## More on Multiplying Decimals

Read this text. Pay attention to the "How To" overview of the steps needed to multiply decimals. Complete the practice problems and check your answers.

### Multiply Decimals

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 in Table 5.3. Look for a pattern.

A B
$(0.3)(0.7)$ $(0.2)(0.46)$
Convert to fractions. $\left(\frac{3}{10}\right)\left(\frac{7}{10}\right)$ $\left(\frac{2}{10}\right)\left(\frac{46}{100}\right)$
Multiply. $\frac{21}{100}$ $\frac{92}{1000}$
Convert back to decimals. $0.21$ $0.092$

Table 5.3

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 $(0.01)(0.004)$? 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.

\begin{aligned} (0.\underbrace{01}_{\color{blue}{\text{2 places}}}) (0.\underbrace{004}_{\color{blue}{\text{3 places}}}) &= 0.\underbrace{00004}_{\color{blue}{\text{5 places}}} \\ \left(\frac{1}{100}\right)\left(\frac{4}{1000}\right) & =\frac{4}{100,000} \end{aligned}

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.

#### MULTIPLYING TWO NUMBERS

When multiplying two numbers,

• if their signs are the same, the product is positive.
• if their signs are different, the product is negative.

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.

#### HOW TO

##### Multiply decimal numbers.

Step 1. Determine the sign of the product.

Step 2. Write the numbers in vertical format, lining up the numbers on the right.

Step 3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.

Step 4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.

Step 5. Write the product with the appropriate sign.

In the Example 5.17, we'll need to add several placeholder zeros to properly place the decimal point.

Source: Rice University, https://openstax.org/books/prealgebra/pages/5-2-decimal-operations