- The absolute value is the numerical value of any number.
- On a number line, the distance of any number from zero is the absolute value of the number.
- Since distance is always positive, so the absolute value of any number is always positive.
- We use modulus sign (||) for an absolute value.

**For ****example**: The absolute value of \(\dfrac {-3}{4}\) is represented as \(\left | {\dfrac {-3}{4}}\right|\) which is \(\dfrac {3}{4}\).

A \(\dfrac {3}{7}\)

B \(\dfrac {-3}{7}\)

C \(\dfrac {7}{3}\)

D \(\dfrac {3}{-7}\)

- A rational number is represented in the form of \(\dfrac {p}{q}\), where \(p\) and \(q\) are integers and \(q\neq0\).
- A rational number can be an integer, a fraction or a decimal.
- To understand it, consider an example:

\(2,\;\dfrac {3}{5}\) and \(0.25\) are rational numbers. Check whether the statement is true or not?

- Now, we have to find out whether the given numbers are rational or not.
- Let's consider the \(2\) first.

\(2\) is an integer. It can be written in the form of \(\dfrac {p}{q}\) by putting it over \(1\).

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

Thus, \(2\) is a rational number.

- Now, consider \(\dfrac {3}{5}\).
- \(\dfrac {3}{5}\) is a fraction which is already in the form of \(\dfrac {p}{q}\). So, \(\dfrac {3}{5}\) is a rational number.
- Lastly, \(0.25\) is a decimal number which can also be simplified into a fraction i.e.

\(0.25=\dfrac {25}{100}=\dfrac {1}{4}\)

So, \(0.25\) is a rational number.

**Note: **All perfect square roots are rational numbers.

Example: \(\sqrt 4,\;\sqrt {16},\;\sqrt 9,\;\sqrt {25}.......\)

A \(-2\)

B \(\sqrt 3\)

C \(0.4\)

D \(\dfrac {-3}{7}\)

- Rational numbers (decimal form) can be represented on the number line.
- To do so, first, mark the wholes and then the parts.
- For rational (negative decimal) numbers, move left from zero on the number line.

**For example:**

- Mark \(-1.20\) on the given number line.

- There are \(5\) equal segments between each whole, which means each segment has a scale of \(0.20\)

- First, find the whole of \(-1.20\), i.e. \(-1\).
- To find the '\(-1\)', start from \(0\), move in backward direction and mark '\(-1\)'.

- Next, we find parts of \(-1.20\), i.e. \(0.20\).
- To find the '\(.20\)', start from '\(-1\)', move in backward direction and mark \(-1.20\)
- Hence, the number line represents \(-1.20\)

Hence, the above number line represents \(-1.20\)

A \(-1.75\)

B \(-0.75\)

C \(-0.80\)

D \(-1.80\)

- Rational numbers (fraction form) can be represented on the number line.
- To represent the rational number \(\dfrac {a}{b}\) on a number line, follow the given steps:
- Define the intervals starting from zero.
- Divide each interval on the right side of zero for the positive rational number and left side for the negative rational number.
- The size of each part is \(\dfrac {1}{b}\).
- The rational number \(\dfrac {a}{b}\) represents the combined length of '\(a\)' parts of size \(\dfrac {1}{b}\).

**For example:** Represent \(\dfrac {4}{5}\) on the number line.

**Step 1: **Define the interval \(0\) to \( 1\).

**Step 2: **Divide the interval into \(5\) equal parts.

**Step 3: **Size of each part is \(\dfrac {1}{5}\).

**Step 4: **Thus, \(\dfrac {4}{5}\) represents the combined length of \(4\) parts.

**Step 5: **The resulting number line representation is

A \(\dfrac {-1}{2}\)

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

C \(\dfrac {-1}{4}\)

D \(\dfrac {1}{4}\)

- Rational numbers (mixed number form) can be represented on the number line.
- To represent the rational number '\(a\dfrac {b}{c}\)' on a number line, follow the given steps:
- First, find the whole number.
- Define the intervals starting from '\(a\)'.
- Divide each interval into '\(c\)' equal parts.
- The size of each part is \(\dfrac {1}{c}\).
- The rational number \(a\dfrac {b}{c}\) represents the combined length of '\(b\)' parts of size \(\dfrac {1}{c}\) to the right of \(a\).

**For example:** Represent \(2\dfrac {3}{4}\) on the number line.

In the rational number \(2\dfrac {3}{4}\), \(2\) is the whole and \(\dfrac {3}{4}\) is the fraction.

**Step 1: **Find the whole number, i.e. \(2 \).

**Step 2: **Divide the interval between \(2\) and \(3\) into \(4\) equal parts.

**Step 3: **The size of each part is \(\dfrac {1}{4}\).

**Step 4: **Thus, \(2\dfrac {3}{4}\) represents the combined length of \(3\) parts.

**Step 5: **The resulting number line representation is

A \(-4\dfrac {1}{3}\)

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

C \(-4\dfrac {2}{3}\)

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