1.2-Properties of Rational Numbers
1.2-Properties of Rational Numbers Important Formulae
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1.2-Properties of Rational Numbers
- Rational numbers are of the form $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $.
- Closure Property: Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero).
- Commutative Property:
- Addition: $ a + b = b + a $
- Multiplication: $ a \times b = b \times a $
- Associative Property:
- Addition: $ (a + b) + c = a + (b + c) $
- Multiplication: $ (a \times b) \times c = a \times (b \times c) $
- Existence of Identity:
- Addition: 0 is the additive identity ($ a + 0 = a $)
- Multiplication: 1 is the multiplicative identity ($ a \times 1 = a $)
- Existence of Inverses:
- Additive inverse: $ -a $
- Multiplicative inverse: $ \frac{1}{a} $ for non-zero $ a $
1.2 - Properties of Rational Numbers
Rational numbers are numbers that can be expressed in the form $ \frac{p}{q} $, where $ p $ and $ q $ are integers, and $ q \neq 0 $. These numbers have specific properties that are fundamental to their operations. The major properties of rational numbers include:
1. Closure Property
The set of rational numbers is closed under addition, subtraction, multiplication, and division (except division by zero). This means that when you perform any of these operations between two rational numbers, the result will always be a rational number.
Examples:
- Addition: $ \frac{1}{2} + \frac{3}{4} = \frac{5}{4} $
- Subtraction: $ \frac{5}{6} - \frac{1}{3} = \frac{1}{2} $
- Multiplication: $ \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} $
- Division: $ \frac{3}{4} \div \frac{2}{5} = \frac{15}{8} $
2. Commutative Property
The commutative property states that the order in which we add or multiply two rational numbers does not change the result.
- Addition: $ \frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b} $
- Multiplication: $ \frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b} $
3. Associative Property
The associative property states that the grouping of rational numbers does not affect the result of addition or multiplication.
- Addition: $ \left( \frac{a}{b} + \frac{c}{d} \right) + \frac{e}{f} = \frac{a}{b} + \left( \frac{c}{d} + \frac{e}{f} \right) $
- Multiplication: $ \left( \frac{a}{b} \times \frac{c}{d} \right) \times \frac{e}{f} = \frac{a}{b} \times \left( \frac{c}{d} \times \frac{e}{f} \right) $
4. Identity Property
There are identity elements for both addition and multiplication in the set of rational numbers:
- Addition Identity: The identity element for addition is 0. That is, for any rational number $ \frac{a}{b} $, we have: $ \frac{a}{b} + 0 = \frac{a}{b} $
- Multiplication Identity: The identity element for multiplication is 1. That is, for any rational number $ \frac{a}{b} $, we have: $ \frac{a}{b} \times 1 = \frac{a}{b} $
5. Inverse Property
The inverse property states that every rational number has an additive and a multiplicative inverse.
- Additive Inverse: The additive inverse of a rational number $ \frac{a}{b} $ is $ -\frac{a}{b} $. That is, $ \frac{a}{b} + \left( -\frac{a}{b} \right) = 0 $
- Multiplicative Inverse: The multiplicative inverse of a non-zero rational number $ \frac{a}{b} $ is $ \frac{b}{a} $. That is, $ \frac{a}{b} \times \frac{b}{a} = 1 $
6. Distributive Property
The distributive property states that multiplication distributes over addition and subtraction. In other words, for any rational numbers $ \frac{a}{b}, \frac{c}{d}, \frac{e}{f} $, we have:
- Distributive over addition: $ \frac{a}{b} \times \left( \frac{c}{d} + \frac{e}{f} \right) = \frac{a}{b} \times \frac{c}{d} + \frac{a}{b} \times \frac{e}{f} $
- Distributive over subtraction: $ \frac{a}{b} \times \left( \frac{c}{d} - \frac{e}{f} \right) = \frac{a}{b} \times \frac{c}{d} - \frac{a}{b} \times \frac{e}{f} $
7. Negative Rational Numbers
Rational numbers can be negative. The negative of a rational number is obtained by changing the sign of the numerator or denominator (but not both). For example, $ -\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b} $.
8. Reciprocal
The reciprocal of a rational number $ \frac{a}{b} $ is $ \frac{b}{a} $, provided that $ a \neq 0 $. The product of a rational number and its reciprocal is always 1, i.e., $ \frac{a}{b} \times \frac{b}{a} = 1 $.