some might not even start at zero, like this one that starts at 7 and stops at 15. One way to think about these different number lines is that they're all snapshots of the ultimate ideal number line that has every number on it. It's kinda like trying to take a picture of something really big, like an elephant. If you're too close, you'll only be able to fit a small part of it in each snapshot. Of course, if you back up far enough, you could fit the whole elephant in the shot. But the number line is INFINITELY big. No matter how far you zoom out, you can never fit it all in a single image or drawing. So you only draw the part of it that you need for whatever problem you're working on. For example, instead of a number line that stops at 10, maybe we need one that goes all the way to 20. That wouldn't be too hard to draw. But what if we needed to go all the way to 100? Would we need to draw a number line that shows all the numbers from 0 to 100? Fortunately, no. We actually have two different options that are much easier. The first option would work for situations where we only need to use a portion of the number line that's near 100. In that case, instead of starting our number line at zero, we could have it start at a bigger number, like 95, and increase by one until we get to 105 or so. All the marks are still one apart like the other number lines we've seen. We're just starting and stopping at different values. Notice that we have arrows on both sides now to show that the line keeps on going in either direction. The second option would work for situations where we need to use the entire range from 0 to 100. In that case, we could simply leave out most of the digit marks on the number line and just show a mark for each multiple of 10, like 10, 20, 30, 40 and so on. Let's compare that number line to the one we previously saw that went from 0 to 10. Even though these two number lines look the same size on the screen, the bottom one is actually 10 times longer. If we resize them so that they use the same scale, lining up the 0's and the 10's, you can see that most of the number line going from 0 to 10 fits in between the first two marks on the longer number line. And that helps you realize a very important idea. Just because a number line has marks showing specific numbers doesn't mean that there's nothing in between them. There's actually a LOT in between them as you'll see in a minute. In this example, the space between the 0 and the 10 on the longer number line still contains all the counting numbers 1 through 9, but for simplicity we left them out. But it's important to remember that they're still there, in case you need to use those in-between values. Like if someone asked you to put a point on the number line where 65 is located, you would put it halfway between the 60 and the 70. That's where 65 is, even though its mark isn't shown. Okay, but let's zoom back in on the number line that goes from 0 to 10. You may be wondering, are there any numbers in between these basic whole numbers? There sure are! In the video about decimal place value, we learned that special numbers called fractions are used to represent amounts that are less than 1, like one-half of an apple or one-tenth of an apple. Well, we can divide up the sections of the number line that are between the whole numbers to show exactly where those fractions would go. For example, what if we put an extra mark exactly halfway between each of the marks that we already have? What would these new marks represent? Well, the first one is pretty easy. It's halfway between 0 and 1, so it represents the number one-half, which happens to be the same amount as the decimal, 0.5. The next one is a little trickier since that mark is halfway between 1 and 2. That means that it's one-half more than 1 and one-half less than 2. In other words, it's the number 1 and one-half, which has the decimal value 1.5. The next in-between mark would be 2 and one-half or 2.5, and the next would be 3 and one-half or 3.5. Get the idea? But there's an even more useful way to subdivide the space between each of the whole numbers. If we divide up the space between 0 and 1 into 10 equal parts, you'll see that this works perfectly with our decimal number places. Each one of these new marks is the fraction one-tenth apart. So after 0, this first mark is one-tenth, and this second mark is two-tenths, the third is three-tenths, and so on. And remember, we have a decimal number place immediately to the right of the decimal point that's specifically used for counting tenths. Using that decimal number place also, this first mark has the decimal value 0.0 because it represents 0 ones and 0 tenths… and 0 everything for that matter. The next is 0.1 because we have 0 ones and one-tenth. The next is 0.2 because we have 0 ones and two-tenths. And the pattern continues on like that until we get to the number 1, which can be shown as the number 1 in the ones place and 0 in the tenths place. Because it's a whole number, it doesn't need any decimal digits, but we can put a zero in them if we'd like. As you might expect, we can keep going down the number line like that. The next mark is one-tenth more than the one, so it's 1.1. The next is 1.2 and the next is 1.3. We could go on, but if we wrote all of the tenths in the numbers between the whole numbers, our number line would get very cluttered again. The point is just to realize that the tenths are in between the ones on the number line. See how amazing the number line is? But we don't have to stop there because there are numbers in between the tenths too. If we zoom in further and divide the space in between this first tenth into 10 equal parts, each of those new smaller fractions is called a hundredth because it's the amount you'd get if you divided 1 into 100 equal parts. Again, remember that we have a number place for counting hundredths immediately to the right of the tenths place. So this first mark would be 0.01 because it's one-hundredth, and this next would be 0.02 because it's two hundredths. But why stop at hundredths? If we divide the spaces between each hundredth into 10 equal parts, we'll get thousandths, which are even smaller fractions. Can you see where we're going with this? Yup… to infinity. Just like there's no biggest number because you can always keep adding one more, there's no smallest number either because you can always keep dividing whatever small fraction you have into even smaller and smaller fractions. It's kinda mind-blowing if you think about it. Not only are there an infinite amount of numbers on the number line even if you just stick with the whole numbers and count them forever, which you can't actually do, but there's also an infinite amount of numbers in between each pair of whole numbers if you keep subdividing forever, which you can't actually do either. So what can you do? Well, you can use the number line to understand how numbers relate to each other. It's helpful because it's a visual representation that shows the range of values that get bigger as you move to the right and smaller as you move to the left. And that's the main thing you need to know about the number line. It's a tool to help you see how numbers work. It kind of helps you with the big picture of math. A lot of the time you won't need it, but it's always there when you do. Alright, that's it for this lesson. There aren't many practice problems for this video because the main goal is for you to just understand how the number line works. But don't worry, there'll be plenty of opportunities to practice math in future lessons. As always, thanks for watching Math Antics and I'll see ya next time. Learn more at www.mathantics.com