Rational Numbers – An introduction
A rational number is defined as a fraction of two numbers in the form of \[\dfrac{p}{q}\] where p and q can be any integer but q is not equal to 0. Rational numbers consist of all types of numbers including Natural numbers, Whole numbers, 0 etc. Integers consist of both positive and negative numbers. Rational can also be compared just to fractions and integers.
Comparison of Rational Numbers
Comparison of rational numbers means to determine that the given two or more rational numbers are equal, smaller and greater than each other. It is done in the same way as the fractions and integers are compared. Every positive rational number is always greater than zero and negative rational numbers and in the same way, negative rational numbers are always smaller than zero and positive rational numbers.
Rational numbers can be compared by numerators and denominators. Here, we only compare the values of numerators and denominators. There are two methods to compare rational numbers making either the numerator equal or making the denominator equal. Comparing rational numbers is easy; it is similar to fractions and integers as you have done these in previous classes.
How to Compare Rational Numbers
There are two methods to compare rational numbers, i.e., the Decimal Method and the Same Denominator Method.
Decimal Method
In this method of comparison of rational numbers, we convert the rational numbers into their decimal expansions. Then compare the decimal numbers to find out which is larger and which is smaller.
For example, Suppose we need to check the numbers \[\dfrac{3}{5}\] and \[\dfrac{7}{10}\].
We will find the decimal expansion of both the numbers:
\[\dfrac{3}{5}=0.6\]
and
\[\dfrac{7}{10}=0.7\]
Now, we will compare the decimals.
Clearly, the whole number part is the same in both numbers, so we will focus on the decimal part.
\[7>6\]
Hence,
\[\dfrac{7}{10}>\dfrac{3}{5}\]
Same Denominator Method
In this method of comparison, we check if the denominator of the given numbers is equal. If they have the same denominator, comparing just the numerator will give the result for comparing the numbers. If the denominators are different, then we use the HCF of both denominators and make the denominators the same using multiplicative factor \[\dfrac{x}{x}\] where \[x\] is the number which when multiplied to one of the denominator results in the HCF.
Example: Let us again take the same example of \[\dfrac{3}{5}\] and \[\dfrac{7}{10}\].
Here, the denominators in both fractions are not equal, thus we will find the HCF of the denominators.
\[HCF\left(5,10\right)=10\]
Now, we will convert the fractions so that they both have the same denominator, i.e., their HCF.
1st fraction:
\[\dfrac{3}{5}\times\dfrac{2}{2}=\dfrac{6}{10}\]
2nd fraction:
This fraction already has the same denominator, thus we move on to the next step, which is comparing the numerators.
\[7>6\]
Hence,
\[\dfrac{7}{10}>\dfrac{6}{10}\]
\[\Rightarrow\dfrac{7}{10}>\dfrac{3}{5}\]
Finding Rational Numbers Greater or Smaller than a Given Number
Finding rational numbers greater or smaller than a given number can be done using the same denominator method of comparison. To find the numbers greater than the given number, we can just simply increase or decrease, respectively, the number in the numerator, keeping the denominator the same.
Example: Find five rational numbers greater than and 5 rational numbers smaller than 2.
The number 2 can also be written as \[\dfrac{2}{1}\]. So, to write five rational numbers greater than 2 or \[\dfrac{2}{1}\], we will increase the numerator without changing the denominator. Thus, 5 rational numbers greater than \[\dfrac{2}{1}\] are:
\[\dfrac{3}{1},\dfrac{4}{1},\dfrac{5}{1},\dfrac{6}{1},\dfrac{7}{1}\]
Similarly, to find the rational numbers smaller than 2 or \[\dfrac{2}{1}\], we will decrease the numerator without changing the denominator.
Hence, five rational numbers less than 2 or \[\dfrac{2}{1}\] are:
\[\dfrac{1}{1},\dfrac{0}{1},\dfrac{-1}{1},\dfrac{-2}{1},\dfrac{-3}{1}\]
Facts About Rational Numbers
Rational numbers are expressed as \[\dfrac{p}{q}\] such that \[q \ne 0\] where p and q are coprime integers.
All fractions can be rational numbers but all rational numbers cannot be a fraction.
Rational numbers can also be shown on a number line.
Rational numbers are mainly of 3 types: Positive rational numbers, negative rational numbers and zero.
Whole numbers, integers, fractions and natural numbers are all included in rational numbers.
Uses of Rational Numbers
Rational numbers are used in many instances of daily life and Mathematics; some of them are given below:
Rational numbers are used to calculate simple interest and compound interest.
Rational numbers are used in finding the values of trigonometric values.
Used in sharing of anything like land, food, money etc.
Rational numbers are used to show the proportion of anything like one-third of milk.
Rational numbers are also used for measurements.
Interesting Facts
All fractions can be rational numbers but all rational numbers cannot be a fraction.
Rational numbers are only created for inadequate measurements and are created by the Greek mathematician Pythagoras.
Rational numbers can be extended up to infinity.
Every rational number falls in real numbers.
Solved Problems
Find the positive difference between the rational numbers \[\dfrac{5}{8}\] and \[\dfrac{2}{3}\].
Ans: To find the positive difference between the two numbers, we first need to check which number is larger. We can use the same denominator method to check the larger number.
\[HCF\left(8,3\right)=24\]
We will convert the denominator of both fractions to 24 using respective multiplicative factors.
\[\dfrac{5}{8}\times\dfrac{3}{3}=\dfrac{15}{24}\]
\[\dfrac{2}{3}\times\dfrac{8}{8}=\dfrac{16}{24}\]
Clearly, the fraction \[\dfrac{16}{24}\] is greater than the fraction \[\dfrac{15}{24}\]. Thus, \[\dfrac{2}{3}\] is larger than \[\dfrac{5}{8}\].
Now subtracting \[\dfrac{5}{8}\] from \[\dfrac{2}{3}\], we get
\[\dfrac{2}{3} - \dfrac{2}{3} = \dfrac{16}{24} - \dfrac{15}{24} = \dfrac{16-15}{24} = \dfrac{1}{24}\]
Hence, the positive difference between \[\dfrac{5}{8}\] and \[\dfrac{2}{3}\] is \[\dfrac{1}{24}\].
Find five rational numbers greater than \[-7\] but numbers should be consecutive.
Ans: Rational numbers greater than \[-7\] are as follows: \[-5,-2,0,4,2\], etc. as you can see these rational numbers are not consecutive.
Find out which of the two rational numbers is greater \[\dfrac{2}{3}\] or \[\dfrac{3}{4}\] using the decimal method?
Ans: Given rational numbers are \[\dfrac{2}{3}\], \[\dfrac{3}{4}\].
We will find the decimal expansions of both rational numbers,
\[\dfrac{2}{3} = 0.666…=0.{\overline{6}}\]
and
\[\dfrac{3}{4} = 0.75\]
Clearly, both decimal expansions have the same whole number, i.e., 0. So, it comes down to the decimals, i.e., 7 is greater than 6.
Thus, \[\dfrac{3}{4}\] is greater than \[\dfrac{2}{3}\].
Key Features
A rational number is the ratio of two integers, where the denominator of that ratio should not be zero.
Rational numbers can be compared just like fractions with the help of their numerators and denominators.
Natural numbers, whole numbers, integers and fractions with integers are rational numbers. And all these numbers fall in the category of real numbers.
Practice Questions
1. Find 5 rational numbers greater than \[\dfrac{3}{4}\].
Ans: \[1,2,\dfrac{5}{4},\dfrac{7}{4},4\]
2. Which rational number is smaller \[\dfrac{7}{18}\] and \[\dfrac{5}{9}\]?
Ans: \[\dfrac{5}{9}\]
FAQs on Comparison of Rational Numbers
1. Which is the easier method to compare rational numbers?
Both methods of comparing rational numbers have their own advantages and disadvantages. If the fraction is easier to convert into decimal, i.e., the denominator is a small prime number, thus the decimal method is easier in this case. Whereas if the denominator is a comparatively large number, we can use the same denominator method relatively easily.
2. Which rational number is the greatest between two rational numbers having the same numerators?
Since it is given that numerators are equal, therefore, the rational number which has the least denominator is the greatest rational number.
3. Which rational number is the smallest among all the rational numbers?
Rational numbers span from positive infinity to negative infinity, thus there is no smallest rational number. But if we only look at the value, then zero can be called the smallest rational number.
