8 Math Rational Numbers

Finish Line & Beyond 1. Whole numbers are closed under addition and multiplication, but are not closed under subtraction and division. Let us see what does this mean. Example: 4+7=11 5+3=8 The addition of two or more whole numbers always result in a whole number that is why it is termed as closed. Similarly, multiplication of two or more whole numbers always results in a whole number. Example: 2X3=6, 5X7=35 On the other hand subtraction and division of two whole numbers may not always result in whole number, hence it is not a closed case. Example: 5-7= -2 is not a whole number 5 ÷ 7=

5 is not a whole number. 7

2. Rational numbers are closed under the operations of addition, subtraction and multiplication. As you know rational numbers can be written in the form

p , where p q

and q are integers and q ≠ 0. Rational numbers can either be positive or negative. Following examples illustrate how rational numbers are closed that is result in a rational number after operations of addition, subtraction and multiplication.

1 1 1 + = is a rational number. 2 2 1 3 1 1 − = Subtraction: is a rational number. 4 4 2 3 1 3 = Multiplication: × is a rational number. 4 4 16 Division: For any rational number a, a ÷ 0 is not defined, so this is not a closed case. Addition:

Numbers

Rational Numbers Integers Whole Numbers Natural Numbers

Closed Under Addition Subtraction Yes Yes Yes Yes Yes No Yes No

Multiplication Yes Yes Yes Yes

Division No No No No

2. The operations addition and multiplication are (i) commutative for rational numbers, which means that for any two rational numbers a and b, a+b=b+a and aXb=bXa Example:

3 1 1 3 + = + 4 4 4 4 3 1 1 3 × = × 4 4 4 4

www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond Numbers Rational Numbers Integers Whole Numbers Natural Numbers

Commutative for Yes No Yes No Yes No Yes No Yes No Yes No Yes No Yes No

(ii) associative for rational numbers, which means that for any three rational numbers a, b and c, a+(b+c) = (a+b)+c Example:

3 5+   − 2 3  3 + 5  + − 2 + 3

Numbers Rational Numbers Integers Whole Numbers Natural Numbers

 − 5  − 2 +   = 3  6   − 5 − 1  +    =  6  15 

 − 7  − 27 − 9 =   = 30 10  30  − 5  − 27 − 9 =  = 6  30 10

Associative for Addition Subtraction Yes No Yes No Yes No Yes Yes

Multiplication Yes Yes Yes Yes

Division No No No No

3. The rational number 0 is the additive identity for rational numbers. For given rational number a, a+0= a Zero is called the identity for the addition of rational numbers, integers and whole numbers. 4. The rational number 1 is the multiplicative identity for rational numbers. 5. The additive inverse of the rational number a and b is –a and –b and vice versa is also true. 6. The reciprocal or multiplicative inverse of the rational number is the numbers which result in 1 after multiplication with the number. Example:

a b × = 1, b a

7. Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac 8. Rational numbers can be represented on a number line. 9. Between any two given rational numbers there are countless rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.

www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond Exercise 1 1. Using appropriate properties find

2 3 5 3 1 × + − × 3 5 2 5 6 2 3 3 1 5 Answer: − × − × + 3 5 5 6 2 3 2 1 5 − + = − (distributive property) 5 3 6 2 3 5 5 = × − + 5 6 2 1 5 4 + = = 2 = − 2 2 2 2  3 1 3 1 2 + × (ii) ×  −  − × 5  7  6 2 14 5 2 3 1  1 3 + − × = − (distributive property) 5  7 14  6 2 2  − 6 + 1 1 − =  5  14  4 2 5 1 − = × − 5 14 4 − 1 1 − 4− 7 11 − = = − = 7 4 28 28 (i)

−

2. Write the additive inverse of following: (i)

2 8

Answer: (ii)

− 5 9

Answer: (iii)

− 6 − 5

Answer:

−

2 2 2 + − = 0 ; Proof: 8 8 8

5 − 5 5 + = 0 ; Proof: 9 9 9

−

6 6 6 + − = 0 ; Proof: 5 5 5

www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond 2 − 9

(iv)

2 2 2 + = 0 ; Proof: 9 − 9 9

Answer:

19 − 6

(v)

19 19 19 19 − 19 + = = 0 ; Proof: 6 − 6 6 − 6

Answer:

3. Verify that –(-x)=x for (i)

x=

11 15

Answer: -(-x) = (ii)

x= −

 11  11 − −  =  15  15

13 17

Answer: -(-x) =

  13   13  13  −  −  −   = −   = − 17  17    17  

4. Find the multiplicative inverse of the following: (i) -13 Answer: (ii)

1 13

−

9 13

13 9

−

Answer: (iii)

−

1 5

Answer: 5

5 − 3 × 8 7 5 3 15 = Answer: − × − 8 7 56 (iv)

−

So multiplicative inverse= (v)

− 1× −

Answer:

56 15

2 5 − 1× −

2 2 = 5 5

www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond So multiplicative inverse=

5 2

(Vi) -1 Answer: 1 5. Name the property under multiplication used in each of the following: (i)

− 4 − 4 4 × 1 = 1× = − 5 5 5

Answer: Here, 1 is the multiplicative identity. (ii)

−

13 − 2 − 2 − 13 × = × 17 7 7 17

Answer: Here commutativity of multiplication is shown. (iii)

− 19 29 × = 1 29 − 19

Answer: Here, multiplicative inverse is used.

6 7 by reciprocal of − 13 16 7 16 = − Answer: Reciprocal of − 16 7 6 16 96 × − = − Multiplication: 13 7 91 6. Multiply

7. What property allows you to compute

1  4 1  4 ×  6 ×  as  × 6  × 3  3 3  3 Answer: Here, associativity is being used.

8 1 the multiplicative inverse of − 1 ? Why or why not? 9 8 8 9 = − 1≠ 1 Answer: × − 9 8 8. Is

Hence, this is not a case of multiplicative inverse. 9. Is 0.3 the multiplicative inverse of Answer:

0.3 =

1 3 ? Why or why not? 3

3 1 10 and 3 = 10 3 3

3 10 × = 1 10 3 www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond Hence, this is a case of multiplicative inverse. 10. Write. (i) The rational number that does not have a reciprocal. Answer: 0 does not have a reciprocal. Because a number divided by 0 is undefined. (ii) The rational numbers that are equal to their reciprocals. Answer: 1 and -1 are equal to their reciprocals. (iii) The rational number that is equal to its negative. Answer: 0 is the number equal to its negative. 11. Fill in the blanks. (i) Zero has no reciprocal. (ii) The numbers 1 and -1 are their own reciprocals (iii) The reciprocal of – 5 is (iv) Reciprocal of

−

1 5

1 , where x ≠ 0 is x . x

(v) The product of two rational numbers is always a rational number (vi) The reciprocal of a positive rational number is positive Exercise 2 1. Represent these numbers on the number line: (i)

7 4

www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond

(ii)

−

5 6

Answer:

www.excellup.com ©2009 send your queries to [email protected]

Finish Line & Beyond 2. Represent

−

2 5 9 , − and − on the number line. 11 11 11

Answer:

3.Write five rational numbers which is smaller than 2. Answer: To write such type of numbers either put a number less than 2 in the numerator or greater than 2 in the denominator.

1 2 2 2 2 , , , , ,............................. 2 3 4 5 6 4. Find five rational numbers between

−

2 1 and 5 2

Answer: Keep on increasing denominator while writing such numbers

−

2 2 2 2 1 1 1 1 1 ,− ,− ,− ,0, , , , .............. 6 7 8 9 9 8 7 6 3

www.excellup.com ©2009 send your queries to [email protected]