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

• 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 \(-1.20\)

• 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

• 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