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Rational Numbers Formula
Before learning rational numbers formulas, let us recall what are rational numbers. A rational number is a fraction whose numerator is an integer and the denominator is a nonzero integer. But all fractions are not rational numbers as a fraction may also have its numerator and/or denominator to be an irrational number(s). Let us learn the rational numbers formulas in detail in the next section. The set of rational numbers is denoted by 'Q' and it includes:
 The set of natural numbers, N
 The set of whole numbers, W
 The set of integers, W
 fractions of integers where the denominator is not zero.
What Is Rational Numbers Formula?
Using the definition of a rational number, which we discussed in the previous section, a rational number is of the form \(\dfrac p q\), where 'p' and 'q' are integers and q≠0. So some examples of rational numbers can be 2, 1, 3/2, 1/3, 0, etc. We operate the rational numbers in just the way as we operate the fractions. Thus, the rational numbers formulas are:

\(\mathbb{Q} = \left\{\dfrac{p}{q} \,\,:\,\, p,q \in \mathbb{Z}; \,\, q\neq 0\right\}\)

\(\dfrac{x}{y} \pm \dfrac{m}{n}=\dfrac{x n \pm y m}{y n}\)

\(\dfrac{x}{y} \times \dfrac{m}{n}=\dfrac{x m}{y n}\)

\(\dfrac{x}{y} \div \dfrac{m}{n}=\dfrac{x n}{y m}\)
Note: The set of rational numbers is closed, associative, and commutative under addition and multiplication. The additive identity, 0, and the multiplicative identity, 1 are present in the set of rational numbers. All rational numbers have their additive inverses in the set of rational numbers. All rational numbers other than 0 have their multiplicative inverses in the set of rational numbers.
Let us see the usage of the rational numbers formulas in the following solved examples.
Examples Using the Rational Numbers Formula
Example 1: Identify which of the following are rational numbers using the rational numbers formula: 2, \(\sqrt{2}\), \(\dfrac 1 2\), \(\dfrac 1 3\), and \(\dfrac {1}{\sqrt 2}\).
Solution:
The given numbers can be written as:
2 = \(\dfrac {2} 1\), here both 2 and 1 are integers where 1 \(\neq \) 0.
\(\sqrt 2 \) = \(\dfrac {\sqrt 2} 1\), but here \(\sqrt 2\) is NOT an integer.
\(\dfrac 1 2\), here both 1 and 2 are integers where 2 \(\neq \) 0.
\(\dfrac 1 3\) = \(\dfrac {1} 3\), here both 1 and 3 are integers where 3 \(\neq 0\).
\(\dfrac {1}{\sqrt 2}\), though it is a fraction, \(\sqrt 2\) is NOT an integer.
Thus, only 2, \(\dfrac 1 2\), and \(\dfrac 1 3\) are rational numbers among the given numbers.
Answer: 2, \(\dfrac 1 2\), and \(\dfrac 1 3\) are rational numbers.
Example 2: Find the sum, difference (in the given order), product, and the quotient (in the given order) of the following rational numbers: \(\dfrac 1 3\) and \(\dfrac 2 5\).
Solution: We will find the sum, difference, product, and the quotient using the rational numbers formulas.
\( \dfrac{1}{3}+ \dfrac{2}{5}= \dfrac{5}{15}+ \dfrac{6}{15} = \dfrac{11}{15}\)
\( \dfrac{1}{3} \dfrac{2}{5}= \dfrac{5}{15} \dfrac{6}{15} = \dfrac{1}{15}\)
\(\dfrac{1}{3}\times \dfrac{2}{5}= \dfrac{1 \times 2}{3 \times 5} = \dfrac{2}{15}\)
\(\dfrac{1}{3}\div \dfrac{2}{5}= \dfrac{1}{3} \times \dfrac{5}{2} = \dfrac{5}{6}\)
Answer: Sum = \( \dfrac{11}{15}\), Difference = \( \dfrac{1}{15}\), Product = \( \dfrac{2}{15}\), and Quotient = \( \dfrac{5}{6}\).
Example 3: Find five rational numbers between 1 and 2?
Solution: When any two integers, are expressed in the form of p/q, where q ≠ 0, it is known as a rational number.
Let's get the solution step by step.
Let's represent 1 and 2 as rational numbers:
1 can be written as 10/10 and 2 can be written as 20/10
So,
The rational numbers between 10/10 and 20/10 or 1 and 2 are {11/10, 12/10, 13/10, 14/10, 15/10, 16/10, 17/10, 18/10, 19/10}
Answer: Hence, any five rational numbers between 1 and 2 are 11/10, 12/10, 13/10, 14/10, and 15/10.
FAQs on Rational Numbers Formula
What Are Rational Numbers And Rational Number Formulas?
A rational number is a number that is in the form of p/q, where p and q are integers, and q is not equal to 0. Examples of rational numbers include 1/2, 2/7, 1/9, 9/3, and so on. The rational numbers formula applies to rational numbers. Rational numbers formulas are:

\(\mathbb{Q} = \left\{\dfrac{p}{q} \,\,:\,\, p,q \in \mathbb{Z}; \,\, q\neq 0\right\}\)

\(\dfrac{x}{y} \pm \dfrac{m}{n}=\dfrac{x n \pm y m}{y n}\)

\(\dfrac{x}{y} \times \dfrac{m}{n}=\dfrac{x m}{y n}\)

\(\dfrac{x}{y} \div \dfrac{m}{n}=\dfrac{x n}{y m}\)
What Is the Difference Between Rational and Irrational Numbers?
A rational number is a number that can be expressed as the ratio of two integers in the form p/q, where the denominator (q) should not be equal to zero. An irrational number cannot be expressed in the form of fractions. Rational numbers are terminating decimals whereas irrational numbers are nonterminating. An example of a rational number is 11/2, and Pi(π) which is equal to 3.141592653589…….
Check If 0 a Rational Number Using Rational Numbers Formula?
Yes, 0 is a rational number because it is an integer and it can be written in the form: p/q = 0/1. Hence, 0 is a rational number. The rational numbers formula applies to 0.
Find a Rational Number Between 1 and 2 Using Rational Numbers Formula.
Rational number between 1 and 2 = (1+2)/2
= 3/2
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