Recommended
More Related Content
What's hot
What's hot (20)
Similar to Properties of Rational Numbers
Similar to Properties of Rational Numbers (20)
Recently uploaded
Recently uploaded (20)
Properties of Rational Numbers
- 4. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 + 𝑐 𝑑 is always a rational number. Example :- 2 3 + 1 4 = 11 12 ∴ rational numbers are closed under addition
- 6. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 + 𝑐 𝑑 = 𝑐 𝑑 + 𝑎 𝑏 . Example :- 2 3 + 1 4 = 1 4 + 2 3 ∴ Addition of rationa numbers is commutative.
- 8. If 𝑎 𝑏 , 𝑐 𝑑 𝑎𝑛𝑑 𝑒 𝑓 are rational numbers then ( 𝑎 𝑏 + 𝑐 𝑑 ) + 𝑒 𝑓 = 𝑐 𝑑 + ( 𝑎 𝑏 + 𝑒 𝑓 ). Example :- ( 2 3 + 1 4 ) + 5 6 = 2 3 + ( 1 4 + 5 ) ∴ Addition of ratio numbers is associative.
- 9. Additive Identity & Additive Inverse
- 10. Additive identity for rational number is 0 as 𝑎 𝑏 + 0 = 𝑎 𝑏 Example :- 2 3 + 0 = 2 3 ∴thus when we add 0 to rational number its answer is the number itself. Additive identity for rational number is 0 as 𝑎 𝑏 + 0 = 𝑎 𝑏 Example :- 2 3 + 0 = 2 3
- 11. Additive inverse for rational number is 𝑎 𝑏 +(- 𝑎 𝑏 )=0 Example :- 2 3 + (- 2 3 )=0 ∴thus for an rational number 𝑎 𝑏 there exists its opposite (- 𝑎 𝑏 ) such that there sum is zero, (- 𝑎 𝑏 ) is additive inverse
- 13. Closure Property
- 14. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 − 𝑐 𝑑 is always a rational number. Example :- 2 3 − 1 4 = 5 12 ∴ rational numbers are closed under subtraction.
- 16. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 − 𝑐 𝑑 ≠ 𝑐 𝑑 − 𝑎 𝑏 . Example :- 2 3 − 1 4 ≠ 1 4 − 2 3 ∴ Subtraction of rational numbers is not commutative.
- 18. If 𝑎 𝑏 , 𝑐 𝑑 𝑎𝑛𝑑 𝑒 𝑓 are rational numbers then ( 𝑎 𝑏 − 𝑐 𝑑 ) − 𝑒 𝑓 ≠ 𝑐 𝑑 − ( 𝑎 𝑏 − 𝑒 𝑓 ). Example :- ( 2 3 − 1 4 ) − 5 6 ≠ 2 3 − ( 1 4 − 5 6 ) ∴Subtraction of rational numbers is not associative.
- 20. Closure Property
- 21. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 × 𝑐 𝑑 is always a rational number. Example :- 2 3 × 1 4 = 1 6 ∴ rational numbers are closed under multiplication
- 23. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 × 𝑐 𝑑 = 𝑐 𝑑 × 𝑎 𝑏 . Example :- 2 3 × 1 4 = 1 4 × 2 3 ∴ Multiplication of rational numbers is commutative.
- 25. If 𝑎 𝑏 , 𝑐 𝑑 𝑎𝑛𝑑 𝑒 𝑓 are rational numbers then ( 𝑎 𝑏 × 𝑐 𝑑 ) × 𝑒 𝑓 = 𝑐 𝑑 × ( 𝑎 𝑏 × 𝑒 𝑓 ). Example :- ( 2 3 × 1 4 ) × 5 6 = 2 3 × ( 1 4 × 5 6 ) ∴ Multiplication of rational numbers is associative.
- 27. ∴thus when we multiply 1 to rational number its answer is the number Multiplicative identity for rational number is 1 as 𝑎 𝑏 × 1 = 𝑎 𝑏 Example :- 2 3 × 1 = 2 3
- 28. If 𝑎 𝑏 is a rational number then 𝑎 𝑏 × 𝑏 𝑎 =1 Example :- 2 3 × 3 2 =1 ∴thus for an rational number 𝑎 𝑏 there exists its opposite 𝑏 𝑎 such that there product is 1, 𝑏 𝑎 is additive inverse for 𝑎 .
- 30. Closure Property
- 31. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 ÷ 𝑐 𝑑 is always a rational number. Example :- 2 3 ÷ 1 4 = 8 3 ∴ rational numbers are closed under Division.
- 33. If 𝑎 𝑏 𝑎𝑛𝑑 𝑐 𝑑 are rational numbers then 𝑎 𝑏 ÷ 𝑐 𝑑 ≠ 𝑐 𝑑 − 𝑎 𝑏 . Example :- 2 3 ÷ 1 4 ≠ 1 4 ÷ 2 3 ∴ Division of rational numbers is not commutative.
- 35. If 𝑎 𝑏 , 𝑐 𝑑 𝑎𝑛𝑑 𝑒 𝑓 are rational numbers then ( 𝑎 𝑏 ÷ 𝑐 𝑑 ) ÷ 𝑒 𝑓 ≠ 𝑐 𝑑 ÷ ( 𝑎 𝑏 ÷ 𝑒 𝑓 ). Example :- ( 2 3 − 1 4 ) − 5 6 ≠ 2 3 − ( 1 4 − 5 6 ) ∴Division of rational numbers is not associative.
- 37. If 𝑎 𝑏 , 𝑐 𝑑 𝑎𝑛𝑑 𝑒 𝑓 are rational numbers then 𝑎 𝑏 × 𝑐 𝑑 + 𝑒 𝑓 = 𝑎 𝑏 × 𝑒 𝑓 + 𝑎 𝑏 × 𝑐 𝑑 𝑎 𝑏 × 𝑐 𝑑 − 𝑒 𝑓 = 𝑎 𝑏 × 𝑒 𝑓 − 𝑎 𝑏 × 𝑐 𝑑 Example :- 2 3 × 4 5 + 3 4 = 2 3 × 3 4 + 2 3 × 4 5