Comparison of Rational Numbers: Definition, Examples, and More

Comparison of Rational Numbers: A rational number is a number of the type \(\frac{p}{q}\) or a number that may be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q ≠ 0\). To put it another way, a rational number is any number that can be expressed as the quotient of two integers with the divisor not being zero. We have practised comparing two integers and two fractions. Every positive integer is more than zero, and every negative integer is smaller than zero. In addition, every positive integer exceeds every negative integer.

We have the following facts which show that the comparison of rational numbers is similar to the comparison of integers; i.e., every positive rational number greater than zero is called a positive rational number. Every negative rational number has a value that is less than zero.

Comparison of Rational Numbers Definition

A comparison of rational numbers is the same as a comparison of integers and fractions.
The following are some facts about rational number comparisons:

1. Every positive rational number exceeds zero.
2. Every rational number less than \(0\) is a negative rational number.
3. Every positive rational number is greater than every negative rational number.
4. Every rational number represented by a point on the number line is bigger than all rational numbers represented by points on the number line’s left.
5. Every rational number represented by a point on the number line is less than every rational number represented by a number line.

Example: Which is the greater of the two rational numbers \(\frac{3}{5}\) and \(\frac{-2}{3}\)?
Ans: \(\frac{3}{5}\) is a positive rational number, while \(\frac{-2}{3}\) is a negative rational number. As we all know, every positive rational number is greater than every negative rational number.
Therefore, \(\frac{3}{5} > \frac{-2}{3}\)

Comparison of Rational Numbers Examples

Example: Which is the greater of the two rational numbers \(\frac{-4}{9}\) and \(\frac{5}{-12}\)?
Solution: First, we use a positive denominator to write each of the given rational numbers.
Here, the denominator of the rational number \(\frac{-4}{9}\) is positive. The denominator of the rational number \(\frac{5}{-12}\) is negative.

We can use the steps below to compare any two rational numbers:
Step 1: Obtain the rational numbers that have been provided.
Step 2: Change the denominators of the specified rational numbers to positive.
Step 3: Calculate the LCM of the positive denominators of the rational numbers from step 2.
Step 4: As the common denominator, express each rational number (obtained in step 2) using the LCM (obtained in step 3).
Step 5: Compare the rational numbers’ numerators acquired in step 4.
The greater rational number is the one with the larger numerator.
The process will be demonstrated using the examples below.

As a result, we express it as follows with a positive denominator:
\(\frac{5}{{ – 12}} = \frac{{5 \times \left( { – 1} \right)}}{{\left( { – 12} \right) \times \left( { – 1} \right)}} = \frac{{ – 5}}{{12}}\)
Now, the LCM of denominators of the rational numbers \(9\) and \(12\) is \(36\).
The rational numbers are written as follows, with a common denominator of \(36\):
\(\frac{{ – 4}}{9} = \frac{{ – 4 \times 4}}{{9 \times 4}} = \frac{{ – 16}}{{36}}\) and \(\frac{{ – 5}}{{12}} = \frac{{ – 5 \times 3}}{{12 \times 3}} = \frac{{ – 15}}{{36}}\)
Therefore, \(- 15 > – 16 \Rightarrow \frac{{ – 15}}{{36}} > \frac{{ – 16}}{{36}}\)
\( \Rightarrow \frac{{ – 5}}{{12}} > \frac{{ – 4}}{9} \Rightarrow \frac{5}{{ – 12}} > \frac{{ – 4}}{9}\)

Comparison of Rational Numbers by Cross Multiplication

Rational numbers can be compared using the cross multiplication method.
Property of cross products
For comparing the two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\), we do \(a × d\) and \(c × b\).
When, \(a × d < c × b\), we say \(\frac{a}{b} < \frac{c}{d}.\)
Example: Compare \( – \frac{2}{5}\) and \(- \frac{3}{7}\)
To compare the given rational numbers by the cross multiplication method, cross multiply the denominator of the first rational number by the numerator of the second rational number and cross multiply the denominator of the second rational number by the numerator of the first rational number.
That is, \(\left( { – 2} \right) \times 7 > \left( { – 3} \right) \times 5\)
\(⟹ -14 > -15\)
Therefore, \(- \frac{2}{5} > – \frac{3}{7}.\)

Comparison of Negative Rational Numbers

We compare two negative rational numbers by disregarding their negative signs and then reversing the order of inequality (\(<\) or \(>\)).
For example, to compare \( – \frac{7}{5}\) and \( – \frac{5}{3}\), we first compare \(\frac{7}{5}\) and \(\frac{5}{3}.\)
We get \(\frac{7}{5} < \frac{5}{3}\) and conclude that \(\frac{-7}{5} > \frac{-5}{3}.\)

It is self-evident to compare a negative and a positive rational number. A negative rational number is to the left of zero on a number line, and a positive rational number is to the right of zero. As a result, a negative rational number is always less than a positive rational number.
Thus, \(- \frac{2}{7} < \frac{1}{2}\)

Comparison of Rational and Irrational Numbers

Rational numbers can be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q\) is not zero. The category of rational numbers includes decimal numbers that are ending and non-repeating. Irrational numbers, on the other hand, cannot be expressed in the \(\frac{p}{q}\) format since they are non-terminating and non-repeating decimals.

We can quickly compare rational numbers by comparing the numerators of rational numbers (in the case of rational numbers) or by taking LCM and comparing the numerators (unlike rational numbers).
We will learn about the comparison between rational and irrational numbers here.

Example: Compare \(3\) and \(\sqrt 5 \).
Solution: To compare the above numbers, we must get the square of both values before proceeding with the comparison.
Thus, \(3^2 = 9\) and \({\left( {\sqrt 5 } \right)^2} = 5.\)
Here, \(9\) is greater than \(5\).
Therefore, \(3 > \sqrt 5.\)

Solved Examples – Comparison of Rational Numbers

Q.1. Arrange the rational numbers \(\frac{{ – 7}}{{10}},\frac{{ – 5}}{8},\frac{{ – 2}}{3}\) in ascending order:
Ans: We begin by writing the specified rational integers with positive denominators.
We have
\(\frac{5}{{ – 8}} = \frac{{5 \times ( – 1)}}{{ – 8 \times ( – 1)}} = \frac{{ – 5}}{8}\) and \(\frac{2}{{ – 3}} = \frac{{2 \times ( – 1)}}{{( – 3) \times ( – 1)}} = \frac{{ – 2}}{3}\)
Thus, the LCM of the denominators \(10,\,8\) and \(3\) is \(2 × 2 × 5 × 2 × 3 = 120\)
Now we will write the numbers in such a way that they all have the same denominator \(120\) as follows:
\(\frac{{ – 7}}{{10}} = \frac{{ – 7 \times 12}}{{10 \times 12}} = \frac{{ – 84}}{{120}}\), \(\frac{{ – 5}}{8} = \frac{{ – 5 \times 15}}{{8 \times 15}} = \frac{{ – 75}}{{120}}\) and \(\frac{{ – 2}}{3} = \frac{{ – 2 \times 40}}{{3 \times 40}} = \frac{{ – 80}}{{120}}\)
Comparing the numerators of these rational numbers, we get
\(-84 < -80 < -75\)
Therefore, \(\frac{{ – 84}}{{120}} < \frac{{ – 80}}{{120}} < \frac{{ – 75}}{{120}}\)
\( \Rightarrow \frac{{ – 7}}{{10}} < \frac{{ – 2}}{3} < \frac{{ – 5}}{8}\)

Q.2. Which of the two rational numbers \(\frac{5}{7}\) and \(\frac{3}{5}\) is greater?
Ans: Clearly, the denominators of the given rational numbers are positive. The denominators are \(7\) and \(5\). The LCM of \(7\) and \(5\) is \(35\). So, we first express each rational number with \(35\) as a common denominator.
Therefore, \(\frac{5}{7} = \frac{{5 \times 5}}{{7 \times 5}} = \frac{{25}}{{35}}\) and \(\frac{3}{5} = \frac{{3 \times 7}}{{5 \times 7}} = \frac{{21}}{{35}}.\)
Now, we compare the numerators of these obtained rational numbers.
That is, \(25 > 21 \Rightarrow \frac{{25}}{{35}} > \frac{{21}}{{35}}\)
Therefore, \(\frac{{5}}{{7}} > \frac{{3}}{{5}}.\)

Q.3. Compare \(\frac{-6}{13}\) and \(\frac{1}{13}.\)
Ans: We have the rational numbers \(\frac{-6}{13}\) and \(\frac{1}{13}.\)
We know that every positive rational number is greater than every negative rational number.
Here, the rational number \(\frac{-6}{13}\) has a negative numerator and the rational number \(\frac{1}{13}\) has a positive numerator.
Therefore, \(\frac{-6}{13} < \frac{1}{13}.\)

Q.4. Compare the following rational numbers \(\frac{-3}{8}\) and \(0.\)
Ans: We have the rational numbers \(\frac{-3}{8}\) and \(0\).
We know that every negative rational number is always less than \(0\).
Here, the rational number \(\frac{-3}{8}\) has a negative numerator.
Therefore, \(\frac{-6}{13} < 0.\)

Q.5. Compare the following rational numbers \(\frac{5}{2}\) and \(0\).
Ans: We have the rational numbers \(\frac{5}{2}\) and \(0\).
We know that every positive rational number is always greater than \(0\).
Here, the rational number \(\frac{5}{2}\) is a positive rational number.
Therefore, \(\frac{5}{2} > 0\).

FAQs on Comparison of Rational Numbers

Below are some of the most frequently asked questions related to the Comparison of Rational Numbers

Q.1. How to compare rational numbers?
Ans: Comparison of rational numbers is the same as comparison of integers and fractions.
The following are some facts about rational number comparisons.
1. Every positive rational number exceeds zero.
2. Every rational number less than \(0\) is a negative rational number.
3. Every positive rational number is greater than every negative rational number.
4. Every rational number represented by a point on the number line is bigger than all rational numbers represented by points on the number line’s left.
5. Every rational number represented by a point on the number line is less than every rational number represented by a number line.

Q.2. What are the three ways in comparing and ordering rational numbers?
Ans: The three ways to compare rational numbers are:
1. Cross-multiply the rational numbers and compare their products.
2. Convert both rational numbers to rational numbers with a common denominator and compare.
3. Divide the two rational numbers to get decimals and compare the decimals.

Q.3. How do you compare two rational numbers by cross multiplication?
Ans: For comparing the two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\), we do \(a × d\) and \(c × b\).
When, \(a × d < c × b\), we say \(\frac{a}{b} < \frac{c}{d}.\)
For example, for comparing \(\frac{-2}{5}\) and \(\frac{-3}{7}\) by cross multiplication method, we cross multiply the denominator of the first rational number by the numerator of the second rational number and cross multiply the second rational number denominator by the numerator of the first rational number. After cross multiplying, compare the products.

Q.4. What is comparing and ordering rational numbers?
Ans: To compare and sort rational numbers, first convert them to the same form to be compared more easily. It is usually simpler to convert each number to a decimal. The numbers can then be ordered using a number line. To arrange rational numbers from greatest to least, or least to greatest, inequality signs must be used.

Q.5. How do you compare two rational numbers?
Ans: We can use the steps below to compare any two rational numbers:
Step 1: Obtain the rational numbers that have been provided.
Step 2: Change the denominators of the specified rational numbers to positive.
Step 3: Calculate the LCM of the positive denominators of the rational numbers from step 2.
Step 4: As the common denominator, express each rational number (obtained in step 2) using the LCM (obtained in step 3).
Step 5: Compare the rational numbers’ numerators acquired in step 4.
The greater rational number is the one with the larger numerator.

Summary

In this article, we learnt about the definition of comparison of rational numbers, examples of comparison of rational numbers, comparison of rational numbers by cross multiplication, comparison of negative rational numbers, comparison of rational and irrational numbers, solved examples of comparison of rational numbers and FAQs on comparison of rational numbers.
The article's learning outcome is how to compare positive rational numbers, negative rational numbers, positive and negative rational numbers, and rational numbers with zero.

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