Decimals

Name and write decimals

Decimals are another way of writing fractions whose denominators are powers of $10$.

So $0.1$ is “one tenth,” $0.01$ is “one hundredth,” $0.001$ is “one thousandth,” and $0.0001$ is “one ten-thousandth.” Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one — the “th” at the end of the name tells you the number is smaller than one.

When we name a whole number, the name corresponds to its place value based on the powers of ten: we read $10{,}000$ as “ten thousand” and $10{,}000{,}000$ as “ten million.” Likewise, the names of the decimal places correspond to their fraction values. The table below shows the names of the place values to the left and right of the decimal point.

Example. Name the decimal $4.3$.

We name the number to the left of the decimal point — four — write “and” for the decimal point, then name the “number” to the right of the decimal point as if it were a whole number, followed by its decimal place. $4$ is to the left of the decimal point, so we start with “four.” $3$ is to the right of the decimal point, in the tenths place, so we finish with “three tenths.” Altogether, $4.3$ is “four and three tenths.”

Example. Name the decimal $-15.571$.

The number to the left of the decimal point is negative fifteen. Write “and” for the decimal point: negative fifteen and. The number to the right of the decimal point, named as a whole number, is five hundred seventy-one. The last digit, $1$, is in the thousandths place. Putting it together, $-15.571$ is “negative fifteen and five hundred seventy-one thousandths.”

When we write a check we write both the numerals and the name of the number. Let’s see how to write a decimal from its name.

Example. Write “fourteen and twenty-four thousandths” as a decimal.

The word “and” locates the decimal point — place a decimal point under the word “and,” and translate the words before “and” into the whole number placed to its left: $14.\underline{\ \ }\underline{\ \ }\underline{\ \ }$. The last word is “thousandths,” so we need three decimal places to the right of the decimal point. Translate the words after “and” into the number to the right of the decimal point, putting the final digit in the last place: $14.\ \_2\ \_4$, filling in a zero for the empty tenths place: $14.024$. So “fourteen and twenty-four thousandths” is written $14.024$.

Round decimals

Rounding decimals is very much like rounding whole numbers — 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 $7$ is in the hundredths place of $18.379$. Underline the digit to the right of the hundredths place — the $9$. Since $9$ is greater than or equal to $5$, add $1$ to the $7$, giving $18.38$. Rewrite the number, removing all digits to the right of the rounding digit: $18.379$ rounded to the nearest hundredth is $18.38$.

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

(a) The tenths place holds the $3$. The digit to its right is $7$, which is greater than or equal to $5$, so we add $1$ to the $3$. Rewriting and deleting all digits to the right of the rounding digit — without replacing them with zeros — gives $18.4$. So $18.379$ rounded to the nearest tenth is $18.4$.

(b) The ones place holds the $8$. The digit to its right is $3$, which is not greater than or equal to $5$, so we do not add $1$ to the $8$. Rewriting and deleting all digits to the right of the rounding digit gives $18$. So $18.379$ rounded to the nearest whole number is $18$.

Add and subtract decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

Example. Add: $23.5 + 41.38$.

Write the numbers so the decimal points line up vertically, putting a $0$ as a placeholder after the $5$ in $23.5$ (remember, $\tfrac{5}{10} = \tfrac{50}{100}$, so $0.5 = 0.50$):

Adding the numbers as if they were whole numbers and placing the decimal point in the sum gives $23.5 + 41.38 = 64.88$.

Example. Subtract: $20 - 14.65$.

Write the numbers so the decimal points line up vertically, remembering that $20$ is a whole number, so its decimal point goes right after the $0$: $20.$ Put in zeros to the right as placeholders: $20.00$. Subtracting and placing the decimal point in the answer gives $20.00 - 14.65 = 5.35$.

Multiply and divide decimals

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

For example, $(0.3)(0.7)$ converts to $\tfrac{3}{10} \cdot \tfrac{7}{10} = \tfrac{21}{100}$, which converts back to $0.21$: two factors with one decimal place each produced a product with two decimal places. Likewise, $(0.2)(0.46)$ converts to $\tfrac{2}{10} \cdot \tfrac{46}{100} = \tfrac{92}{1{,}000}$, which converts back to $0.092$: a factor with one decimal place times a factor with two decimal places produced a product with three decimal places.

We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal places in the factors, and that sum tells us the number of decimal places in the product.

The rules for multiplying positive and negative numbers apply to decimals, too, of course! When multiplying two numbers, if their signs are the same the product is positive; if their signs are different the product is negative. When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.

Example. Multiply: $(-3.9)(4.075)$.

The signs are different, so the product will be negative. Writing in vertical format, lining up the numbers on the right, and multiplying as if they were whole numbers gives $158{,}925$. Adding the number of decimal places in the factors ($1 + 3 = 4$) and placing the decimal point four places from the right gives $15.8925$. Since the signs are different, the product is negative: $(-3.9)(4.075) = -15.8925$.

In many of your other classes, especially in the sciences, you will multiply decimals by powers of $10$ ($10, 100, 1{,}000$, etc.). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of $10$ to the number of decimal places you move the decimal point to the right to get the product.

Example. Multiply $5.63$ by (a) $10$ (b) $100$ (c) $1{,}000$.

(a) There is $1$ zero in $10$, so move the decimal point $1$ place to the right: $5.63(10) = 56.3$.

(b) There are $2$ zeros in $100$, so move the decimal point $2$ places to the right: $5.63(100) = 563$.

(c) There are $3$ zeros in $1{,}000$, so move the decimal point $3$ places to the right, adding a zero at the end since there are only $2$ digits after the decimal point: $5.63(1{,}000) = 5{,}630$.

Just as with multiplication, division of decimals is very much like dividing whole numbers — we just have to figure out where the decimal point must be placed. To divide decimals, determine what power of $10$ to multiply the denominator by to make it a whole number. Then multiply the numerator by that same power of $10$. Because of the Equivalent Fractions Property, we haven’t changed the value of the fraction! The effect is to move the decimal points in the numerator and denominator the same number of places to the right. For example, $\tfrac{0.8}{0.4} = \tfrac{0.8(10)}{0.4(10)} = \tfrac{8}{4}$.

We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.

We review the notation and vocabulary for division: for $a \div b = c$, $a$ is the dividend, $b$ is the divisor, and $c$ is the quotient.

Example. Divide: $-25.65 \div (-0.06)$.

The signs are the same, so the quotient is positive. Make the divisor a whole number by moving the decimal point in $-0.06$ all the way to the right, and move the decimal point the same number of places in the dividend: $0.06\overline{)25.65}$ becomes dividing $2{,}565$ by $6$, giving $427.5$. Placing the decimal point in the quotient above the decimal point in the (shifted) dividend and writing the quotient with the appropriate sign: $-25.65 \div (-0.06) = 427.5$.

A common application of dividing whole numbers into decimals is when we want to find the price of one item that is sold as part of a multi-pack. For example, suppose a case of $24$ water bottles costs $\text{\textdollar}3.99$. To find the price of one water bottle, we would divide $\text{\textdollar}3.99$ by $24$. In calculations with money, we round the answer to the nearest cent (hundredth).

Example. Divide: $\text{\textdollar}3.99 \div 24$.

Placing the decimal point in the quotient above the decimal point in the dividend and dividing as usual — carrying the division to the thousandths place so we can round to the nearest cent — gives a quotient of $0.166\ldots$. Rounding $\text{\textdollar}0.166$ to the nearest cent gives $\text{\textdollar}0.17$, so $\text{\textdollar}3.99 \div 24 \approx \text{\textdollar}0.17$.

Convert decimals, fractions, and percents

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal $0.03$ the $3$ is in the hundredths place, so $100$ is the denominator of the fraction equivalent: $0.03 = \tfrac{3}{100}$. Notice that when the number to the left of the decimal is zero, we get a fraction whose numerator is less than its denominator — fractions like this are called proper fractions.

Example. Write $0.374$ as a fraction.

The place value of the final digit, $4$, is thousandths, so the denominator is $1{,}000$. The numerator is $374$: $0.374 = \tfrac{374}{1{,}000}$. Simplifying by dividing out the common factors gives $0.374 = \tfrac{187}{500}$. Did you notice that the number of zeros in the denominator of $\tfrac{374}{1{,}000}$ is the same as the number of decimal places in $0.374$?

We’ve learned to convert decimals to fractions. Now we will do the reverse — convert fractions to decimals. Remember that the fraction bar means division, so $\tfrac{4}{5}$ can be written $4 \div 5$ or $5\overline{)4}$.

Example. Write $-\tfrac{5}{8}$ as a decimal.

Since a fraction bar means division, we begin by writing $\tfrac{5}{8}$ as $8\overline{)5}$. Dividing gives $0.625$, so $-\tfrac{5}{8} = -0.625$.

When we divide, we will not always get a zero remainder. Sometimes the quotient ends up with a decimal that repeats.

Example. Write $\tfrac{43}{22}$ as a decimal.

Dividing $43$ by $22$ gives a quotient whose digits begin to repeat: $1.95454545\ldots$. The digits $5$ and $4$ repeat endlessly, so $\tfrac{43}{22} = 1.9\overline{54}$.

Sometimes we may have to simplify expressions with fractions and decimals together.

Example. Simplify: $\tfrac{7}{8} + 6.4$.

First we must change one number so both numbers are in the same form — we can change the fraction to a decimal, or change the decimal to a fraction. Usually it is easier to change the fraction to a decimal. Changing $\tfrac{7}{8}$ to a decimal gives $0.875$. Adding, $0.875 + 6.4 = 7.275$. So $\tfrac{7}{8} + 6.4 = 7.275$.

A percent is a ratio whose denominator is $100$. Percent means per hundred. We use the percent symbol, %, to show percent.

Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per $100$, so the denominator of the fraction is $100$. We then change the fraction to a decimal by dividing the numerator by the denominator: $6\% = \tfrac{6}{100} = 0.06$, $78\% = \tfrac{78}{100} = 0.78$, and $135\% = \tfrac{135}{100} = 1.35$. Do you see the pattern? To convert a percent number to a decimal number, we move the decimal point two places to the left.

Example. Convert each percent to a decimal: (a) $62\%$ (b) $135\%$ (c) $35.7\%$.

(a) Moving the decimal point two places to the left: $62\% = 0.62$.

(b) Moving the decimal point two places to the left: $135\% = 1.35$.

(c) Moving the decimal point two places to the left: $35.7\% = 0.357$.

Converting a decimal to a percent makes sense if we remember the definition of percent and keep place value in mind. To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is $100$, it is easy to change that fraction to a percent: $0.83 = \tfrac{83}{100} = 83\%$, $1.05 = 1\tfrac{5}{100} = 105\%$, and $0.075 = \tfrac{7.5}{100} = 7.5\%$. Recognize the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

Example. Convert each decimal to a percent: (a) $0.51$ (b) $1.25$ (c) $0.093$.

(a) Moving the decimal point two places to the right: $0.51 = 51\%$.

(b) Moving the decimal point two places to the right: $1.25 = 125\%$.

(c) Moving the decimal point two places to the right: $0.093 = 9.3\%$.

Key terms

decimal — a way of writing a fraction whose denominator is a power of $10$, using a decimal point to separate the whole-number part from the fractional part. repeating decimal — a decimal in which the last digit or group of digits repeats endlessly, indicated with a bar over the repeating block. percent — a ratio whose denominator is $100$.

This section is adapted from Elementary Algebra 2e, Section 1.7: 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 as a table, condensed the worked examples and How To boxes into narrative prose, omitted the Be Prepared quiz, Manipulative Mathematics callouts, media links, and end-of-section exercises, and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.