Decimals

Name 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 $3.45, the bottle of water costs $1.25, and the total sales tax is $0.33, what is the total cost of your lunch?

The total is $5.03. Suppose you pay with a $5 bill and 3 pennies. Should you wait for change? No, $5 and 3 pennies is the same as $5.03.

Because $100$ pennies $= \text{\textdollar}1$, each penny is worth $\tfrac{1}{100}$ of a dollar. We write the value of one penny as $0.01, since $0.01 = \tfrac{1}{100}$.

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. We read $10{,}000$ as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. The table below lays out the place-value names to the left and right of the decimal point.

Notice two important facts:

• 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 $5.03 lunch? We read $5.03 as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number $5.03$ as five and three hundredths.

We sometimes need to translate a number written in decimal notation into words — for example, when writing the amount on a check in both words and numbers.

Let’s try naming a decimal, such as $15.68$.

The number $15.68$ is read fifteen and sixty-eight hundredths.

Example. Name each decimal: (a) $4.3$ (b) $2.45$ (c) $0.009$ (d) $-15.571$.

(a) Name the number to the left of the decimal point: four. Write “and” for the decimal point: four and. Name the number to the right of the decimal point as if it were a whole number: four and three. Name the decimal place of the last digit: four and three tenths.

(b) Following the same steps: two and forty-five hundredths.

(c) There is a zero to the left of the decimal point; it is not included in the name. Name the number to the right of the decimal point as if it were a whole number: nine. Name the decimal place of the last digit: nine thousandths.

(d) Name the number to the left of the decimal point: negative fifteen. Write “and” for the decimal point: negative fifteen and. Name the number to the right of the decimal point as if it were a whole number: negative fifteen and five hundred seventy-one. Name the decimal place of the last digit: negative fifteen and five hundred seventy-one thousandths.

Naming a decimal in words is good practice, but since word answers can’t be graded automatically here, try the reverse: what decimal number is named “negative two and fifty-three thousandths”?

Write decimals

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

Let’s start by writing the number six and seventeen hundredths.

Example. Write fourteen and thirty-seven hundredths as a decimal.

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: $14.$ Mark two places to the right of the decimal point for “hundredths.” Translate the words after “and” and write the number to the right of the decimal point: $14.37$.

Fourteen and thirty-seven hundredths is written $14.37$.

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.

Example. Write twenty-four thousandths as a decimal.

There is no “and,” so start with $0.$ To the right of the decimal point, put three decimal places for thousandths. Write the number $24$ with the $4$ in the thousandths place, and put zeros as placeholders in the remaining decimal places:

So, twenty-four thousandths is written $0.024$.

Before we move on, think about money again. We know that $1 is the same as $1.00. The way we write $1 (or $1.00) 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:

Convert decimals to fractions or mixed numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that $5.03 means 5 dollars and 3 cents. Since there are $100$ cents in one dollar, 3 cents means $\tfrac{3}{100}$ of a dollar, so $0.03 = \tfrac{3}{100}$.

We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal $0.03$, the $3$ is in the hundredths place, so $100$ is the denominator of the fraction equivalent to $0.03$:

For our $5.03 lunch, we can write the decimal $5.03$ as a mixed number:

Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.

Example. Write each of the following decimal numbers as a fraction or a mixed number: (a) $4.09$ (b) $3.7$ (c) $-0.286$.

(a) There is a $4$ to the left of the decimal point, so write “4” as the whole number part of the mixed number. The final digit, $9$, is in the hundredths place. Write $9$ in the numerator and $100$ in the denominator:

Did you notice that the number of zeros in the denominator is the same as the number of decimal places?

(b) There is a $3$ to the left of the decimal point, so write “3” as the whole number part. The final digit, $7$, is in the tenths place:

(c) There is a $0$ to the left of the decimal point, so write a negative sign before the fraction. The final digit, $6$, is in the thousandths place. Write $286$ in the numerator and $1000$ in the denominator, then remove a common factor of $2$ to simplify:

Locate decimals on the number line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Example. Locate $0.4$ on a number line.

The decimal $0.4$ is equivalent to $\tfrac{4}{10}$, so $0.4$ is located between $0$ and $1$. On a number line, divide the interval between $0$ and $1$ into $10$ equal parts and place marks to separate the parts. Label the marks $0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0$. We write $0$ as $0.0$ and $1$ as $1.0$, so that the numbers are consistently in tenths. Finally, mark $0.4$ on the number line.

Example. Locate $-0.74$ on a number line.

The decimal $-0.74$ is equivalent to $-\tfrac{74}{100}$, so it is located between $0$ and $-1$. On a number line, mark off and label the multiples of $-0.10$ in the interval between $0$ and $-1$ ($-0.10, -0.20$, etc.) and mark $-0.74$ between $-0.70$ and $-0.80$, a little closer to $-0.70$.

Order decimals

Which is larger, $0.04$ or $0.40$? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,

In previous chapters, we used the number line to order numbers: $a < b$ means “$a$ is less than $b$” when $a$ is to the left of $b$ on the number line, and $a > b$ means “$a$ is greater than $b$” when $a$ is to the right of $b$ on the number line. Since $0.40$ is to the right of $0.04$ on the number line, $0.40 > 0.04$.

How does $0.31$ compare to $0.308$? This doesn’t translate into money to make the comparison easy. But if we convert $0.31$ and $0.308$ to fractions, we can tell which is larger. We need a common denominator to compare them:

Because $310 > 308$, we know that $\tfrac{310}{1000} > \tfrac{308}{1000}$. Therefore, $0.31 > 0.308$.

Notice what we did in converting $0.31$ to a fraction — we started with the fraction $\tfrac{31}{100}$ and ended with the equivalent fraction $\tfrac{310}{1000}$. Converting $\tfrac{310}{1000}$ back to a decimal gives $0.310$. So $0.31$ is equivalent to $0.310$. Writing zeros at the end of a decimal does not change its value.

We say $0.31$ and $0.310$ are equivalent decimals. Remember, writing zeros at the end of a decimal does not change its value.

Example. Order the following decimals using $<$ or $>$: (a) $0.64$ __ $0.6$ (b) $0.83$ __ $0.803$.

(a) Both numbers do not have the same number of decimal places, so write one zero at the right of $0.6$: $0.64$ __ $0.60$. Compare the numbers to the right of the decimal point as if they were whole numbers: $64 > 60$. So $0.64 > 0.60$, which means $0.64 > 0.6$.

(b) Write one zero at the right of $0.83$: $0.830$ __ $0.803$. Compare: $830 > 803$. So $0.830 > 0.803$, which means $0.83 > 0.803$.

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $-2$ lies to the right of $-3$ on the number line, we know that $-2 > -3$. Similarly, smaller numbers lie to the left on the number line. For example, because $-9$ lies to the left of $-6$ on the number line, we know that $-9 < -6$. If we zoomed in on the interval between $0$ and $-1$, we would see in the same way that $-0.2 > -0.3$ and $-0.9 < -0.6$.

Example. Use $<$ or $>$ to order: $-0.1$ __ $-0.8$.

Write the numbers one under the other, lining up the decimal points. They have the same number of digits. Since $-1 > -8$, $-1$ tenth is greater than $-8$ tenths:

Round decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at $3.279 per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $3.28, because the final price would be rounded to the nearest cent. We saw in an earlier chapter that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round $2.72 to the nearest dollar — is it closer to $2 or to $3? We see that $2.72$ is closer to $3$ than to $2$, so $2.72$ rounded to the nearest whole number is $3$. What if we wanted to round $2.72 to the nearest ten cents — is it closer to $2.70 or to $2.80? We see that $2.72$ is closer to $2.70$ than $2.80$, so $2.72$ rounded to the nearest tenth is $2.7$.

Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

Example. Round $18.379$ to the nearest hundredth.

Locate the hundredths place and mark it with an arrow: the hundredths digit is $7$. Underline the digit to the right of the hundredths place, the $9$: $18.37\underline{9}$. Because $9$ is greater than or equal to $5$, add $1$ to the $7$. Rewrite the number, deleting all digits to the right of the hundredths place:

$18.38$ is $18.379$ rounded to the nearest hundredth.

Example. Round $18.379$ to the nearest (a) tenth (b) whole number.

(a) Locate the tenths place and mark it with an arrow: the tenths digit is $3$. Underline the digit to the right of the tenths digit, the $7$. Because $7$ is greater than or equal to $5$, add $1$ to the $3$. Rewrite the number, deleting all digits to the right of the tenths place:

So, $18.379$ rounded to the nearest tenth is $18.4$.

(b) Locate the ones place and mark it with an arrow: the ones digit is $8$. Underline the digit to the right of the ones place, the $3$. Since $3$ is not greater than or equal to $5$, do not add $1$ to the $8$. Rewrite the number, deleting all digits to the right of the ones place:

So $18.379$ rounded to the nearest whole number is $18$.

Key terms

decimal notation — a way of writing a number as a whole-number part and a fractional part, where the fractional part has a denominator that is a power of ten. equivalent decimals — two decimals that convert to equivalent fractions; writing zeros at the end of a decimal does not change its value.

This section is adapted from Prealgebra 2e, Section 5.1: Decimals by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the place-value chart and decimal/fraction tables as markdown tables, the number lines as accessible inline graphics, and the step-by-step translation tables as simplified prose and tables; omitted the Be Prepared quiz, Figure 5.3 (a check image), the Media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.