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Multiplication of decimals

Multiplication of decimals (real numbers: \mathbb{R}) is done in the same way as is the multiplication of integers. It’s pretty much the same from a theoretical standpoint, because the same rules and properties apply to both cases. The only slight complication is the decimal point, which we have to keep track of. Because of this, we’ll skip the intro, do a couple of examples together, and explain things along the way. So, let’s continue with our first example.

Example 1.
Let’s say we want to multiply decimal number 3.758 by 2.

This is a very easy example. We start by multiplying the last numeral of the first factor (that’s the numeral 8) by the first (and only) numeral of the second factor. Then we work our way through the first factor from right to left. Like this:

  • 2 \cdot 8 = 16. We write down 6, and remember 1.
  • 2 \cdot 5 = 10, and when we add the 1, we get 11. We write down 1, and remember 1.
  • 2 \cdot 7 = 14, and when we add the 1, we get 15. We write down 5, and remember 1.
  • We’ve “reached” the decimal point, so we write it down in this spot.
  • 2 \cdot 3 = 6, and when we add the 1, we get 7. We write down 7.
  • So, the final result of the multiplication of the factors 3.758 and 2 is 7.516.

Well, this one was pretty much as simple as they come. Only one of the numbers was a decimal number and we didn’t really have to try to keep an eye on the decimal point. Now, let’s see what happens when we try to multiply two decimal numbers.

Example 2.

Let’s say we want to multiply number 7.21 by 2.31. This one works in a similar way, but should take a bit longer to do. So, to avoid long blocks of text, we’ll explain this one using this picture.

multiplication of two decimals

We can see that decimal number 16.6551 has for decimal numerals. The reason of that fact is that numbers 7.21 and 2.31 has in sum 4 decimal numerals.

Example 3.

Let’s complicate things a bit more, by multiplying two decimal numbers, one of which is smaller than 1, but larger than 0. For example, the numbers 5.723 and 0.986.

multiplication of small and big decimals

The first thing we notice is that the result of this multiplication is actually smaller than the first factor. The reason for this is simple. When we multiply a number by 1, we get that same number. Also, when we multiply a number by 0, we get 0 as a result. Therefore, it stands to reason that, if we multiply a number by another number whose value is between 0 and 1, we get a part of that number as a result. But, what happens if we multiply two such numbers? We’ll find out in the next example.

 

Example 4.

Let’s multiply two numbers whose values are between 0 and 1. For example, the numbers 0.3854 and 0.1586.
As we can see, the concept is the same as in the previous example, but we have to take a bit of extra care with the decimal point.

So, that about covers the multiplication of real numbers. Although you will probably use a calculator for more complex multiplication of decimal numbers, learning to do it well this way will sharpen your intuition and give you the confidence to master more difficult lessons. If you would like to practice this multiplying real numbers a bit more, please use the worksheets below.

 

 

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