Operation on rational numbers:
- Addition of two rational numbers:
Case 1: When the denominators are the same.
If the denominators of the rational numbers are the same, add their numerators and retain the common denominator.
If the denominators of the rational numbers are the same, add their numerators and retain the common denominator.
That is, \(\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}\).
Case 2: When the denominators are different.
If the denominators are different, first convert the rational numbers into equivalent fractions with a common denominator.
To do this:
(i) Find the Least Common Multiple \(LCM\) of the denominators.
(ii) Multiply each rational number by the appropriate factor so that both fractions have the same denominator.
(iii) Once the denominators are made equal, add the numerators and keep the common denominator.
This gives the required sum of the rational numbers.
This gives the required sum of the rational numbers.
- Subtraction of two rational numbers:
Case 1: When the denominators are the same.
If the denominators of the rational numbers are the same, subtract their numerators and retain the common denominator.
If the denominators of the rational numbers are the same, subtract their numerators and retain the common denominator.
That is, \(\frac{a}{b} - \frac{c}{b} = \frac{a+c}{b}\).
Case 2: When the denominators are different.
If the denominators are different, first convert the rational numbers into equivalent fractions with a common denominator.
To do this:
(i) Find the Least Common Multiple \(LCM\) of the denominators.
(ii) Multiply each rational number by the appropriate factor so that both fractions have the same denominator.
(iii) Once the denominators are made equal, subtract the numerators and keep the common denominator.
This gives the required difference of the rational numbers.
This gives the required difference of the rational numbers.
- Multiplication of two rational numbers:
If \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers then, \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\), provided that \(b \neq 0\) and \(d \neq 0\).
- Division of two rational numbers:
If \(\frac{a}{b}\) and \(\frac{c}{d}\) are any two rational numbers then, \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\), provided that \(b \neq 0\), \(c \neq 0\) and \(d \neq 0\).
Properties of rational number:
1. Equality property:
Any two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) are said to be equal if \(a \times d = b \times c\).
2. Closure property:
If \(a\) and \(b\) are two rational numbers, then:
- \(a+b\) is also a rational number.
That is, rational numbers are closed under addition.
- \(a-b\) is also a rational number.
That is, rational numbers are closed under subtraction.
- \(a \times b\) is also a rational number.
That is, rational numbers are closed under multiplication.
- \(a \div b\) is also a rational number.
That is, rational numbers are closed under division, provided that \(b \neq 0\).
3. Commutative property:
If \(a\) and \(b\) are two rational numbers, then:
- \(a+b = b+a\)
- \(a \times b = b \times a\)
Important!
Commutativity is not always true for subtraction and division.
4. Distributive property:
Distributive property of multiplication over addition:
If \(a\), \(b\) and \(c\) are rational numbers, then:
- \(a \times (b + c) = (a \times b) + (a \times c)\)
Distributive property of multiplication over subtraction:
If \(a\), \(b\) and \(c\) are rational numbers, then:
- \(a \times (b - c) = (a \times b) - (a \times c)\)