Rational Numbers: Classify and Convert to Decimals | Grade 7 Math | LevelKite

You already work with integers such as $-6$ , $0$ and $15$ , with fractions such as $\frac{3}{5}$ , and with decimals such as $0.4$ . All of these belong to one large family called the rational numbers. The name comes from the word ratio: a rational number is a number you can write as a fraction of two integers, as long as the bottom number is not $0$ .

In this lesson you will learn what makes a number rational and how to sort, or classify, numbers inside that family. Then you will use long division to change fractions and mixed numbers into decimals. A mixed number, such as $2\frac{1}{2}$ , has a whole-number part and a fraction part: it means $2 + \frac{1}{2}$ .

Key concept

What makes a number rational

A rational number is any number that can be written as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is not $0$ . Here $p$ is the numerator (the top number) and $q$ is the denominator (the bottom number).

The fraction $\frac{p}{q}$ means the division $p \div q$ . You cannot divide by $0$ , so $q$ is never $0$ .

When you carry out that division, the decimal you get either terminates (it stops) or repeats (from some point on, the same digit or group of digits comes back forever).

\frac{p}{q} = p \div q \quad \text{for integers } p \text{ and } q,\ q \neq 0

When you classify a number, you say which sets, or groups, of numbers it belongs to. This lesson uses three sets:

Whole numbers: $0, 1, 2, 3, \ldots$
Integers: the whole numbers together with their opposites (the opposite of $3$ is $-3$ ): $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$
Rational numbers: every number that can be written as $\frac{p}{q}$ with integers $p$ and $q$ , where $q \neq 0$

Each set sits inside the next one. Every whole number is an integer. Every integer $n$ is rational, because $n = \frac{n}{1}$ : a fraction of two integers whose denominator is $1$ , not $0$ . So one number can have several names.

It does not work in the other direction. The integer $-4$ is not a whole number, and the rational number $\frac{3}{5}$ is not an integer.

To classify a number, first find its value. Then ask three questions. Can it be written as a fraction of integers with a denominator that is not $0$ ? Is it an integer? Is it a whole number?

One number can belong to more than one set

Number	As a fraction of integers	Whole number?	Integer?	Rational number?
$7$	$\frac{7}{1}$	Yes	Yes	Yes
$0$	$\frac{0}{1}$	Yes	Yes	Yes
$-4$	$\frac{-4}{1}$	No	Yes	Yes
$\frac{12}{3}$	$\frac{12}{3}$ , which equals $4$	Yes	Yes	Yes
$\frac{3}{5}$	$\frac{3}{5}$	No	No	Yes
$2\frac{1}{2}$	$\frac{5}{2}$ , since $2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2}$	No	No	Yes
$-0.25$	$\frac{-25}{100}$ , since $0.25$ is $25$ hundredths	No	No	Yes

Negative fractions are rational too. The negative sign can sit in three places: in front of the fraction, in the numerator or in the denominator. Test all three with $12$ and $4$ :

$-\frac{12}{4} = -(12 \div 4) = -3$
$\frac{-12}{4} = (-12) \div 4 = -3$
$\frac{12}{-4} = 12 \div (-4) = -3$

All three give $-3$ . You know why from dividing integers: a negative integer divided by a positive one is negative, and so is a positive integer divided by a negative one.

The same sign rules hold when the quotient, the answer to a division, is not an integer. So $-\frac{3}{4}$ , $\frac{-3}{4}$ and $\frac{3}{-4}$ are three ways to write one number. That is why $-\frac{3}{4}$ is rational: written as $\frac{-3}{4}$ , it is a fraction of the integers $-3$ and $4$ , and $4 \neq 0$ .

In general, for any integers $p$ and $q$ with $q \neq 0$ :

-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q} \qquad (p \text{ and } q \text{ integers},\ q \neq 0)

Worked example

Classify a fraction with a negative denominator

Which of these names fit the number $\frac{24}{-6}$ : whole number, integer, rational number?

Start with the definition of a rational number. The numerator and the denominator are both integers, and the denominator is not 0. So the number is rational.
$\frac{24}{-6}: \quad p = 24, \quad q = -6, \quad q \neq 0$
Find the value. The fraction means the numerator divided by the denominator. A positive integer divided by a negative integer is negative, and 24 divided by 6 is 4 because 4 times 6 is 24.
$\frac{24}{-6} = 24 \div (-6) = -4$
Check the integers. They are the whole numbers together with their opposites. The value is the opposite of the whole number 4, so it is an integer.
$-4 \text{ is in the list } \ldots, -5, -4, -3, \ldots$
Check the whole numbers. They start at 0 and count up, so none of them is negative. The value is negative, so it is not a whole number.
$-4 \text{ is not in the list } 0, 1, 2, 3, \ldots$

Answer

The number $\frac{24}{-6}$ equals $-4$ , so it is an integer and a rational number, but it is not a whole number.

Now turn fractions into decimals. Because $\frac{p}{q}$ means $p \div q$ with $q \neq 0$ , you find the decimal form by dividing the numerator by the denominator. The number being divided is the dividend. The number you divide by is the divisor.

For a negative fraction, use the sign rule first: $-\frac{3}{8} = -(3 \div 8)$ . Divide the positive numbers, then put the negative sign on the decimal. That way the long division itself uses only whole numbers, with a divisor that is not $0$ .

Long division finds the quotient one digit at a time. When the dividend runs out of digits, write a decimal point and zeros after it. This does not change its value: $3 = 3.000$ . At each step you divide, write one digit of the quotient, subtract, and bring down the next $0$ . What is left after you subtract is the remainder, a whole number smaller than the divisor.

Once the only digits left to bring down are zeros you added, a remainder of $0$ means nothing is left to divide. The division ends, and the decimal terminates.

Worked example

Convert a fraction to a decimal

Write $\frac{3}{8}$ as a decimal using long division.

Rewrite the fraction as a division: the numerator divided by the denominator. The divisor is 8, which is not 0, so the division is allowed. Write the dividend with a decimal point and zeros. Its value stays the same, and now there are digits to bring down.
$\frac{3}{8} = 3 \div 8 = 3.000 \div 8$
The divisor 8 is greater than 3, so there are no eights in 3. Write 0 in the ones place of the quotient, then the decimal point. The 3 ones are the same as 30 tenths. The divisor goes into 30 three times, because 3 times 8 is 24 and 4 times 8 is 32, which is too big. Write 3 in the tenths place and subtract to find the remainder.
$30 - 3 \times 8 = 30 - 24 = 6$
The remainder is 6 tenths. Bring down a 0 to make 60 hundredths. The divisor goes into 60 seven times, because 7 times 8 is 56 and 8 times 8 is 64, which is too big. Write 7 in the hundredths place and subtract.
$60 - 7 \times 8 = 60 - 56 = 4$
The remainder is 4 hundredths. Bring down a 0 to make 40 thousandths. The divisor goes into 40 exactly 5 times, because 5 times 8 is 40. Write 5 in the thousandths place and subtract. The remainder is 0 and only added zeros are left, so the division ends.
$40 - 5 \times 8 = 40 - 40 = 0$
Read the quotient from the digits you wrote: 0 ones, 3 tenths, 7 hundredths and 5 thousandths. Check it by multiplying the quotient by the divisor 8, which is not 0. The product should equal the dividend 3, and it does.
$0.375 \times 8 = 3$

Answer

As a decimal, $\frac{3}{8} = 0.375$ . The remainder reached $0$ , so this decimal terminates.

Sometimes the remainder never reaches $0$ . After the decimal point, every digit you bring down is a $0$ , so each step depends only on the remainder before it. If one of those remainders comes back, the steps that followed it the first time happen again in the same order, without end. The decimal repeats.

One of these two things must happen. Each remainder is a whole number smaller than the divisor, so only a limited number of remainders are possible. When you divide by $6$ , for example, they are $0, 1, 2, 3, 4$ and $5$ . Keep dividing and you must reach $0$ or meet a remainder again. That is why the decimal form of every rational number terminates or repeats.

To write a repeating decimal, put a bar over the digit or digits that repeat, and only over those: $0.8333\ldots = 0.8\overline{3}$ and $0.272727\ldots = 0.\overline{27}$ . The bar says that the pattern goes on forever.

Worked example

Convert a negative mixed number to a decimal

Write $-2\frac{5}{6}$ as a decimal using long division.

The negative sign belongs to the whole mixed number, both the 2 and the five sixths. So this number is the opposite of the positive mixed number. Convert the positive mixed number first, then put the negative sign back at the end. The whole-number part 2 stays as it is, so only the fraction part needs long division.
$-2\frac{5}{6} = -\left(2 + \frac{5}{6}\right)$
Divide the numerator 5 by the denominator 6. The divisor 6 is not 0, so the division is allowed. There are no sixes in 5, so write 0 and the decimal point. The 5 ones are 50 tenths, and the divisor goes into 50 eight times, because 8 times 6 is 48 and 9 times 6 is 54, which is too big. Write 8 in the tenths place and subtract.
$\frac{5}{6} = 5.000 \div 6, \qquad 50 - 8 \times 6 = 50 - 48 = 2$
Bring down a 0 to make 20 hundredths. The divisor goes into 20 three times, because 3 times 6 is 18 and 4 times 6 is 24, which is too big. Write 3 in the hundredths place and subtract.
$20 - 3 \times 6 = 20 - 18 = 2$
The remainder 2 has come back. Bringing down a 0 makes 20 again, so the next digit is 3 with remainder 2, and the same step repeats without end. The digit 3 repeats but the 8 does not, so the bar goes over the 3 only.
$\frac{5}{6} = 0.8333\ldots = 0.8\overline{3}$
Add the whole-number part to the decimal for the fraction part. Then put the negative sign back, because the number is the opposite of the positive mixed number.
$-\left(2 + 0.8\overline{3}\right) = -2.8\overline{3}$
Check by using a single fraction. The positive mixed number equals 17 sixths. Dividing 17 by 6 gives 2 with remainder 5. From there the steps match the ones above: 50 tenths, then 20 hundredths, then 20 again. The digits agree.
$2\frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{17}{6}, \qquad 17 - 2 \times 6 = 5$

Answer

As a decimal, $-2\frac{5}{6} = -2.8\overline{3}$ . The bar shows that the digit $3$ repeats forever.

Takeaway. A rational number is any number you can write as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ . Whole numbers and integers are rational too, so to classify a number, find its value and check each set. For those same $p$ and $q$ , a negative sign can sit in front, in the numerator or in the denominator: $-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$ .

To find the decimal form, divide the numerator by the denominator, which is never $0$ , using long division. Once you are bringing down only added zeros, the decimal terminates if a remainder is $0$ , and it repeats if a remainder comes back. Mark a repeat with a bar instead of cutting it off.

Next, you will learn to identify terminating and repeating decimals and to convert terminating decimals back into fractions or mixed numbers.