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Rational Numbers

Natural numbers (N): The counting numbers {1, 2, 3, …}, are called natural numbers. Some authors include 0, so that the natural numbers are {0, 1, 2, 3, …}.

Whole numbers(W): The numbers {0, 1, 2, 3, …}.

Integers (Z): Positive and negative counting numbers, as well as zero:{…, -2, -1, 0, 1, 2,…}.

Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

Real numbers (R): Numbers that have decimal representations that have a finite or infinite sequence of digits to the right of the decimal point. All rational numbers are real, but the converse is not true.

For all natural numbers m, n and p the following hold.

m + n = n + m                        (Commutative property of addition)

m + (n + p) = (m + n) + p     (Associative property of addition)

m x n = n x m               (Commutative property of multiplication)

m x (n x p) = (m x n) x p       (Associative property of multiplication)

m x (n + p) = (m x n) + (m x p)               (distributive property)

Every non-empty subset of natural numbers of N (or W) has the smallest element.

This is called the well ordering property of natural numbers.

We know, the set of all Positive and negative counting numbers, as well as zero:{…, -2, -1, 0, 1, 2,…}. is called whole numbers. Denoted by Z. If m and n are two whole numbers, with the extension of addition and multiplication, we have to following properties:

1. Closure property: for all integers a, b, both a + b and b + a also integers;
2. Commutative property: for all integers a, b

a + b = b + a

a x b = b x a

1. Associative property: for all integers a, b, c

a + (b + c) = (a + b) + c

a x (b x c) = (a x b) x c

1. Distributive property: for all integers a, b, c

a x (b + c) = (a x b) + (a x c)

1. Cancellation law: if a, b, c are integers such that, c ≠ 0 and ac = bc

Then a = b.

EXERCISE 1.3.1

1. Identify the property in the following statements:

(i) 2+(3+4)=(2+3)+4

Solution:

Associative property of addition

(ii) 2.8=8.2

Solution:

Commutative property of multiplication.

(iii) 8.(6+5)=(8.6)+(8.5)

Solution:

Distributive property

1. Find the additive inverses of the following integers:

(i) 6

Solution:

(-6) is the additive inverse of 6.

(ii) 9

Solution:

(-9) is the additive inverse of 9.

(iii) 123

Solution: (-123) is the additive inverse of 123.

(iv) -76

Solution:

76 is the additive inverse of -76.

(v) -85

Solution:

-85 is the additive inverse of 85.

(vi) 1000

Solution:

(-1000) is the additive inverse of 1000.

1. Find the integer m in the following:

(i) m + 6 = 8

Solution:

m = 8 – 6

Thus, m = 2

(ii) m + 25 = 15

Solution:

m = 15 – 25

Thus, m = -10

(iii) m – 40 = 26

Solution:

m = 26 – 40

Thus, m = + 14

(iv) m + 28 = -49

Solution:

m = -49 -28

Thus, m = -77

1. Write in the following in increasing order:

21,-8, 26, 85, 38, -333, -210, 0, 2011

Solution:

-333, -210, -26, -8, 0, 21, 33, 85, 2011

1. Write the following in decreasing order: 85, 210, -58, 2011, -1024, 528, 364, -10000, 12

Solution:

2011, 528, 364, 210, 85, 12, -58, -1024, -10000

1.3.2 Rational numbers

Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

Clearly, the numbers of the form p/q, where p and q are natural numbers.

Example 1: Add and multiply 1/3 and 8/5

Solution:

1/3 + 8/5 = (5+8×3)/15 = (5+24)/15  = 29/15

Their product is,

1/3 x 8/5 = (1×8)/15 = (8)/15  = 8/15

EXERCISE 1.3.2

1. Write down ten rational numbers which are equivalent to 𝟓/𝟕 and the denominator not exceeding 80.

Solution:

5/x  2/2 = 10/14

5/x  3/3 = 15/21

5/x  4/4 = 20/28

5/x  5/5 = 25/35

5/x  6/6 = 30/42

5/x  7/7 = 35/49

5/x  8/8 = 40/56

5/x  9/9 = 45/63

5/x  10/10 = 50/70

5/x  11/11 = 55/77

1. Write down 15 rational numbers which are equivalent to and the numerator not exceeding 180.

Solution:

11/x  2/2 = 22/10

11/x  3/3 = 33/15

11/x  4/4 = 44/20

11/x  5/5 = 55/25

11/x  6/6 = 66/30

11/x  7/7 = 77/35

11/5  x  8/8 = 88/40

11/x  9/9 = 99/45

11/x  10/10 = 110/50

11/x  11/11 = 121/55

11/x  12/12 = 132/60

11/x  13/13 = 143/65

11/x  14/14 = 154/70

11/x  15/15 = 165/75

11/x  16/16 = 176/80

1. Write down the ten positive numbers such that the sum of numerator and denominator of each is 11. Write them in decreasing order.

Solution:

Number:

10/1,9/2, 8/3, 7/4, 6/5,5/6, 4/7, 3/8,2/9, 1/10

Decreasing order:

10/1,9/2, 8/3, 7/4, 6/5,5/6, 4/7, 3/8,2/9, 1/10

1. Write down the ten positive numbers such that the numerator and denominator for each them is -2. Write them in increasing order.

Solution:

Increasing order:

1/3,2/4, 3/5 , 4/6, 5/7,6/8, 7/9, 8/10,9/11, 10/12

1. Is 3/(-2) a rational number? If so, how do you write it in a form conforming to the definition of a rational number (that is, the denominator as a positive integer)?

Solution:

3/(-2)  is a rational number. It should be written as -3/2  to be rational number.

1. Earlier you have studied decimals 0.9,0.8. Can you write these as rational numbers?

Solution:

Yes, we can write decimals like 0.9, 0.8 as rational numbers. i.e.,

0.9 = 9/10  and 0.8 = 8/10

1.3.3 Properties of rational numbers

Closure property:

We have learnt earlier, that, for all integers a, b, both a + b and b + a also integers;

Example1: Let us find the sum of  5/6 and 11/13

Solution:

5/6 + 11/13 = [(5×13)+(11×6)]/(6×13)

= (65+66)/78

= 131/78

Example 2: The product of  2/11 and 8/7 :

Solution:

2/11 + 8/7 = (2×8)/(11×7)

= 16/77

The set of all rational numbers is closed under addition and multiplication.

Associative property:

We have learnt earlier, that, the associative property is, for all integers a, b, c

a + (b + c) = (a + b) + c

a x (b x c) = (a x b) x c

Example 3: Consider three rational numbers 1/2 , 4/5 , -6/

Solution:

1/2 +( 4/5 + -6/7) = 1/2 +( (1×5+4×2)/(5×7))

= 1/2 + (-2/35 )

= (35×1+(-2)x2)/(35×2)

= 31/70

On the other hand,

(1/2 + 4/5) + (-6)/7 = (1×5+4×2)/(2×5) + (-6)/7

= (13)/(10) + (-6)/7

= [13×7+(-6)x10]/(10×7)

= 31/70

Similarly, we can find for multiplication.

Thus, we can come to the conclusion that,

Addition and multiplication are associative on the set of all rational numbers.

Commutative property

We have learnt earlier, that, the commutative property is, for all integers a, b

a + b = b + a

a x b = b x a

Example 4: Let us take two rational numbers, say 8/11 and (-16)/9

Solution:

8/11 + (-16)/9 = (8×9+(-16)x11)/99

= 72-176/99

= -104/99

On the other hand,

(-16)/9 + 8/11 = ((-16)x11+8×9)/99

= -176+72/99

= -104/99

Example 5: Similarly, we can verify that,

8/11 x (-16)/9 = (-16)/9 x 8/11

Addition and multiplication are commutative on the set of all rational numbers.

Distributive property

We have studied earlier, that, distributive property is, for all integers a, b, c

a x (b + c) = (a x b) + (a x c)

Consider the rational numbers, 3/2, 1/2 and 1/9. Observe that,

3/2 x ( 1/2 + 1/9) = 3/2 x ( 11/18 ) = 22/54 = 11/27

(3/2 x 1/2 ) + ( 3/2 x 1/9) = (2/6 ) + ( 2/27 ) = 11/27

In the set of all rational numbers, multiplication is distributive over addition.

Consider the rational number  0/1, observe that,

7/8 + 0/1 = (7×1+0x8)/8 = 7

0/1 + 7/8 = (0+7)/8 = 7/8

Thus, the rational number 0/1 acts as additive identity. We denote this by 0.

The set of all rational numbers has 0 as additive identity; that is r + 0 = 0 + r = r, for all rational numbers.

Multiplicative identity

Again consider, the rational number, 1/1. We have, for example,

11/12 x 1/1 = 11/12

1/1 x 11/12 = 11/12

Thus, the rational number 1/1  is identity with respect to multiplication.

The set of all rational numbers has 1 as multiplicative identity, that is r x 1 = 1 x r = r, for all rational numbers.

Take 8/13 and -8/13. If we add these two, we get

8/13 + (-8)/13 = (8-8)/13 = 0/13 = 0.

This is true for all rational numbers.

For each rational number r, there exists a rational number, denoted by – r, such that r + (-r) = 0 = (-r) + r.

Multiplicative inverse

We have studied earlier about multiplication inverse. Consider a rational number, 7/5 , we see that,

7/5 + 5/7 = 35/35 = 1

This is true for all rational numbers.

For each rational number r ≠ 0, there exists a rational number, denoted by r-1 (or r x r-1 = r-1 x r = 1

The only fundamental operations are addition and multiplication. The subtraction and division are defined in terms of addition and multiplication.

EXERCISE 1.3.3

1. Name the property indicated in the following:

(i) 315+115 = 430

Solution:

Closure property of addition

(ii) 3/4 + 9/5 = 27/20

Solution:

Closure property of multiplication

(iii) 5 + 0 = 0 + 5 = 5

Solution:

0 is the additive identity

(iv) 8/9 x 1 = 8/9

Solution:

1 is the multiplicative identity

(v) 8/17 + -8/17 = 0

Solution:

(vi) 22/23 + 22/23 = 1

Solution:

Multiplication inverse

1. Check the commutative property of addition for the following pairs:

(i) 102/201 , 3/4

Solution:

We know, commutative property, a + b = b + a

Therefore, 102/201 + 3/4 = 3/4 + 102/201

LHS,

102/201 + 3/4 = (102×4+3×201)/(201×4) = (408+603)/804 = 1011/804

RHS,

3/4 + 102/201 = (102×4+201×3)/(201×4) = 603+408/804 = 1011/804

Therefore, RHS = LHS, Commutative property proved.

(ii) -8/13 , 23/27

Solution:

We know, commutative property, a + b = b + a

Therefore, -8/13 + 23/27 = -8/13 + 23/27 LHS,

-8/13 + 23/27 = -8/13 + [(-8)x27+23×13]/(13×27) = -216+299/351 = 83/351

RHS,

-8/13 + 23/27 = [23×13+(-8)x27]/(13×27) = 299-216/351 = 83/351

Therefore, RHS = LHS, Commutative property proved.

(iii) -7/9 , -18/19

Solution:

We know, commutative property, a + b = b + a

Therefore,  -7/9 + -18/19 = (-18)/19 + (-7)/9

LHS,

-7/9 + -18/19 =  [(-7)x19+(-18)x9]/(9×19) =  -133-162/171 = -295/171

RHS,

(-18)/19 + (-7)/9  = [(-18)x9+(-7)x19]/(9×19) =  -162-133/171 = -295/171

Therefore, RHS = LHS, Commutative property proved.

1. Check the commutative property of multiplication for the following pairs:

(i) 22/45 , 3/4

Solution:

We know commutative property multiplication, a×b = b×a

Therefore, 22/45 x 3/4 = 3/4 x 22/45

LHS,

22/45 x 3/4 = (22×3)/(45×4) = 66/180

RHS,

3/4 x 22/45 = (22×3)/(45×4) = 66/180

Therefore, RHS = LHS, Commutative property proved.

(ii) -7/13 , 25/27

Solution:

We know commutative property multiplication, a×b = b×a

Therefore,  -7/13 x 25/27 = 25/27 x -7/13

LHS,

-7/13 x 25/27 = [25x(-7)]/27×13 = -175/351

RHS,

25/27 x -7/13 = [25x(-7)]/27×13 = -175/351

Therefore, RHS = LHS, Commutative property proved.

(iii) -8/9, 17/19

Solution:

We know commutative property multiplication, a×b = b×a

Therefore, -8/9 x 17/19 = 17/19 x (-8)/9

LHS,

-8/9 x 17/19 = (-8)x17/9×19 = 136/171

RHS,

17/19 x (-8)/9 = (-8)x17/9×19 = 136/171

Therefore, RHS = LHS, Commutative property proved

1. Check the distributive property for the following triples of rational numbers:

(i) 1/8 , 1/9 , 1/10

Solution:

We know distributive property, a (b + c) = ab + ac

Therefore, 1/8 x (1/9 + 1/10) = (1/8 x 1/9) + (1/8 x 1/10)

LHS,

1/8 x (1/9 + 1/10) = 1/8 x ((10+9)/(9×10))

= 1/8 x (19/90)

= (19/720)

RHS,

(1/8 x 1/9) + (1/8 x 1/10) = (1/9×8) + (1/10×8)

= (1/72) + (1/40 )

=(1+18/720) = 19/720

Therefore, RHS = LHS, distributive property proved.

(ii) -4/9 ,6/5 ,11/10

Solution:

We know distributive property, a (b + c) = ab + ac

Therefore, -4/9 x (6/5 + 11/10) = (-4/9 x 6/5 ) + (-4/9 x 11/10)

LHS,

-4/9 x (6/5 + 11/10) = -4/9 x (11+12/10×5) = -4/9 x (23/10) = (-92/90)

RHS,

(-4/9 x 6/5 ) + (-4/9 x 11/10)  = (-24/45 ) + (-44/90)= (-48-44)/90 = -92/90

Therefore, RHS = LHS, distributive property proved.

(iii) 3/8 ,0 , 13/7

Solution:

We know distributive property, a (b + c) = ab + ac

Therefore,  3/8 x (0 + 13/7) = (3/8 x 0) + (3/8 x13/7)

LHS,

3/8 x (0 + 13/7) = 3/8 x (13/7) = 13×3/7×8 = 39/56

RHS,

(3/8 x 0) + (3/8 x13/7) = 3/8 x (13/7) = 13×3/7×8 = 39/56

Therefore, RHS = LHS, distributive property proved.

1. Find the additive inverse of each of the following numbers:

8/5 , 6/10, -3/8 , -16/3, -4/1

Solution:

Additive inverse of 8/5 , 6/10, -3/8 , -16/3, -4/1 are -8/5 , -6/10, 3/8 , 16/3, 4/1 respectively.

1. Find the multiplicative inverse of each of the following numbers:

2 , 6/11, -8/15 , 19/18, 1/1000

Solution:

Multiplicative inverse of 2 , 6/11, -8/15 , 19/18, 1/1000 are   1/2, 11/6 , -15/8, 18/19 , 1000 respectively.

1.3.4 Representation of rational numbers on the Number.

Earlier, we have seen how to represent integers on a line. We choose an infinite line and fix some point on the line. This is denoted by 0. Fix a unit of length and on both sides of 0; go on marking points at equal unit distance. We can also use the same number line to represent rational numbers.

For example, Between any two distinct rational numbers, there is another rational number.

Exercise 1.3.4

1. Represent the following rational numbers on the number line:

-8/5 , 3/8 , 2/7, 12/5 , 45/13

Solution:

-8/5 3/8 2/7 12/5 45/13 1. Write the following rational numbers in ascending order:

3/4 ,   7/12 , 15/11 ,   22/19 , 101/100 ,   -4/5 , -102/81 ,   -13/7

Solution:

Ascending order

-13/7 , -102/81 ,   -4/57/12 , 3/4,  , 101/100,   22/19 , 15/11

Method:

We know,

3/4 ,   7/12 , 15/11 ,   22/19 , 101/100 ,   -4/5 , -102/81 ,   -13/7 is equal to 0.75, 0.5, 1.3, 1.1, 1.01, -0.3, -1.8, -1.2 respectively.

1. Write 5 rational number between 2/5 and 3/5  having the same denominators.

Solution:

We know,

2/5 = 0.4 and   3/5 = 0.6

Now, we have to find 5 rational numbers between 0.4 and 0.6. We get,

2/5 = 0.4 < 2.1/5 = 0.42 < 2.2/5 = 0.44 <   2.4/5 = 0.48 <   2.6/5 = 0.52 <2.8/5 = 0.56 < 3/5 = 0.6

Therefore, the rational numbers between 2/5 and   3/5    are 2.1/5  , 2.2/5 ,   2.4/5 , 2.6/5 and   2.8/5

1. How many positive rational numbers less than 1 are there such that the sum of the numerator and denominator does not exceed 10?

Solution:

1/2 ,1/3 , 1/4 ,1/5 , 1/6 ,1/7 , 1/8 ,1/9 ,2/3 ,2/5 ,2/7 ,3/4 , 3/7 ,3/5 , 1/2 ,4/5 ,

Therefore, only 15 positive rational number are possible such that they are less than 1 and the sum of the numerator and denominator does not exceed 10.

1. Suppose m/n and p/q are two positive rational numbers. Where does (m+n)/(n+q) lie, with respect to m/n and p/q?

Solution:

It is given that, m/n and p/ q are 2 numbers.

If m/n = p/q i.e., if m/n = p/q =1/2

Then,

(m+n)/(n+q) = (1+1)/(2+2) = 2/4 = 1/2

Therefore, if m/n = p/q then (m+n)/(n+q) = m/n = p/q

If  m/n and p/q are 2 distinct numbers. i.e., m/n = 1/5 and p/q = 1/3 then,

(m+n)/(n+q) = (1+1)/(5+3) = (2)/(8) = 1/4

Therefore, if m/n and p/q are 2 distinct numbers then (m+n)/(n+q)  lies between m/n and p/q

i.e, 1. How many rational numbers are there strictly between 0 and 1 such that the denominator of the rational number is 80?

Solution:

1/80 , 2/80 , 3/80 , 4/8078/80 , 79/80

Therefore, there are 79 positive rational numbers.

1. How many rational numbers are there strictly between 0 and 1 with the property that the sum of the numerator and denominator is 70?

Solution:

1/69 , 2/68 , 3/67 , 4/6668/2 , 69/1

Therefore, there are 69 rational numbers.