- A number that can be written in the form of \(\dfrac{a}{b}\), is called rational number, where \(a\) and \(b\) are integers and \(b\neq\) zero.

**Note:**

An integer can be written in the form of \(\dfrac{a}{b}\) by putting it over one \((1)\).

Thus, integers are rational numbers.

**For example:**

\(2=\dfrac{2}{1}\)

\(-3=\dfrac{-3}{1}\)

\(0=\dfrac{0}{1}\)

A \(5\)

B \(\dfrac{-3}{2}\)

C Zero

D None of these

- A rational number can be positive, negative or zero.
- A rational number is positive if both numerator and the denominator have same signs, i.e. \((+)\) or \((-)\).

**For example:** \(\dfrac{3}{4},\;\dfrac{-3}{-4}\)

Numerator \((3)\) and denominator \((4)\) of \(\dfrac{3}{4}\) have same signs \((+)\).

Therefore, \(\dfrac{3}{4}\) is a positive rational number.

Similarly, numerator \((-3)\) and the denominator \((-4)\) of \(\dfrac{-3}{-4}\) have same signs\((-)\). Therefore, \(\dfrac{-3}{-4}\) is also a positive rational number.

- A rational number is negative if both numerator and the denominator have opposite signs.

**For example:** \(\dfrac{-5}{8},\;\dfrac{5}{-8}\)

Numerator \((-5)\) and denominator \((8)\) of \(\dfrac{-5}{8}\) have opposite signs (– and + respectively).

Therefore, \(\dfrac{-5}{8}\) is a negative rational number.

Similarly, numerator \((5)\) and denominator \((-8)\) of \(\dfrac{5}{-8}\) have opposite signs (+ and – respectively).

Therefore, \(\dfrac{5}{-8}\) is a negative rational number.

**Note:**

If '\(a\)' and '\(b\)' are integers then

\(-\left(\dfrac{a}{b}\right)=\dfrac{(-a)}{b}=\dfrac{a}{(-b)}\)

where \(b\neq\) zero

Positive and negative rational numbers are used to describe quantities having opposite directions or values.

**For example:**

'\($500\) debited from the account', represents a negative value, i.e.

\(-500\)

While '\($500\) credited to the account', represents a positive value, i.e.

\(500\)

A \(\dfrac{4}{-5}\)

B \(-\dfrac{4}{5}\)

C \(\dfrac{-4}{-5}\)

D

- Two or more rational numbers are equivalent to each other if their simplest forms are equal.
- When the numerator and denominator of a rational number do not have any common factor other than one \((1)\), it is called the simplest form of a rational number.

**For example:** \(\dfrac{3}{6}\) and \(\dfrac{9}{18}\)

Simplest form of \(\dfrac{3}{6}=\dfrac{3\div3}{6\div3}\)

\(=\dfrac{1}{2}\)

Simplest form of \(\dfrac{9}{18}=\dfrac{9\div9}{18\div9}\)

\(=\dfrac{1}{2}\)

Here, the simplest form of \(\dfrac{3}{6}\) and \(\dfrac{9}{18}\) are equal, i.e. \(\dfrac{1}{2}\).

\(\therefore\;\dfrac{3}{6}\) and \(\dfrac{9}{18}\) are equivalent rational numbers.

\(\dfrac{3}{6}=\dfrac{9}{18}\)

- We can obtain an equivalent of a rational number by dividing or multiplying both numerator and the denominator by same non-zero number.

**For example: **\(-\dfrac{4}{6}\)

\(-\dfrac{4}{6}=-\dfrac{4\div2}{6\div2}=-\dfrac{2}{3}\)

\(-\dfrac{4}{6}=-\dfrac{4×3}{6×3}=-\dfrac{12}{18}\)

Thus, \(-\dfrac{4}{6},\;-\dfrac{2}{3}\) and \(-\dfrac{12}{18}\) are equivalent rational numbers, i.e.

\(-\dfrac{4}{6}=-\dfrac{2}{3}=-\dfrac{12}{18}\)

A \(\dfrac{-3}{4}=\dfrac{-6}{2}\)

B \(\dfrac{8}{7}=\dfrac{4}{-21}\)

C \(\dfrac{16}{28}=\dfrac{4}{7}\)

D \(\dfrac{13}{53}=\dfrac{1}{4}\)

- Plotting a rational number is similar to plotting a fraction on the number line.
- Consider the following two cases:

**Case 1:** For positive rational numbers:

Let's plot \(\dfrac{5}{8}\) on the number line.

Both numerator and the denominator have the same signs.

\(\therefore\;\dfrac{5}{8}\) is a positive number.

First draw a number line and divide the interval \(0\) to \(1\) into \(8\) parts.

Now, count \(5\) units from zero to its right. It is also called moving up or moving forward from zero up to \(5\) units.

**Note:** All positive numbers are at the right side of zero on the number line.

**Case 2:** For negative rational numbers:

Let's plot \(\dfrac{-5}{8}\) on the number line.

Both numerator and the denominator have the opposite signs.

\(\therefore\;\dfrac{-5}{8}\) is a negative number.

First, draw a number line and divide the interval \(0\) to \(-1\) into \(8\) parts.

Now, count \(5\) units from zero to its left. It is also called moving down or moving backward from zero up to \(5\) units.

**Note: **All negative numbers are at the left side of zero on the number line.

**Note : **All negative numbers are at the left side of zero on the number line.

- When two points are plotted on a number line, they are known as endpoints.
- The distance between two endpoints is always positive whether they are positive or negative.

**Method I:** By using distance formula

- If '\(a\)' and '\(b\)' are two rational numbers, then the distance between them on the number line is:

\(|a-b|\;\text{or}\;|b-a|\)

**For example:** Distance between \(\dfrac{4}{5}\) and \(\dfrac{6}{5}=\left|\dfrac{4}{5}-\dfrac{6}{5}\right|=\left|-\dfrac{2}{5}\right|=\dfrac{2}{5}\) units

or \(\left|\dfrac{6}{5}-\dfrac{4}{5}\right|=\left|\dfrac{2}{5}\right|=\dfrac{2}{5}\) units

**Method II:** By using number line

Consider an example to understand it.

Here, \(\dfrac{-4}{3}\) and \(2\) are rational numbers that are representing the endpoints on the number line.

**Step 1:** Count the number of units from one endpoint to another.

**Step 2:** There are \(10\) units of \(\dfrac{1}{3}\;\left(10\:\text{times}\;\dfrac{1}{3}\right)\) between endpoints, i.e. \(\dfrac{-4}{3}\) and \(2.\)

Thus, distance between \(\dfrac{-4}{3}\) and \(2\)

\(=10×\dfrac{1}{3}\)

\(=\dfrac{10}{3}\) units

A \(P=\dfrac{-3}{4},\;Q=\dfrac{-4}{3},\;\text{Distance}=\dfrac{-2}{4}\text{units}\)

B \(P=\dfrac{-2}{3},\;Q=\dfrac{-7}{3},\;\text{Distance}=\dfrac{5}{3}\text{units}\)

C \(P=\dfrac{-7}{3},\;Q=\dfrac{-2}{3},\;\text{Distance}=\dfrac{-5}{3}\text{units}\)

D \(P=\dfrac{-4}{3},\;Q=\dfrac{-3}{4},\;\text{Distance}=\dfrac{1}{3}\text{units}\)

- An absolute value of a rational number is the distance of that number from zero on the number line.
- We can show this distance by a sign of arrow.
- An arrow points to right for positive rational numbers and to left for negative rational numbers.

**For example: **We will find the absolute value of \(\dfrac{-8}{5}\).

Since, distance is always positive, therefore the absolute value of a rational number is always positive.

Hence, absolute value of \(\dfrac{-8}{5}\) is \(\dfrac{8}{5}\).

or absolute value of \(\dfrac{-8}{5}=\left|\dfrac{-8}{5}\right|=\dfrac{8}{5}\)

Here, '\(|\;\;\;|\)' sign is used to represent the absolute value.

**Note:** Since, distance is always positive, therefore the absolute value of a rational number is always positive.

Hence, absolute value of \(\dfrac{-8}{5}\) is \(\dfrac{8}{5}\)

or absolute value of \(\dfrac{-8}{5}=\left|\dfrac{-8}{5}\right|=\dfrac{8}{5}\)

Here, '\(|\;\;\;|\)' sign is used to represent the absolute value.

A Absolute value of \(\dfrac{13}{15}\) is \(\dfrac{13}{-15}\).

B \(\left|\dfrac{-31}{42}\right|\) \(=\dfrac{42}{31}\)

C Absolute value of \(\dfrac{11}{-2}\) is \(\dfrac{-2}{11}\).

D Absolute value of \(\dfrac{-14}{25}\) is \(\dfrac{14}{25}\).

- An opposite of a rational number is the number itself but with a different sign.

**For ****example:** Opposite of \(3=-3\)

Opposite of \(\dfrac{-4}{3}=\dfrac{4}{3}\)

- All positive numbers are opposites of their negatives and all negative numbers are opposites of their positives.
- On the number line, the distance of opposite numbers from zero is same although they are on the opposite sides.

**Example:** Are \(-3\) and \(3\) opposite numbers?

Distance between \(0\) and \(3=|0-3|=3\)

Distance between \(0\) and \(-3=|0-(-3)|=3\)

Distance of both the numbers from zero is same, i.e. \(3\).

\(\therefore\) They are opposite numbers.

- Zero doesn't have an opposite as it is neither negative nor positive.
- The opposite of an opposite of a number is the number itself, i.e.

\(-(-9)=9\)

Here, \(-9\) is the opposite of \(9.\)

Opposite of \(-9=9\)

Other examples of opposites:

- The temperature above sea level is opposite to the temperature below sea level and vice versa.
- The amount debited from an account is opposite to the amount credited to the account and vice versa.
- Loss is opposite to gain and vice versa.
- Giving something is opposite to taking something and vice versa.

A Marc has a loss of \($25.00\) and gain of \($50\) on purchasing some items.

B The temperature above sea level is \(40°C\) and temperature below sea level is \(38°C\).

C Kevin gives \(5\) chocolates and take \(6\) chocolates.

D \($40.00\) is debited from an account and \($40.00\) is credited to an account.