Arithmetic with Decimal Numbers

Arithmetic with Decimal Numbers

Watch this video lesson and see how our little professor goes about adding, subtracting, multiplying, and dividing decimal numbers all without the use of a calculator!

Decimal Numbers

Mr. Prof is our little professor for this video. He's about to show us how he adds, subtracts, multiplies and divides decimal numbers. He's picking up his chalk, and he's getting ready.

First order of business he wants to take care of is the difference between decimal numbers and regular numbers. Decimal numbers, he says, are numbers with a decimal point, while regular numbers are the numbers you normally use to count with. A decimal point is a dot used to show that a particular number has a part of it that is less than 1. For example, the number 1.5 means that we have a whole one plus half a one. The 0.5 is the half a one.

All of these are examples of decimal numbers because all of them have a decimal point:

1.5, 2.01, 0.7, 0.95, 3.14, 1.2349

Now that we've talked about what decimal numbers are, let's see how we go about adding and subtracting decimals. Watch how Mr. Prof does it.

Adding and Subtracting

Let's add 0.5 and 0.81 together and see what happens. Mr. Prof has just written these on his board. He's written the 0.5 on the first line and immediately under, he's written 0.81. But, something looks different. Instead of the 5 and the 1 lining up because they are the last digits, the 5 and the 8 are lined up.

Why is that? This is because when you are working with decimals, the decimal point matters. You want to line things up according to the decimal, like the 8 and the 5 here.

When adding and subtracting decimals, line the numbers up according to the decimal point.

Now that we have our problem correctly set up and written down, our next step is to drop that decimal point down into our answer area. So, we write a decimal point in line with the other two decimal points.

Once we have our decimal point where we want it, we will now go ahead and add these numbers the way we know how. I don't see a number above the 1, but since it's after the decimal point, I can go ahead and add a zero there. This is exactly what Mr. Prof has done. So, I add 0 plus 1 equals 1. So, I put a 1 underneath the 1 in the answer field.

Next, I have 5 plus 8 equals 13. Because 13 is greater than 9, I carry the 1 over and write a little 1 on top of the 0, and I write 3 underneath the 8 in my answer field. Next, I have 1 plus 0 plus 0, which equals 1. So, I write 1 underneath the 0 in my answer field. So my answer, as Mr. Prof also shows, is 1.31.

So, when adding decimal numbers together, what you have to remember is to line them up with the decimal point when you set up your problem. After that, you add the way you know how.

Now, what about subtracting decimal numbers? Yes, you guessed it, we subtract in a similar way - by setting up our problem with the decimal points lined up. Let's subtract 0.5 from 0.81 to see how it works. Since I am subtracting 0.5 from 0.81, I will write 0.81 on the top line and 0.5 on the second line, remembering to line them up according to the decimal point. Mr. Prof, I see, has already done that.

With everything lined up, I then write a decimal point in the answer field in line with the other decimal points. Then I go ahead and subtract the usual way. Since I am working on numbers after the decimal point, if I don't see a number, I can add a 0. So, I have 1 minus 0, which is 1. Then I have 8 minus 5, which is 3. Okay. So, my answer is 0.31.

Also, for subtracting, we remember to always subtract the smaller number from the larger. If the problem is asking us to subtract a larger number from a smaller number, then our answer will be negative. For example, if we were subtracting 0.81 from 0.5, then our answer is still 0.31, but negative 0.31. Mr. Prof gives us a high five for making it this far. Now, let's go on to multiplying.

Multiplying

Multiplying decimals is similar to multiplying our regular numbers with the exception of the last step, where we place our decimal point in our answer. Up until that point, we essentially ignore the decimal points. When we set up our problem for multiplication, instead of lining it up according to the decimal, we line it up according to the last digit.

So, to multiply 0.81 and 0.5 together, we line up the 1 and the 5 because they are the last digits. Once we have the problem set up, we go ahead and multiply the way we know how. I have 5 times 1 equals 5. Then, I have 5 times 8, which equals 40. 40 is greater than 9, but since we are multiplying our last two digits together, I don't have to carry the 4, and I can write 40 down in my answer line.

I'm not done yet, though. I need to figure out where my decimal point goes for my answer. To figure this out, I'm going to count how many decimal places I have from the numbers I'm multiplying together. 0.81 has two decimal places, and 0.5 has one decimal place. So, that means I have a total of three decimal places. That means I need to have three digits after my decimal point in the answer. So, I look at the number in the answer - the 405 - and I count three spaces to the left starting from the right. I end up before the 4, so that is where I put my decimal. So, my answer is 0.405.

Mr. Prof wants another high five for doing this one correctly!

Dividing

Dividing decimals is not the same as multiplying. While we left all the decimal points in place while multiplying, we might have to move the decimal points when dividing. This is because, for our final result to make sense, we need to divide by a whole number if we are to do this by hand without a calculator. We will be dividing 0.81 by 0.5. Since we are dividing by 0.5, we have to move the decimal point one place to the right to change that 0.5 to a 5, a whole number.

Because we moved the decimal point one place to the right in the 0.5, we also have to move the decimal point one place to the right in the 0.81. If we don't do this, we would be changing the problem. So, moving the decimal point one place to the right in the 0.81 gives us 8.1. So, now we are dividing 8.1 by 5.

We set up our division problem like we normally do, with the 8.1 inside the division bracket thing and the 5 outside. At this point, we can put our decimal point into place in our answer field. We located the decimal point between the 8 and the 1, and we write another decimal point in line with it on top in our answer field.

Now that we have our decimal point in place in our answer, we can go ahead and divide the way we normally do. 8 divided by 5 is 1 with a remainder of 3. The 3 is smaller than 5, so I know 1 is the correct digit for that part of my answer. I multiply 1 by 5 to get 5, and I put 5 underneath the 8. I subtract 5 from the 8 to get 3. I bring down the 1. Now I divide 31 by 5 to get 6 with a remainder of 1. 1 is smaller than 5, so 6 is correct for that digit of my answer.

I multiply 6 by 5 to get 30, and I write 30 on the bottom. I subtract 30 from 31 to get 1. I don't have any more numbers to bring down, but I have a remainder that I still need to finish dividing. Because I've already placed my decimal point in my answer, I can go ahead and add a 0 and drop that down to make 10. I then divide 10 by 5 to get 2 with no remainder. No remainder means I am done. So, my answer is 1.62.

High five, Mr. Prof! Some things to remember about division are that we need to move the decimal point over until the decimal number we are dividing by is a whole number. However many spaces we move the decimal point in that number, we have to move the decimal point over the same number of spaces in the other number. Once we set up our problem, we place our decimal point on top in the answer field in line with the decimal point in the divisor - the number inside the division bracket. We then go ahead and divide like we normally do.

Lesson Summary

To recap what we've learned, decimal numbers are those numbers that have a decimal point in them. A decimal point is that dot placed before the part of the number that shows it has a portion that is less than a whole 1.

When adding and subtracting decimals, we set up our problem by lining up the numbers according to the decimal point and not according to the last digit. Right away, we place our decimal point in our answer in line with the other decimal points. Then, we go ahead and add and subtract the way we normally do. We remember to always subtract the smaller number from the larger even if the problem is asking us to subtract a larger number from a smaller. If that is the case, then our answer will be negative.

When multiplying, we set up the problem like normal, lining up the numbers by the last digit. We go ahead and multiply the way we normally do. And, to find where the decimal point goes in the answer, we count the number of decimal places in the numbers we are multiplying together, and we count that many spaces from right to left in our answer to find the location of our decimal point.

When dividing, if we are dividing by a decimal number, we need to move the decimal point over to the right as many spaces as needed to turn that number into a whole. We then move the decimal point over the same number of spaces in the other number. We then set up our problem in long division form using the new numbers that we came up with. We place the decimal point on top in our answer field in line with the decimal point inside the division bracket. We then divide as we normally would.

Learning Outcome

You could be able to correctly line up and add, subtract, multiply and divide numbers that have decimals after watching this video lesson.