In mathematics, a rational number is a number which can be expressed as a fraction or a quotient, i.e., in the form of p/q where p and q are the two integers and ‘p’ is the numerator and ‘q’ is the non-zero denominator and p/q is in the lowest form, i.e. p and q have no common factors.
“Rational numbers (Q) are included in the real numbers, and in turn include the integers (Z), which include the natural numbers (N).” Thus, integers as well as fractions can be expressed in this form. So, all integers and fractions are rational numbers.
In p/q form q can be any number. So, q can also be equal to 1. When q = 1, every integer formed a rational number. Set of all rational number is denoted by ‘Q’.
Note: N ⊂ W ⊂ Z ⊂ Q
Some of the following are the examples of rational numbers:
(i) 2/3 is a rational number of form p/q where p = 2 and q = 3.
(ii) Similarly, 4/9 is a rational number of form p/q where p = 4 and q = 9.
(iii) 2 (i.e. 2/1) is a rational number of form p/q where p = 2 and q = 1. Also, 2 is a real number, a natural number, and an integer.
(iv) 0/5 (i.e. 0) is a rational number of form p/q where p = 0 and q = 5.
Arithmetical Operations on Rational numbers:
1. Addition of Rational Numbers:
(i) Addition of proper fractions:
1/2 + 3/2 = 4/2 = 2
4/9 + 5/9 = 9/9 = 1
(ii) Addition of improper fractions:
For addition of improper fractions we need to take LCM of the denominators and solve accordingly.
2/3 + 2/5
= 2 x 5/15 + 2 x 3/15
= 10/15 + 6/15
2. Subtraction of Rational Numbers:
(i) For proper fractions:
5/9 - 4/9 = 1/9
(ii) For improper rational fractions:
Step involved will same as in case of addition of improper rational numbers
2/5 – 4/15
= 2 x 3/5 x 3 – 4/15
= 6/15 – 4/15
3. Multiplication of Rational Numbers:
For multiplication of proper or improper rational fraction process is same for both. The process id that numerator of both is multiplied and is divided by the product of both denominators.
(i) 3/7 x 4/7 = 12/49
(ii) 4/9 x 7/5 = 28/45
4. Division of Rational Numbers:
(i) Divide 4/6 with 2.
In above example both numerator and denominator will be separately divided by 2. After dividing we get the answer 2/3.
(ii) Divide 2/3 by 3/4:
In such cases questions will be solved as-
2/3 × 4/3 = 8/9