Informative line

Representation Of Rational Numbers On Number Line

Absolute Value

  • 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 \(\dfrac {3}{4}\).

Illustration Questions

Which one of the following is the absolute value of \(\dfrac {-3}{7}\)?

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

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

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

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

×

The absolute value of any number is the numerical value of that number and is always positive.

So, the numerical value of \(\dfrac {-3}{7}\) is \(\dfrac {3}{7}\)  i.e.

 \(\left |\dfrac {-3}{7}\right|=\dfrac {3}{7}\) 

Hence, option (A) is correct.

Which one of the following is the absolute value of \(\dfrac {-3}{7}\)?

A

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

.

B

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

C

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

D

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

Option A is Correct

Rational Number

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

Illustration Questions

Which one of the following is NOT a rational number?

A \(-2\)

B \(\sqrt 3\)

C \(0.4\)

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

×

In option (A), 

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

So, \(-2=\dfrac {-2}{1}\)

Thus, \(-2\) is a rational number.

Hence, option (A) is incorrect.

In option (B), 

\(\sqrt 3\) is not a perfect square root.

So, \(\sqrt 3\) is not a rational number.

Hence, option (B) is correct.

In option (C), 

\(0.4\) is a decimal number which can be written in the form of  \(\dfrac {p}{q}\) as 

\(0.4=\dfrac {4}{10}\) or \(\dfrac {2}{5}\)

So, \(0.4\) is a rational number.

Hence, option (C) is incorrect.

In option (D), 

\(\dfrac {-3}{7}\) is a fraction which is already in the form of \(\dfrac {p}{q}\).

So, \(\dfrac {-3}{7}\) is a rational number.

Hence, option (D) is incorrect.

Which one of the following is NOT a rational number?

A

\(-2\)

.

B

\(\sqrt 3\)

C

\(0.4\)

D

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

Option B is Correct

Representation of Rational Numbers (Decimal Form) on Number line 

  • 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 \(-1.20\)

Illustration Questions

Which one of the following options represents the location of point \(P\) on the given number line?

A \(-1.75\)

B \(-0.75\)

C \(-0.80\)

D \(-1.80\)

×

On the given number line, there are four equal segments between each whole, which means each segment has a scale of \(0.25\).

image

To find the location of point \(P\) on the given number line, start from \(0\) by moving in the backward direction and continue moving till we reach point \(P\).

Count the number of segments we moved to reach the point \(P\).

Here, we moved \(3\) segments, each of length \(0.25\)

Thus, point \(P\) represents the point  \(0.75(=3\times0.25)\) on the number line.

image

Hence, option (B) is correct.

Which one of the following options represents the location of point \(P\) on the given number line?

image
A

\(-1.75\)

.

B

\(-0.75\)

C

\(-0.80\)

D

\(-1.80\)

Option B is Correct

Representation of Rational Numbers (Fraction Form) on Number Line

  • 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

Illustration Questions

Which one of the following options represents the location of point \(S\) on the given number line?

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

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

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

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

×

On the given number line, point \(S\) lies in between \(0\) and \(-1\).

image

Since the interval between \(0\) and  \(-1\) is divided into four equal parts, so the size of each part represents \(\dfrac {1}{4}\).

image

Point \(S\) represents the combined length of \(2\) parts from \(0\) in the leftward direction.

So, point \(S\) represents \(-\dfrac {2}{4}\) or \(-\dfrac {1}{2}\).

image

Hence, option (A) is correct.

Which one of the following options represents the location of point \(S\) on the given number line?

image
A

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

.

B

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

C

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

D

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

Option A is Correct

Representation of Rational Numbers (Mixed Numbers) on Number line 

  • 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

Illustration Questions

Which one of the following options represents the location of point \(T\) on the given number line?

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

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

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

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

×

On the given number line, point \(T\) lies in between \(-3\) and \(-4\).

So, the whole is \(-3\), as point \(T\) lies to the left of \(-3\)

image

The size of each part between \(-3\) and \(-4\) represents \(\dfrac {1}{3}\) as it is divided into \(3\) equal parts.

image

Point \(T\) represents the combined length of \(2\) parts.

image

Thus, \(-3\dfrac {2}{3}\) represents the location of point \(T\) on the given number line.

Hence, option (D) is correct.

Which one of the following options represents the location of point \(T\) on the given number line?

image
A

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

.

B

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

C

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

D

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

Option D is Correct

Practice Now