Now that we’ve learned what real numbers () or, as they are sometimes called – decimal numbers are, let’s learn how to use them in basic arithmetic operations. First in line are addition and subtraction of decimals.
So, the basic principles behind the addition of real numbers are pretty much the same as the ones with the whole numbers. And the decimal point actually doesn’t make as much of a difference as we might think. Let’s take a look at this example:
The most important thing to remember is that the numbers on the right side of the decimal point add up the same way as the numbers on the left side of the decimal point. That means that we add the tenths with the tenths, the hundredths with the hundredths, etc. When we add the numbers, write one beneath the other in a way that their decimal points align. This will help us keep track of which parts we’re adding.
As in any other addition, we start by adding the rightmost numerals of both numbers, and keep going from right to left. If the sum of two added numerals adds up to more than ten, we will just increase the sum of the first numerals to the left by one. Don’t be alarmed if one of the numbers has more numerals behind the decimal point then the other one – we can add as many zeroes as we want behind the rightmost numeral of our number. That’s basically the same thing as adding zeroes in front of the leftmost numeral to the left of the decimal point – it won’t change the value of the number at all, but it will make calculations easier.
And the same principles apply to subtraction. We start by subtracting the rightmost numeral behind the decimal point of the subtrahend from the rightmost numeral of the minuend, and work our way to their respective leftmost numerals. Sounds complicated? Well, it’s easier than it seems. Follow the basic rules for subtraction, and then apply what you learned about the addition of real numbers.
Let’s do a couple of examples together – an addition and a subtraction – to start things off. We’ll do it step by step to make things as clear as possible. And as the first example, let’s add the numbers 107.445 and 224.581.
We begin by adding the last decimals of both numbers together. 5 and 1 is 6. Now we add together the next two decimals, 4 and 8. Since 4 and 8 add up to 12, and 12 is greater than 10 by 1, we’ll write down the 2 and remember 1 as well.
Now we add together the next two decimals, 4 and 5, and increase their sum by the 1 we remembered earlier. Since 4 + 5 + 1 is 10, we write down the 0, and remember 1 once more.
Now that we’re past the decimal point, we’re in familiar territory. Now it’s time to add together the 7 and the 4, and increase their sum by the 1 we remembered from the previous addition. Since 7 + 4 + 1 is 12, which is bigger than 10, we write down the 2, and remember 1 once more. In next decimals, 0 and 2 with 1 from previous addition is 3. Now, you can remember 0, if it makes it easier to keep track, and we will add the 0 in the next addition.
After that, we just write down the results of our additions to their appropriate places. Therefore, the final result of our addition is 2.971.
Subtraction works in a similar fashion. The biggest difference is that, instead of “remembering the 1”, we “borrow the 1”.
Well, that’s it for the addition and subtraction of real numbers. If you would like to practice a bit more and get a better hang of it, try solving the examples from the worksheets below.