Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.

Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.

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 below. Look for a pattern.

There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.

How many decimal places would you expect for the product of [latex]\left(0.01\right)\left(0.004\right)?[/latex] If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five.

Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.

The rules for multiplying positive and negative numbers apply to decimals, too, of course.

When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.

example

Multiply: [latex]\left(3.9\right)\left(4.075\right)[/latex]

Solution

[latex]\left(3.9\right)\left(4.075\right)[/latex]
Determine the sign of the product. The signs are the same.	The product will be positive.
Write the numbers in vertical format, lining up the numbers on the right.
Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
Place the decimal point. Add the number of decimal places in the factors [latex]\left(1+3\right)[/latex]. Place the decimal point 4 places from the right.
The product is positive.	[latex]\left(3.9\right)\left(4.075\right)=15.8925[/latex]

example

Multiply: [latex]\left(-8.2\right)\text{(}5.19\text{)}[/latex]

Show Solution

Solution

[latex]\left(-8.2\right)\left(5.19\right)[/latex]
The signs are different.	The product will be negative.
Write in vertical format, lining up the numbers on the right.	[latex]\begin{array}{c}\hfill 5.19\\ \hfill \underset{\text{_____}}{\times 8.2}\end{array}[/latex]
Multiply.	[latex]\begin{array}{c}\hfill 5.19\\ \hfill \underset{\text{_____}}{\times 8.2}\\ \hfill 1038\\ \underset{\text{_____}}{4152}\\ \hfill 42558\end{array}[/latex]
	[latex]\begin{array}{c}\hfill 5.19\\ \hfill \underset{\text{_____}}{\times 8.2}\\ \hfill 1038\\ \underset{\text{_____}}{4152}\\ \hfill 42.558\end{array}[/latex]
The product is negative.	[latex]\left(-8.2\right)\left(5.19\right)=-42.558[/latex]

In the following video we show another example of how to multiply two decimals.

In the next example, we’ll need to add several placeholder zeros to properly place the decimal point.

example

Multiply: [latex]\left(0.03\right)\text{(}0.045\text{)}[/latex]

Show Solution

Solution

[latex]\left(0.03\right)\left(0.045\right)[/latex]
The product is positive.
Write in vertical format, lining up the numbers on the right.
Multiply.
Add zeros as needed to get the [latex]5[/latex] places.
The product is positive.	[latex]\left(0.03\right)\left(0.045\right)=0.00135[/latex]

Multiply by Powers of [latex]10[/latex]

In many fields, especially in the sciences, it is common to multiply decimals by powers of [latex]10[/latex]. Let’s see what happens when we multiply [latex]1.9436[/latex] by some powers of [latex]10[/latex].

Look at the results without the final zeros. Do you notice a pattern?

[latex]\begin{array}{ccc}1.9436\left(10\right)\hfill & =& 19.436\hfill \\ 1.9436\left(100\right)\hfill & =& 194.36\hfill \\ 1.9436\left(1000\right)\hfill & =& 1943.6\hfill \end{array}[/latex]

The number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below summarizes the results.

We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of [latex]10[/latex] and then move the decimal point that same of places to the right.

So, for example, to multiply [latex]45.86[/latex] by [latex]100[/latex], move the decimal point [latex]2[/latex] places to the right.

Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply [latex]2.4[/latex] by [latex]100[/latex]. We need to move the decimal point [latex]2[/latex] places to the right. Since there is only one digit to the right of the decimal point, we must write a [latex]0[/latex] in the hundredths place.

example

Multiply [latex]5.63[/latex] by factors of

1. [latex]10[/latex]

2. [latex]100[/latex]

3. [latex]1000[/latex]

Show Solution

Solution
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.

1.
[latex]56.3\left(10\right)[/latex]
There is [latex]1[/latex] zero in [latex]10[/latex], so move the decimal point [latex]1[/latex] place to the right.
[latex]56.3[/latex]

2.
[latex]5.63\left(100\right)[/latex]
There are [latex]2[/latex] zeros in [latex]100[/latex], so move the decimal point [latex]2[/latex] places to the right.
[latex]563[/latex]

3.
[latex]5.63\left(1000\right)[/latex]
There are [latex]3[/latex] zeros in [latex]1000[/latex], so move the decimal point [latex]3[/latex] places to the right.
A zero must be added at the end.	[latex]5,630[/latex]

In the following video we show more examples of how to multiply a decimal by 10, 100, and 1000.