Rational numbers

Discover the basics about Rational Numbers

1. NAME : DEEPANSHA SINGH CLASS : VIII – A ROLL NO : ⑭ Rational Numbers

2. What Are Rational Numbers? All numbers including natural numbers, whole number and integers A number of the form p/q, where q is not equal to 0, is called a ‘Rational Number’. Here, p and q both are integers. For example, (-5)/6 , 11/5 , 2/15 , etc.

3. History Of Rational Numbers The history of rational numbers goes way back to the beginning of historical times. It is believed that knowledge of rational numbers precedes history but no evidence of this survives today. The earliest evidence is in the ancient Egyptian document ‘The Kahuna Papyrus’ .

4. Ancient Greeks also worked on rational numbers as a part of their number theory. Euclid elements date to around 300 BC.

5. Zero As Rational Number ‘0’ divided by any integer results in the rational number Zero. So, zero can be written in the form of p/q. Therefore, Zero is also a Rational Number. For example, 0/11 , 0/(-8) , 0/8 , 0/5 , etc.

6. Negative And Positive Rational Numbers If p and q both are positive, the rational number is positive. For example, 7/8 , 56/145 , 6/1259 , etc. If p and q both are negative, the rational number is positive. For example, (-56)/(-65) = 56/65 (as when we simplify the rational number the ‘minus’ signs get cut ).

7. If any of p or q is negative, the rational number is negative. For example, 98/(-5) , (-1)/2 , etc.

8. Properties Of Rational Numbers Properties of rational numbers lie under the four operations of arithmetic : Addition of rational numbers Subtraction of rational numbers Multiplication of rational numbers Division of rational numbers

9. Addition Of Rational Numbers Closure Property : Sum of two rational numbers is also a rational number. For example, 1/2 + 3/4 = 5/4 Commutative Property : The sum of two rational numbers does not depend on the order in which they are added. (a + b = b + a) For example, 1/2 + 3/4 = 3/4 + 1/2 = 4/6

10. Associative Property : The sum of three or more rational numbers does not depend on the way they are added. (a + b) + c = a + (b + c) Identity Property (Property of 0) : Zero added to any rational numbers the number does not change, so zero is called Identity Element for addition of rational numbers. (0 + a = a) Additive Inverse : If the sum of two rational numbers is 0 then the two numbers are called additive inverse of each other. For example, 2/3 + (-2)/3 = 0

11. Subtraction Of Rational Numbers Closure Property : The difference of two rational numbers is also a rational number. For example, 1/2 – 3/4 = (2-3)/4 = (- 1)/4 Commutative and Associative Property : Rational numbers do not hold commutative and associative property under subtraction.

12. Property of Zero : Zero subtracted from any rational number leaves it unchanged and any rational number subtracted from 0 gives its additive inverse. For example, 3/2 – 0 = 3/2 0 – 3/2 = (-3)/2

13. Multiplication Of Rational Numbers Closure Property : The product of two rational numbers is also a rational number. For example, 3/2 x 1/3 = 1/2 Commutative Property : Two rational numbers can be multiplied in any order. For example, 1/2 x 1/3 = 1/3 x 1/2 = 1/6

14. Associative Property : The product of three or more numbers does not depend on the order they are multiplied in or they are grouped as. Multiplicative Identity : The product of any rational number with one is the number itself, so 1 is called the multiplicative identity of rational numbers. Zero Property : The product of any rational number and 0 is 0. For example, 9874561253 x 0 = 0

15. Distributive Property of Multiplication over Addition and Subtraction : [a x (b + c) = a x b + a x c] [ a x (b - c) = a x b – a x c ]

16. Division Of Rational Numbers Closure Property : A rational number divided by a rational number may or may not be a rational number. Commutative and Associative Property : Rational numbers do not hold these properties under division.

17. Representation Of Numbers On The Number Line Numbers can be represented on the number line according to their types. The most ways of representing the numbers on the number line are as follows : Natural numbers Whole numbers Integers Rational numbers

18. Number Line Of Natural Numbers The line extends indefinitely only to the right side of 1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

19. Number Line Of Whole Numbers The line extends indefinitely to the right, but from 0. There are no numbers to the left of the 0. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

20. Number Line Of Integers The line extends indefinitely on both the sides. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

21. Number Line Of Rational Numbers The line extends indefinitely to both the sides. But you can now see numbers between -1, 0 ; 0, 1 ; etc. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 -1

22. Rational Numbers Between Two Rational Numbers Method I Method II Find the rational numbers between 1/4 and 1/2. 1/4 x 2/2=2/8 1/2 x 4/4=4/8 2/8<3/8<4/8 Find the rational numbers between 1/4 and 1/2. (1/4 + 1/2) ÷ 2 [(1+2)/4] ÷ 2 3/4 ÷ 2/1 3/8

23. Operation On Rational Numbers Rational numbers provide the first number system in which all the operations of arithmetic, addition, subtraction, multiplication and division are possible. Multiplication “makes a number bigger” and division “makes a number smaller”. The arithmetical operations are reduced to operations between two rational numbers .

24. Addition : It is the first operation. This operation uses only one sign [+]. Subtraction : It is the second operation. This operation uses only one sign [-]. Multiplication : it is often described as a sort of short hand for addition. This operation uses sign [x]. Division : It is the last and an important operation. The operation uses the sign [÷].

25. Adding Rational Numbers With Common Denominators To add rational numbers that have a common denominator, we add the numerators, but we do not add the denominators. For example,

26. Adding Rational Numbers With Different Denominators To add rational numbers with different denominators, first we equalize the denominators by enlarging each rational number by the “lowest common multiple” (LCM) as the denominator. Then we add the numerators.

27. Subtracting Rational Numbers With Common Denominators Subtraction is the inverse operation of addition. To subtract rational numbers that have a common denominator, we subtract the numerator, but we do not subtract the denominators. For example,

28. Subtracting Rational Numbers With Different Denominators To subtract rational numbers with different denominators, first equalize the denominators by enlarging each rational numbers by the “lowest common multiple” (LCM) as the denominator. Then subtract the numerators.

29. Multiplying Rational Numbers To multiply two rational numbers, we multiply the numerators to get the new numerator and multiply the denominators to get the new denominator. For example, x x x

30. Division In Rational Numbers To divide two rational numbers we take the reciprocal of the second rational number and multiply it by the first number: x

31. Application And Uses Rational numbers are important !!! They are used in the real world EVERYDAY !!! Even though we are not thinking about it if the number is rational or not, we still use them in our everyday lives. At school or in the kitchen. We even see them on TV !!!

32. For example, Baking: Ingredients in the recipes are often listed as fractions to show the measurements such as, a 1/2 cup of flour going into a batch of cookie dough. 1/2 is a rational number. Commercials: Many commercials use rational numbers as statistics to get you to buy their products such as, 4/5 dentists approve this toothpaste, or 9/10 women like this lipstick best.

33. Did You Know ???!!! Short Forms : Real Numbers – R Nos. Natural Numbers – N Nos. Whole Numbers – W Nos. Integers – Z Nos. Rational Numbers – Q Nos.

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