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So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side. Look for a pattern!
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| Convert to fractions. |
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| Multiply. |
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| Convert to decimals. |
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Notice, in the first example, we multiplied two numbers that each had one digit after the decimal point and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places.
We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product.
The rules for multiplying positive and negative numbers apply to decimals, too, of course!
When multiplying two numbers,
When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.
Multiply:
| (−3.9)(4.075) | |
| The signs are different. The product will be negative. | |
| Write in vertical format, lining up the numbers on the right. |
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| Multiply. |
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Add the number of decimal places in the factors (1 + 3).
Place the decimal point 4 places from the right. |
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| The signs are different, so the product is negative. | (−3.9)(4.075) = −15.8925 |
In many of your other classes, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc.). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.
Multiply 5.63 ⓐ by 10 ⓑ by 100 ⓒ by 1,000.
By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.
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| 5.63(10) | |
| There is 1 zero in 10, so move the decimal point 1 place to the right. |
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| 5.63(100) | |
| There are 2 zeros in 100, so move the decimal point 2 places to the right. |
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| 5.63(1,000) | |
| There are 3 zeros in 1,000, so move the decimal point 3 places to the right. |
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| A zero must be added at the end. |
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Multiply 2.58 ⓐ by 10 ⓑ by 100 ⓒ by 1,000.
ⓐ 25.8 ⓑ 258 ⓒ 2,580
Multiply 14.2 ⓐ by 10 ⓑ by 100 ⓒ by 1,000.
ⓐ 142 ⓑ 1,420 ⓒ 14,200
Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.
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