You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs [latex]\text{\$3.45}[/latex] , the bottle of water costs [latex]\text{\$1.25}[/latex] , and the total sales tax is [latex]\text{\$0.33}[/latex] , what is the total cost of your lunch?

The total is [latex]$5.03[/latex]. Suppose you pay with a [latex]$5[/latex] bill and [latex]3[/latex] pennies. Should you wait for change? No, [latex]\text{\$5}[/latex] and [latex]3[/latex] pennies is the same as [latex]\text{\$5.03}[/latex].

Because [latex]\text{100 pennies}=\text{\$1}[/latex], each penny is worth [latex]{\Large\frac{1}{100}}[/latex] of a dollar. We write the value of one penny as [latex]$0.01[/latex], since [latex]0.01={\Large\frac{1}{100}}[/latex].

Writing a number with a decimal is known as decimal notation. It is a way of showing parts of a whole when the whole is a power of ten. In other words, decimals are another way of writing fractions whose denominators are powers of ten. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten. The table below shows the counting numbers.

How are decimals related to fractions? The table below shows the relation.

When we name a whole number, the name corresponds to the place value based on the powers of ten. In Whole Numbers, we learned to read [latex]10,000[/latex] as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in the first table relate to the names of the fractions from the second table.

This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in the tables.

• The “th” at the end of the name means the number is a fraction. “One thousand” is a number larger than one, but “one thousandth” is a number smaller than one.
• The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal.

Remember that [latex]$5.03[/latex] lunch? We read [latex]$5.03[/latex] as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number [latex]5.03[/latex] as five and three hundredths.
We sometimes need to translate a number written in decimal notation into words. As shown in the image below, we write the amount on a check in both words and numbers.

When we write a check, we write the amount as a decimal number as well as in words. The bank looks at the check to make sure both numbers match. This helps prevent errors.

The number [latex]15.68[/latex] is read fifteen and sixty-eight hundredths.

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used.

In the following video we show more examples of how to write the name of a decimal using a place value chart.

The second bullet in Step 1 is needed for decimals that have no whole number part, like ‘nine thousandths’. We recognize them by the words that indicate the place value after the decimal – such as ‘tenths’ or ‘hundredths.’ Since there is no whole number, there is no ‘and.’ We start by placing a zero to the left of the decimal and continue by filling in the numbers to the right, as we did above.

example

Write twenty-four thousandths as a decimal.

Show Solution

Solution

twenty-four thousandths
Look for the word “and”.	There is no “and” so start with 0 [latex]0[/latex].
To the right of the decimal point, put three decimal places for thousandths.
Write the number [latex]24[/latex] with the [latex]4[/latex] in the thousandths place.
Put zeros as placeholders in the remaining decimal places.	[latex]0.024[/latex]
So, twenty-four thousandths is written [latex]0.024[/latex]

In the next video we will show more examples of how to write a decimal given its name in words.

Before we move on to our next objective, think about money again. We know that [latex]$1[/latex] is the same as [latex]$1.00[/latex]. The way we write [latex]$1\left(\text{or}$1.00\right)[/latex] depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal.

[latex]\begin{array}{cccc}5=5.0\hfill & & & -2=-2.0\hfill \\ 5=5.00\hfill & & & -2=-2.00\hfill \\ 5=5.000\hfill & & & -2=-2.000\hfill \end{array}[/latex]
and so on [latex]\dots[/latex]