Decimals: Learn It 1

  • Write and name decimals
  • Turn a decimal into a fraction
  • Place decimals on a number line and order them
  • Solve equations using decimals

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?

A vertical addition problem. The top line shows $3.45 for a sandwich, the next line shows $1.25 for water, and the last line shows $0.33 for tax. The total is shown to be $5.03.

 

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.

decimals

Decimals represent fractions or parts of a whole, based on powers of ten, using a point known as a decimal point. They provide a way to express numbers between whole numbers and allow for accurate representation of values, especially in measurement and calculations.

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

Decimal Fraction Name
[latex]0.1[/latex] [latex]{\Large\frac{1}{10}}[/latex] One tenth
[latex]0.01[/latex] [latex]{\Large\frac{1}{100}}[/latex] One hundredth
[latex]0.001[/latex] [latex]{\Large\frac{1}{1,000}}[/latex] One thousandth
[latex]0.0001[/latex] [latex]{\Large\frac{1}{10,000}}[/latex] One ten-thousandth

 

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.

A vertical addition problem. The top line shows $3.45 for a sandwich, the next line shows $1.25 for water, and the last line shows $0.33 for tax. The total is shown to be $5.03.

 

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.

How To: Name a Decimal Number

  1. Name the number to the left of the decimal point.
  2. Write “and” for the decimal point.
  3. Name the “number” part to the right of the decimal point as if it were a whole number.
  4. Name the decimal place of the last digit.

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

How To: Write a Decimal Number From Its Name

  1. Look for the word “and”—it locates the decimal point.
  2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
    • Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
    • If there is no “and,” write a “[latex]0[/latex]” with a decimal point to its right.
  3. Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
  4. Fill in zeros for place holders as needed.