Find any 5 rational numbers between 2 7 and 4 7 .

A rational number between 2 7 and 4 7 is 1 2 2 7 + 4 7 = 3 7 Then, 2 7 < 3 7 < 4 7 A rational number between 2 7 and 3 7 = 1 2 2 7 + 3 7 = 5 14 A rational number between 3 7 and 4 7 = 1 2 3 7 + 4 7 = 7 14 = 1 2 A rational number between 2 7 and 5 14 = 1 2 2 7 + 5 14 = 9 28 A rational number between 3 7 and 1 2 = 1 2 3 7 + 1 2 = 13 28 Therefore, 2 7 < 9 28 < 5 14 < 3 7 < 13 28 < 1 2 < 4 7 Hence, 9 28 , 5 14 , 3 7 , 13 28 and 1 2 are the five rational numbers lying between 2 7 and 4 7 .

Find 3 rational numbers between 1 5 and 1 7 .

We need to find the three rational numbers between 1 5 and 1 7 . Take the LCM of the denominators LCM 5 , 7 = 35 Now multiply and divide both the numerator and denominator of the first rational number by 7 and the second rational number by 5 , we get 1 5 × 7 7 = 7 35 and 1 7 × 5 5 = 5 35 But now there is only a single rational number between them. So we should multiply them by any number which creates three or more rational numbers between them. Multiply both numerator and denominator by 2 . 7 35 × 2 2 = 14 70 and 5 35 × 2 2 = 10 70 Therefore, 14 70 > 13 70 > 12 70 > 11 70 > 10 70 Hence, 11 70 , 12 70 and 13 70 are the three rational numbers lying between 1 5 and 1 7 .

Find 11 rational numbers between - 1 3 and 1 2 .

We need to find 11 rational numbers between - 1 3 and 1 2 . Take the LCM of the denominators LCM 3 , 2 = 6 Now multiply and divide both the numerator and denominator of the first rational number by 2 and the second rational number by 3 , we get - 1 3 × 2 2 = - 2 6 and 1 2 × 3 3 = 3 6 But now there are four rational numbers between them. So we should multiply them by any number which creates three or more rational numbers between them. Multiply both numerator and denominator by 3 . - 2 6 × 3 3 = - 6 18 and 3 6 × 3 3 = 9 18 Therefore, - 6 18 < - 5 18 < - 4 18 < - 3 18 < - 2 18 < - 1 18 < 0 < 1 18 < 2 18 < 3 18 < 4 18 < 5 18 < 9 18 Hence, - 5 18 , - 4 18 , - 3 18 , - 2 18 , - 1 18 , 0 , 1 18 , 2 18 , 3 18 , 4 18 and 5 18 are the eleven rational numbers lying between - 1 3 and 1 2 .

Complete the following table. Numbers Commutativity under Addition Subtraction Multiplication Division Natural numbers Yes No Yes . . . Whole numbers . . . . . . . . . No Integers . . . . . . . . . . . . Rational numbers . . . . . . . . . No

Natural numbers belong to the set of Whole numbers, integers and rational numbers. So, the properties which hold for natural numbers are also hold for whole numbers, integers and rational numbers. The properties which do not hold for natural numbers are not applicable for whole numbers, integers and rational numbers. 1 Commutativity with respect to division: Consider two natural numbers 7 & 21 . 7 ÷ 21 = 7 21 = 1 3 and 21 ÷ 7 = 21 7 = 3 , i.e. 7 ÷ 21 ≠ 21 ÷ 7 ⇒ 7 & 21 are not commutative with respect to division. ⇒ Natural numbers are not commutative with respect to division. ⇒ Whole numbers and integers are not commutative with respect to division. 2 Commutativity with respect to addition. Consider two whole numbers 0 & 18 . We have 0 + 18 = 18 + 0 = 18 ⇒ 0 & 18 are commutative with respect to addition. ⇒ Whole numbers are commutative with respect to addition. ⇒ Integers and rational numbers are commutative with respect to addition. 3 Commutativity with respect to subtraction. We have 0 - 18 = - 18 and 18 - 0 = 18 , i.e. 0 - 18 ≠ 18 - 0 ⇒ 0 & 18 are not commutative with respect to subtraction. ⇒ Whole numbers are not commutative with respect to subtraction. ⇒ Integers and rational numbers are also not commutative with respect to subtraction. 4 Commutativity with respect to multiplication. Consider two whole numbers 14 & 18 .We have 14 × 18 = 252 and 18 × 14 = 252 . i.e. 14 × 18 = 18 × 14 = 252 ⇒ 14 & 18 are commutative with respect to multiplication. ⇒ Whole numbers are commutative with respect to multiplication. ⇒ Integers and rational numbers are commutative with respect to multiplication. Now, we can complete the given table as follows: Numbers Commutativity under Addition Subtraction Multiplication Division Natural numbers Yes No Yes No Whole numbers Yes No Yes No Integers Yes No Yes No Rational numbers Yes No Yes No Thus, we proved the commutative properties of natural numbers, whole numbers, integers and rational numbers.

Complete the given table. Numbers Associativity under Addition Subtraction Multiplication Division Natural numbers Yes No . . . . . . Whole numbers . . . . . . . . . No Integers . . . No Yes . . . Rational numbers . . . . . . . . . . . .

Rational numbers include natural numbers, whole numbers and integers. So, all the properties which hold for natural numbers hold for whole numbers, integers and rational numbers. 1 Associative property with respect to addition. Consider three whole numbers 8 , 10 & 24 . we have 8 + 10 + 24 = 18 + 24 = 42 and 8 + 10 + 24 = 8 + 34 = 42 . ⇒ 8 + 10 + 24 = 8 + 10 + 24 ⇒ 8 , 10 & 24 are associative with respect to addition. ⇒ Whole numbers and hence, integers and rational numbers are associative with respect to addition. 2 Associative property of numbers with respect to subtraction. 8 - 10 - 24 = - 2 - 24 = - 26 and 8 - 10 - 24 = 8 - - 14 = 22 ⇒ 8 - 10 - 24 ≠ 8 - 10 - 24 ⇒ 8 , 10 & 24 are not associative with respect to subtraction. ⇒ Whole numbers and hence, integers and rational numbers are not associative with respect to subtraction. 3 Associative property of numbers with respect to multiplication. 8 × 10 × 24 = 80 × 24 = 1920 and 8 × 10 × 24 = 8 × 240 = 1920 ⇒ 8 × 10 × 24 = 8 × 10 × 24 ⇒ 8 , 10 & 24 are associative with respect to multiplication. ⇒ Whole numbers, hence natural numbers, integers and rational numbers are associative with respect to multiplication. 4 Associative property with respect to division. 8 ÷ 10 ÷ 24 = 8 10 × 1 24 = 1 30 and 8 ÷ 10 ÷ 24 = 8 ÷ 10 24 = 8 × 24 10 = 96 5 ⇒ 8 ÷ 10 ÷ 24 ≠ 8 ÷ 10 ÷ 24 ⇒ 8 , 10 & 24 are not associative with respect to division. ⇒ Natural numbers, integers and rational numbers are not associative with respect to division. Now, we can complete the table as follows: Numbers Associativity under Addition Subtraction Multiplication Division Natural numbers Yes No Yes No Whole numbers Yes No Yes No Integers Yes No Yes No Rational numbers Yes No Yes No Thus, we checked the properties of the numbers and completed the given table.

E M B I B E

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q 1

EASY

9th Chhattisgarh Board

IMPORTANT

Earn 100

How many integers lie between $5$ and $15$ ?

Important Questions on Real Numbers (Rational and Irrational Numbers)

EASY

9th Chhattisgarh Board

IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q 2

How many integers lie between

- 3

and

8

.

MEDIUM

9th Chhattisgarh Board

IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q 1

Find any

5

rational numbers between

\frac{2}{7}

and

\frac{4}{7}

.

EASY

9th Chhattisgarh Board

IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q 2

Find $3$ rational numbers between $\frac{1}{5}$ and $\frac{1}{7}$ .

MEDIUM

9th Chhattisgarh Board

IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q 3

Find $11$ rational numbers between $\frac{- 1}{3}$ and $\frac{1}{2}$ .

EASY

9th Chhattisgarh Board

IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q (1)

If we remove zero from the set of integers, is it closed with respect to division? Similarly, checkout for whole numbers.

EASY

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IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q (1)

Complete the following table.

Numbers	Commutativity under
Numbers	Addition	Subtraction	Multiplication	Division
Natural numbers	Yes	No	Yes	$. . .$
Whole numbers	$. . .$	$. . .$	$. . .$	No
Integers	$. . .$	$. . .$	$. . .$	$. . .$
Rational numbers	$. . .$	$. . .$	$. . .$	No

EASY

9th Chhattisgarh Board

IMPORTANT

Mathematics Class - 9>Chapter 2 - Real Numbers (Rational and Irrational Numbers)>Try This>Q (1)

Complete the given table.

Numbers	Associativity under
Numbers	Addition	Subtraction	Multiplication	Division
Natural numbers	Yes	No	$. . .$	$. . .$
Whole numbers	$. . .$	$. . .$	$. . .$	No
Integers	$. . .$	No	Yes	$. . .$
Rational numbers	$. . .$	$. . .$	$. . .$	$. . .$