Math Antics - Decimal Arithmetic

Hi, I'm Rob. Welcome to Math Antics. In this lesson, we're gonna learn about decimal arithmetic. But before we get started, if you don't already know how to do multi-digit arithmetic with regular whole numbers, be sure to watch our videos that cover those subjects first. That's really important because I'm just going to show you how you can modify the procedures that we already learned in those videos so that they work for decimal numbers. So if you don't know how to do those procedures already, this video won't make very much sense. Specifically, you should make sure you've watched the videos about multi-digit addition, subtraction, multiplication, and long division. If you know how to do the problems in those videos, then decimal arithmetic won't be too hard. That's because the procedures for decimal arithmetic are basically the same as they are for whole numbers. But there's a few important differences that you need to know about, and that's what I'm going to show you in this video. Are you ready? Let's start with an easy one… multi-digit addition. When adding multi-digit whole numbers, the key was to stack the numbers up so that the ones-place digits line up in a column, which ensured that all the other number places lined up in columns too. Then, you just add up the digits in each column, starting with the ones-place and working to the left. Well, adding multi-digit decimal numbers works the same way. The main difference is that instead of lining up the ones-place digits when we stack the numbers, we line up the decimal points instead. But wait a minute… I mean, isn't that the same thing as lining up the ones-place digits? Ah… yes it is! And that's because the decimal is our reference mark that always goes between the ones-place and the tenths place. So, lining up the decimal points is the same thing as lining up the ones-places. It makes sure all the number places line up in columns. Now, you've probably noticed that decimal numbers can have different numbers of decimal digits. For example, 10.8 has only one decimal digit, but 5.34 has two decimal digits. And what that means is that when you line up the decimal points of the two decimal numbers, they might not form a nice column on the right edge. Some of the digits might be missing. But that's no problem. Remember, if there's not a digit in a particular number place, you can just put a zero there to help you keep track of things. Now that these numbers are lined up by their decimal points, we can add them column by column. But instead of starting with the ones-place, like we always did with whole numbers, we start with whatever number-place column is the furthest to the right. In this case, that's the hundredths place, so we'll start there. So, we add the digits in each column, carrying or re-grouping as needed, and we get 1614. So we're done, right? WRONG! There's one last, REALLY important step. Remember, we're doing decimal addition, so we can't just forget about that decimal point. We need to bring a copy of it straight down into our answer line, so we keep the same reference point for our number places. Now we can see that the answer is really 16.14. That's not so hard, is it? And I've got more good news! Decimal subtraction works the same way. You start by lining up the decimal points of the two numbers. Remember that the order of the numbers matters in subtraction, so be sure that the number you're taking away is on the bottom. Then, starting with whatever column is furthest to the right, you subtract the digits column by column, borrowing if you need to. After that, you just bring down a copy of the decimal point and you have your answer. Okay, so decimal addition and subtraction are pretty easy. Let's move on to something a little harder… decimal multiplication. Now, as you know, multi-digit multiplication is more complicated because there are so many multiplication steps. But the good news is that decimal numbers don't really make the procedure much harder than it is with whole numbers. That's because there's a clever way that we can make decimal multiplication look exactly like the multi-digit multiplication with whole numbers that you already know how to do. The key is to pretend that the decimal points are not really there. Hold on a minute… I mean, I like pretending as much as you do, but if we just pretend that the decimal points aren't even there, we aren't going to get the right answer, are we? Well, no, but the only thing that will be wrong with the answer is that the decimal point won't be in the right spot, so we'll need to fix that at the end. I know it sounds a little confusing, so here's an example that should help you understand. I KNEW you would say that! Let's say that you need to multiply 3.65 by 2.4. Now that seems a little tricky, but what if we just pretend that the decimal points are not there for now? In other words, what if we pretended that the numbers were 365 and 24? You already know how to do that problem. You would just follow the procedure that we learned in Multi-Digit Multiplication Part 2 and you'd get the answer 8,760. But that's the answer to 365 times 24, not 3.65 times 2.4. So it's time to stop pretending. To get the correct answer for the decimal problem, we've got to understand what's going on with those decimal points and why we just pretended they weren't there. The truth is, when we pretended that the decimal points weren't there, what we were really doing is pretending that they had been shifted until both of our numbers became whole numbers. Remember, the numbers 365 and 24 technically DO have decimal points. They're right there next to the ones place. We just don't need to show them since there aren't any decimal digits. So, by ignoring the decimal points, what we were really doing is mentally shifting the decimal points to the right. We shifted the top decimal point two places to the right, and we shifted the bottom decimal point one place to the right. But doing that changed the numbers. It made the top number 100 times bigger than the decimal version, and it made the bottom number 10 times bigger. That's because every time you shift the decimal point one number place to the right, it's like multiplying by a factor of 10. And that means the answer we got is WAY too big. It's too big by 3 factors of 10, because the decimal points in our problem got shifted a total of 3 places to the right. Two on the top and one on the bottom. So to fix that, we're going to have to shift the decimal point in our answer the same amount in the opposite direction. In other words, we need to move the decimal point in our answer 3 places to the left, which will make it smaller by 3 factors of 10. So, starting right here, where the decimal point would be if our answer was 8760, we shift it 3 places to the left and we end up with 8.760, or just 8.76. And that is the answer to 3.65 x 2.4 That's a cool trick, huh? It means that you can do decimal multiplication just like regular multi-digit multiplication. You start by setting up your multiplication problem exactly like you would if the decimal points were invisible. But don't just erase them, because you'll need them at the end to figure out how many places to shift the decimal point in the answer. Then, keep ignoring the decimal points while you follow the multiplication procedure. Once you have an answer, count up how many places the decimal points are shifted in the problem you're working. Don't forget, it's the total shift of both the top and bottom decimal points. And then, shift the decimal point in your answer to the left that same number of places. So decimal multiplication turns out to be not too bad after all. But what about decimal division? That's gotta be hard, right? Well, multi-digit division is always a little hard, but luckily, decimals don't really make it very much harder. In fact, it's only when there's a decimal divisor that the procedure is a little different. If you just have a decimal dividend and the divisor is a whole number, it's really simple. That's because you can just do the long division procedure that we learned in the long division videos and the decimal point doesn't affect it at all. You just need to make sure that you bring a copy of the decimal point up into the answer line when you're done. So, if you have the division problem, 12.64 divided by 4, you would follow the division procedure as if the decimal point was not even there, and you'd get 316 in the answer line. But then, you need to bring a copy of the decimal point straight up into the final answer making it 3.16. That's all there is to it! IF it's only the dividend that's a decimal number. But what if BOTH the divisor AND the dividend are decimals? Like, what if you have to divide 6.45 by 1.5? Well, the first step is don't panic. As you'll see, this isn't much harder. It turns out that there's a very simple trick that we can use to make it so our divisor is NOT a decimal number. We can just shift the decimal point in the divisor to the right until it's a whole number. But, if we do that, then we also need to shift the decimal in the dividend the same amount to the right. So in this case, if we want to shift the decimal point in our divisor one place to the right so that it's 15, we can do that as long as we also shift the decimal point in the dividend by the same amount, which will turn it into 64.5. And here's the really cool part… If we do this new division problem, 64.5 divided by 15, we'll get exactly the same answer we would've if we did the problem 6.45 divided by 1.5. That only works because we shifted the decimal point in the divisor and the dividend by the same amount in the same direction. And you'll realize why that works if you remember equivalent fractions. Think about the fraction 1 over 2. That's the same as 1 divided by 2, right? Okay, but what if I multiplied both the top and bottom number by 10? That would give me 10 over 20, which is equivalent to 1 over 2. Even though it uses different top and bottom numbers, both represent the value 1 half. They're equivalent fractions. Well, that's what we did in our decimal division problem when we shifted the decimal point in both the divisor and the dividend by one place. We multiplied each number by 10. And since fractions and division are basically the same, we made equivalent division problems. But now, one of them has a whole number divisor. Pretty cool, huh? That means if we solve 64.5 divided by 15, we get the answer 4.3, which is exactly the same answer we'd get if we did 6.45 divided by 1.5. And you can use that trick to avoid ever having to divide with the decimal divisor, even if the dividend is a whole number. For example, if you have the problem 148 divided by 1.6, you can shift the decimal in both the divisor and the dividend one place to the right. Remember, there's always a decimal point even in a whole number. It's just that when you shift it to the right, you need to put a zero in the place that it shifts past. That gives you the equivalent division problem, 1480 divided by 16. And since these division problems are equivalent, you'll get the same answer for both. Alright, so that's how you can modify all the traditional arithmetic procedures to work with decimal numbers. It can be a little tricky at first since there's a few extra steps that you have to keep track of when the numbers are decimals. But if you practice a lot and check your answers with a calculator, you'll get it. Remember, you can always re-watch this video if you need to, along with the other videos about multi-digit arithmetic. As always, thanks for watching Math Antics and I'll see ya next time. Learn more at www.mathantics.com