Decimals

Round decimals

Decimals are another way of writing fractions whose denominators are powers of ten.

Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. The chart below shows the names of the place values to the left and right of the decimal point.

When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.

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

(a) To the nearest hundredth. Locate the hundredths place, and underline the digit to its right (the $9$). 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 rounding digit. So, $18.379$ rounded to the nearest hundredth is $18.38$. Notice that the deleted digits were NOT replaced with zeros.

(b) To the nearest tenth. Locate the tenths place, and underline the digit to its right (the $7$). Because $7$ is greater than or equal to $5$, add $1$ to the $3$. So, $18.379$ rounded to the nearest tenth is $18.4$.

(c) To the nearest whole number. Locate the ones place, and underline the digit to its right (the $3$). Since $3$ is not greater than or equal to $5$, do not add $1$ to the $8$. 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 or subtract: (a) $-23.5 - 41.38$, (b) $14.65 - 20$.

(a) The difference will be negative. To subtract, we add the numerals. Write the numbers so the decimal points line up vertically, using a $0$ as a placeholder after the $5$ in $23.5$ — remember, $\tfrac{5}{10} = \tfrac{50}{100}$, so $0.5 = 0.50$. Add the numbers as if they were whole numbers, then place the decimal point in the sum, and write the result with the correct sign.

(b) The difference will be negative. To subtract, we subtract $14.65$ from $20$. Write the numbers so the decimal points line up vertically; since $20$ is a whole number, place the decimal point after the $0$ and put in zeros to the right as placeholders. Subtract, place the decimal point in the answer, and write the result with the correct sign.

Multiply and divide decimals

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. 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. Write in vertical format, lining up the numbers on the right, and multiply as if they were whole numbers:

Add the number of decimal places in the factors. The factor $4.075$ has $3$ decimal places and the factor $3.9$ has $1$ decimal place, so the product has $1 + 3 = 4$ decimal places. Place the decimal point $4$ places from the right: $15.8925$. The signs are different, so the product is negative:

Often, especially in the sciences, you will multiply decimals by powers of $10$ ($10$, $100$, $1000$, 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 we move the decimal point to the right to get the product.

Example. Multiply $5.63$ by (a) $10$, (b) $100$, (c) $1000$.

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal point to the right.

(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 $1000$, so move the decimal point $3$ places to the right. A zero must be added to the end: $5.63(1000) = 5630$.

Just as with multiplication, division of signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. 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: in $a \div b = c$, the number $a$ is the dividend, $b$ is the divisor, and $c$ is the quotient. In long-division form, $b\,\overline{\smash{)}\,a}$ places the quotient $c$ above the dividend.

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

Remember, you can “move” the decimals in the divisor and dividend because of the Equivalent Fractions Property. The signs are the same, so the quotient is positive. Make the divisor $0.06$ a whole number by moving the decimal point $2$ places to the right, and move the decimal point in the dividend $2$ places to the right as well ($25.65$ becomes $2565.0$). Then divide, placing the decimal point in the quotient above the decimal point in the dividend:

The signs are the same, so the quotient is positive:

Convert decimals, fractions, and percents

In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.

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 to $0.03$:

Example. Write (a) $0.374$ as a fraction, (b) $-\tfrac{5}{8}$ as a decimal.

(a) Determine the place value of the final digit. The final digit, $4$, is in the thousandths place, so the numerator is $374$ and the denominator is $1000$. Simplify the fraction by dividing out the common factors:

(b) Since a fraction bar means division, we begin by writing $\tfrac{5}{8}$ as $8\,\overline{\smash{)}\,5}$. Now divide:

So, $-\tfrac{5}{8} = -0.625$.

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. After doing this many times, you may see the pattern.

To convert a percent number to a decimal number, we move the decimal point two places to the left:

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. 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: (a) percent to a decimal: $62\%$, $135\%$, and $35.7\%$; (b) decimal to a percent: $0.51$, $1.25$, and $0.093$.

(a) Move the decimal point two places to the left:

(b) Move the decimal point two places to the right and add the percent sign:

Simplify expressions with square roots

Remember that when a number $n$ is multiplied by itself, we write $n^2$ and read it “$n$ squared.” The result is called the square of a number $n$. For example, $8^2$ is read “$8$ squared” and $64$ is called the square of $8$. Similarly, $121$ is the square of $11$ because $11^2$ is $121$. It will be helpful to learn to recognize the perfect square numbers.

What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive:

Because $10^2 = 100$, we say $100$ is the square of $10$. We also say that $10$ is a square root of $100$. A number whose square is $m$ is called a square root of a number $m$.

Notice $(-10)^2 = 100$ also, so $-10$ is also a square root of $100$. Therefore, both $10$ and $-10$ are square roots of $100$. So, every positive number has two square roots — one positive and one negative. The radical sign, $\sqrt{m}$, denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{100} = 10$. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $-\sqrt{100} = -10$. We read $-\sqrt{100}$ as “the opposite of the principal square root of $100$.”

Example. Simplify: (a) $\sqrt{25}$, (b) $\sqrt{121}$, (c) $-\sqrt{144}$.

(a) Since $5^2 = 25$, we have $\sqrt{25} = 5$.

(b) Since $11^2 = 121$, we have $\sqrt{121} = 11$.

(c) The negative is in front of the radical sign, so $-\sqrt{144} = -12$.

Identify integers, rational numbers, irrational numbers, and real numbers

We have already described numbers as counting numbers, whole numbers, and integers:

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

In general, any decimal that ends after a number of digits (such as $7.3$ or $-1.2684$) is a rational number. Simply write the decimal as a mixed number. The decimal for $\tfrac{1}{3}$ is the number $0.\overline{3}$. The bar over the $3$ indicates that the number $3$ repeats infinitely. The number(s) under the bar is called the repeating block and it repeats continuously.

Since all integers can be written as a fraction whose denominator is $1$, the integers (and so also the counting and whole numbers) are rational numbers.

Are there any decimals that do not stop or repeat? Yes! The number $\pi$ (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots ($\dots$) to indicate the decimal does not stop or repeat:

The square root of a number that is not a perfect square is a decimal that does not stop or repeat. A number whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.

Let’s summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number repeats or stops, the number is a rational number. If the decimal form of a number does not repeat and does not stop, the number is an irrational number.

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

Later in this course we will introduce numbers beyond the real numbers. The diagram below illustrates how the number sets we’ve used so far fit together: the counting numbers sit inside the whole numbers, which sit inside the integers, which sit inside the rational numbers; the rational numbers together with the irrational numbers make up the real numbers.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify $\sqrt{-25}$? Is there a number whose square is $-25$? None of the numbers that we have dealt with so far has a square that is $-25$. Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to $\sqrt{-25}$. The square root of a negative number is not a real number.

Example. Given the numbers $-7$, $\tfrac{14}{5}$, $8$, $\sqrt{5}$, $5.9$, $-\sqrt{64}$, list the (a) whole numbers, (b) integers, (c) rational numbers, (d) irrational numbers, (e) real numbers.

(a) Remember, the whole numbers are $0, 1, 2, 3, \dots$, so $8$ is the only whole number given.

(b) The integers are the whole numbers and their opposites (which includes $0$). So the whole number $8$ is an integer, and $-7$ is the opposite of a whole number so it is an integer, too. Also, notice that $64$ is the square of $8$ so $-\sqrt{64} = -8$. So the integers are $-7$, $8$, and $-\sqrt{64}$.

(c) Since all integers are rational, then $-7$, $8$, and $-\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\tfrac{14}{5}$ and $5.9$ are rational. So the list of rational numbers is $-7$, $\tfrac{14}{5}$, $8$, $5.9$, and $-\sqrt{64}$.

(d) Remember that $5$ is not a perfect square, so $\sqrt{5}$ is irrational.

(e) All the numbers listed are real numbers.

Locate fractions and decimals on the number line

We now want to include fractions and 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.

To locate a proper fraction such as $\tfrac{1}{5}$, we note that it has value less than one, so it lies between $0$ and $1$. The denominator is $5$, so we divide the unit from $0$ to $1$ into $5$ equal parts $\tfrac{1}{5}$, $\tfrac{2}{5}$, $\tfrac{3}{5}$, $\tfrac{4}{5}$ and plot $\tfrac{1}{5}$. Similarly, $-\tfrac{4}{5}$ is between $0$ and $-1$. To locate an improper fraction, it is often easier to change it to a mixed number first:

Example. Locate and label the following on a number line: $4$, $\tfrac{3}{4}$, $-\tfrac{1}{4}$, $-3$, $\tfrac{6}{5}$, $-\tfrac{5}{2}$, and $\tfrac{7}{3}$.

Locate and plot the integers $4$ and $-3$. Locate the proper fraction $\tfrac{3}{4}$; it is between $0$ and $1$, so divide that distance into four equal parts and plot $\tfrac{3}{4}$. Similarly plot $-\tfrac{1}{4}$. For the improper fractions, convert to mixed numbers first: $\tfrac{6}{5} = 1\tfrac{1}{5}$, $-\tfrac{5}{2} = -2\tfrac{1}{2}$, and $\tfrac{7}{3} = 2\tfrac{1}{3}$. Then plot each in its interval.

Example. Locate on the number line: (a) $0.4$, (b) $-0.74$.

(a) The decimal number $0.4$ is equivalent to $\tfrac{4}{10}$, a proper fraction, so $0.4$ is located between $0$ and $1$. On a number line, divide the interval between $0$ and $1$ into $10$ equal parts. Now label the parts $0.1$, $0.2$, $0.3$, $0.4$, $0.5$, $0.6$, $0.7$, $0.8$, $0.9$, $1.0$. 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.

(b) 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 $-1$ and place $-0.74$ between $-0.70$ and $-0.80$, closer to $-0.70$.

Key terms

square of a number — the result of multiplying a number by itself; if $n^2 = m$, then $m$ is the square of $n$. square root of a number — a number $n$ whose square is $m$; if $n^2 = m$, then $n$ is a square root of $m$. principal square root — the positive square root, denoted by the radical sign $\sqrt{m}$. radicand — the number under the radical sign. rational number — a number of the form $\tfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$; its decimal form stops or repeats. irrational number — a number that cannot be written as the ratio of two integers; its decimal form does not stop and does not repeat. real number — a number that is either rational or irrational.

This section is adapted from Intermediate Algebra 2e, Section 1.4: Decimals by Lynn Marecek 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, the real-number-sets and number-line figures as accessible inline graphics, and the rounding, operation, and long-division steps as typeset math and prose; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the “Try It” practice problems into interactive exercises with instant feedback.