Representation of Rational Numbers on Number Line: Example

A rational number can be defined as a real number written as a simple fraction or a ratio. All the numbers in mathematics are represented in a number line. A number line is a straight line with all positive and negative real numbers placed to the right and left of a fixed point known as the reference point called 0. The rational number representation on a number line in equal intervals is the rational number on a number line. 

A number of the form \(\frac{p}{q}\) or a number that can be expressed in the form \(\frac{p}{q}\;,\) where \(p\) and \(q\) are integers, and \(q \ne 0\) is called a rational number. Let us learn more about the representation of rational numbers in a number line, with examples and solved questions. 

Learn About Rational Numbers

Rational Number on a Number Line

To represent rational numbers on the number line, we draw a line and mark a point \(O\) on it to represent the rational number zero. The positive rational numbers will be represented by points on the right side of \(O,\) and negative rational numbers will be represented by the points on the left side of \(O.\) If we mark a point \(A\) on the number line to the right side of \(O\) to represent \(1,\) then \(OA = 1\) unit. Similarly, if we chose a point \(A\) on the number line to the left side of \(O\) to represent \( – 1,\) then \(OA’ = 1\) unit. Now, suppose if we want to represent the rational number \(\frac{1}{2}\) on the number line. For this, we divide the line segment \(OA\) into three equal parts. Let \(P\) be the mid-point of the line segment \(OA.\) Then, \(OP = PA = \frac{1}{2}.\) Since \(O\) represent \(0\) and \(A\) represents \(1,\) therefore \(P\) represents the rational number \(\frac{1}{2}\) as shown.

Similarly, if we want to represent the rational number \(\frac{{ – 1}}{3}\) on the number line, we divide the line segment \(OA’\) into three equal parts. Let \(Q\) and \(R\) be the points dividing the segment \(OA’\) into three equal parts. Then, \(OQ = QR = RA’ = \frac{{ – 1}}{3}.\) Since \(O\) represents \(0\) and \(A’\) represents \( – 1.\) Therefore, \(Q\) represents \(\frac{{ – 1}}{3}\) as shown.

Definition of a Number Line

A number line is a straight line with numbers placed at equal intervals or segments along its length. A number line can be extended infinitely in any direction and represented horizontally. A number line is a reference for comparing and ordering numbers. The number line is a way to describe any actual number that includes every whole number and natural number.

Examples of Rational Numbers

1. Each of the numbers \(\frac{3}{{ – 2}},\;\frac{{ – 4}}{{15}},\;\frac{{ – 8}}{5},\;\frac{2}{{19}}\) are  rational numbers.
2. Zero is a rational number since we can write \(0 = \frac{0}{1},\) which is the quotient of two integers with a non-zero denominator.
3. Every natural number is a rational number. We can write \(1 = \frac{1}{1},\;2 = \frac{2}{1},\;3 = \frac{3}{1}\) and so on. In general, if \(n\) is a natural number, then we can write \(n = \frac{n}{1},\) which is a rational number.
4. Every integer is a rational number. If \(m\) is an integer, then we can write it as \(\frac{m}{1},\) which is a rational number.
5. Every fraction is a rational number. Let \(\frac{a}{b}\) be a fraction. Then, \(a\) and \(b\) are whole numbers and \(b \ne 0.\)

Positive and Negative Rational Numbers

A rational number is said to be positive if its numerator and denominator are either both positive or both negative.
Example: \(\frac{7}{5},\;\frac{{ – 8}}{{ – 13}}\)

A rational number is said to be negative if its numerator and denominator are such that one is a positive integer and the other is a negative integer.
Example: \(\frac{{ – 5}}{3},\;\frac{5}{{ – 3}},\;\frac{{ – 19}}{5},\;\frac{5}{{ – 19}}\)

Properties of Rational Numbers

1. If \(\frac{p}{q}\) is a rational number and is a non-zero integer, then \(\frac{p}{q} = \frac{{p \times m}}{{q \times m}}.\) Thus, a rational number remains unchanged if its numerator and denominator are multiplied by the same non-zero integer.
Example: \(\frac{2}{5} = \frac{{2 \times 2}}{{5 \times 2}}, = \frac{{2 \times 3}}{{5 \times 3}} = \frac{{2 \times 4}}{{5 \times 4}}, = ……\)

2. If \(\frac{p}{q}\) is a rational number and \(m\) is a common divisor of \(p\) and \(q,\) then \(\frac{p}{q} = \frac{{p \div m}}{{q \div m}}.\) Thus, on dividing the numerator and denominator of a rational number by a common divisor, the result remains unchanged. Example: \(\frac{{18}}{{81}} = \frac{{18 \div 9}}{{81 \div 9}} = \frac{2}{9}\) [ HCF of \(18\) and \(81\) is \(9\) ]

Equivalent Rational Number

On multiplying the numerator and denominator of a given rational number by the same non-zero number, we get a rational number equivalent to the given rational number.
Similarly, on dividing the numerator and denominator of a given rational number by a common divisor, we get a rational number equivalent to the given rational number.
Thus, two rational numbers are equivalent if one can be obtained from the other by multiplying (or dividing) its numerator and denominator by the same non-zero number. Thus, equivalent rational numbers are equal.

Example: Four rational numbers equivalent to \(\frac{3}{4}\) is \(\frac{{3 \times 2}}{{4 \times 2}} = \frac{{3 \times 3}}{{4 \times 3}} = \frac{{3 \times 4}}{{4 \times 4}} = \frac{{3 \times 5}}{{4 \times 5}}\)

Standard Form of a Rational Number

A rational number \(\frac{p}{q}\) is said to be in standard form if \(q\) is positive and \(p\) and \(q\) have no common divisor except \(1.\) Example: \(\frac{{21}}{{35}} = \frac{{21 \div 7}}{{35 \div 7}} = \frac{3}{5}.\) So, the standard form of \(\frac{{21}}{{35}} = \frac{3}{5}\)

Number Line Representation

We can explain arithmetic operations such as adding and subtracting numbers on a number line. To begin with, one must know to locate numbers on a number line. Zero is the middle point of a number line. Therefore, all positive numbers will be on the right side of the zero, whereas negative numbers will be on the left side of zero on the number line. In other words, if we move to the left side of zero, the value of the number decreases, and if we move to the right side of zero, the value of the number increases.

An example for the number line is given below.

Solved Examples

Q.1. Represent \(\frac{5}{3}\) and \(\frac{{ – 5}}{3}\) on a number line.
Ans: In order to represent \(\frac{5}{3}\) and \(\frac{{ – 5}}{3}\) on a number line, we first draw a number line and mark a point \(O\) on it to represent zero. Now, we find the points \(P\) and \(Q\) on the number line representing the positive integers \(5\) and \( – 5,\) respectively, as shown below. Now, divide the segment \(OP\) into three equal parts. Let \(A\) and \(B\) be the points of division so that \(OA = OB = BP.\) By construction, \(OA\) is one-third of \(OP.\) Therefore, \(A\) represents the rational number \(\frac{5}{3}.\)

Point \(Q\) represents \( – 5\) on the number line. Now, divide \(OQ\) into three equal parts \(OC,\;CD\) and \(DQ.\) The point \(C\) is such that \(OC\) is one-third of \(OQ.\) Since \(Q\) represents the number \(- 5,\) therefore, \(C\) represents the rational number \(\frac{{ – 5}}{3}.\)

Q.2. Represent \(\frac{{13}}{5}\) and \(\frac{{ – 13}}{5}\) on the number line.
Ans: Draw a line. Take a point \(O\) on it, let it represent \(0.\)
Now, \(\frac{{13}}{5} = 2\frac{3}{5} = 2 + \frac{3}{5}\) From \(O,\) set off unit distances \(OA,\;AB,\) and \(BC\) to the right of \(O.\) The points \(A,\;B\) and \(C\) represents the integers \(1,\;2\) and \(3,\) respectively. Now, take \(2\) units \(OA\) and \(AB\) and divide the third unit \(BC\) into \(5\) equal parts to reach a point \(P.\) Then the point \(P\) represents the rational number \(\frac{{13}}{5}.\)

Now, \(\frac{{ – 13}}{5} =  – \left( {2 + \frac{3}{5}} \right)\)

Take \(2\) full unit lengths to the left of \(O.\) Divide the third unit \(B’C\) into \(5\) equal parts. Take \(3\) parts out of these \(5\) parts to reach a point \(P’.\) Then, the point \(P’\) represents the rational number \(\frac{{ – 13}}{5}.\)

Q.3. Represent \(\frac{8}{5}\) and \(\frac{{ – 8}}{5}\) on the number line.
Ans: Draw a number line and mark a point \(O\) on it to represent \(0.\) Now, mark two points \(P\) and \(Q\) representing the integers \(8\) and \(-8\) respectively on the number line. Divide the line segment \(OP\) into five equal parts. Let \(A,\;B,\;C,\;D\) be the points of division so that \(OA = AB = BC = CD = DP.\) By construction, \(OA\) is one-fifth of \(OP.\) So, \(A\) represents the rational number \(\frac{8}{5}.\) Now, \(Q\) represents \(-8\) on the number line. Divide \(OQ\) into five equal parts \(OA’,A’B’,B’C’,C’D’\) and \(D’Q’.\) Since \(Q\) represents \( – 8.\) Therefore, \(A’\) represents the rational number \(\frac{{ – 8}}{5}.\)

Q.4. Draw the number line and represent \(\frac{3}{4}\) on it.
Ans: Draw a line segment. Take a point \(O\) on it. Let it represent \(0.\) Set off unit lengths \(OA\) to the right of \(0.\) Now, divide \(OA\) into \(4\) equal parts. Let \(OP\) be the segment showing \(3\) parts out of \(4.\) Then, the point \(P\) represents the rational number \(\frac{3}{4}.\)

Q.5. Draw the number line and represent \(\frac{{ – 7}}{4}\) on it.
Ans: Draw a line segment. Take a point \(O\) on it. Let it represent \(0.\) Set off unit lengths \(OA’\) to the left of \(0.\) Now, divide \(OA’\) into \(8\) equal parts. Let \(OP\) be the segment showing \(7\) parts out of \(4.\) Then, the point \(P\) represents the rational number \(\frac{{ – 7}}{4}.\)

Summary

The above article covered number line and rational numbers, positive and negative rational numbers, properties, and examples of rational numbers. Also, we discussed the rational number in standard form, even discussed the number line representation and the rational numbers on a number line with few solved examples of the rational number on a number line.

FAQs

Q.1. How do you show \(\frac{2}{7}\) on a number line?
Ans: Since the given rational fraction is positive and is a proper fraction, it will lie on the right side of zero on the number line and between \(0\) and \(1.\) To represent this, we will divide the number line between \(0\) and \(1\) into \(7\) equal parts, and the second part of the seven parts will be \(\frac{2}{7}.\)

Q.2. How do you represent rational numbers on a number line?
Ans: To represent rational numbers on the number line, we draw a line and mark a point \(O\) on it to represent the rational number zero. The positive rational numbers will be represented by points on the right side of \(O,\) and negative rational numbers will be represented by the points on the left side of \(O.\)

Q.3. Explain rational number on a number line with example.
Ans: To represent rational numbers on the number line, we draw a line and mark a point \(O\) on it to represent the rational number zero. The positive rational numbers will be represented by points on the right side of \(O\) and negative rational numbers will be represented by the points on the left side of \(O.\) If we want to represent the rational number \(\frac{{ – 1}}{3}\) on the number line, we divide the line segment \(OA’\) into three equal parts. Let \(Q,\) and \(R\) be the points dividing the segment \(OA’\) into three equal parts. Then, \(OQ = QR = RA’ = \frac{{ – 1}}{3}.\) Since \(O\) represents \(0\) and \(A’\) represents \( – 1.\)

Q.4. How do you represent \(\frac{{ – 5}}{7}\) on a number line?
Ans: Since the given rational number is negative and is a proper fraction, it will lie on the left side of zero on the number line and between \(0\) and \( – 1.\) To represent this, we will divide the number line between \(0\) and \( – 1\) into \(7\) equal parts, and the fifth part of the seven parts will be \(\frac{{ – 5}}{7}.\)

Q.5. How do you represent \(\frac{3}{4}\) on a number line?
Ans: Since the given rational number is positive and is a proper fraction, it will lie on the right side of zero on the number line and between \(0\) and \(1.\) To represent this, we will divide the number line between \(0\) and \(1\) into \(4\) equal parts, and the third part of the four parts will be \(\frac{3}{4}.\)

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