Question 1: Compare the following pairs of rational numbers:

Answer:

(i) and

First take the LCM of and . LCM

Therefore: and

Hence, we see that or we can say that

(ii) and

First take the LCM of and . LCM

Therefore: and

Hence, we see that or we can say that

(iii) and

First take the LCM of and . LCM

Therefore: and

Hence we see that or we can say that

(iv) and

First take the LCM of and . LCM

Therefore: and

Hence we see that or we can say that

Question 2: Arrange in ascending order:

(i) (ii)

(iii) (iv)

Answer:

(i)

LCM of

The fractions can be written as

Therefore}, the order would be

or

(ii)

Note:

LCM of

The fractions can be written as

Therefore}, the order would be

or

(iii)

Note:

LCM of

The fractions can be written as

Therefore}, the order would be

or

(iv)

Note:

LCM of

The fractions can be written as

Therefore}, the order would be

or

Question 3: Represent each of these numbers on a Number Line:

(i) (ii) (iii) (iv) (v)

Answer:

(i) : Divide the unit length between and in equal parts and then mark

(ii) : Divide the unit length between and in equal parts and then mark

(iii) : Divide the unit length between and in equal parts and then mark

(iv) : Divide the unit length between and in equal parts and then mark

(v) : Divide the unit length between and in equal parts and then mark

Question 4: Find the additive inverse of:

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 5: Find the sum:

Answer:

Question 6: Subtract:

Answer:

Question 7: The sum of two rational numbers is . If one of them is then find the other.

Answer:

Question 8: What number should be added to to get

Answer:

Question 9: What number should be subtracted from to get

Answer:

Question 10: Find the products:

Answer:

Question 11: Find the quotient:

Answer:

Question 12: The product of two rational numbers is . If one of the number is , then find the other.

Answer:

Question 13: By what number must be divided to get ?

Answer:

Question 14: Find a rational number between each of the following pairs of rational numbers.

Answer:

First take the LCM of which is .

Convert the numbers with as the denominator.

Hence we get

First take the LCM of which is .

Convert the numbers with as the denominator.

Question 15: Find three rational numbers between:

Answer:

Question 16: Find 5 rational numbers between:

Answer:

The rational numbers are

The rational numbers are

Question 17: Determine whether the numbers are rational or irrational:

(i) (ii) (iii) (iv) (v)

(vi) (vii) (viii) (ix) (x)

(xi) (xii) (xiii) (xiv)

Answer:

Rational numbers are those numbers which can be expressed in the form of where are integers. So, 1 and 17 are respectively and they are integers too.! … Rational numbers are in the form of is not equal to zero.

(ii) : Rational

The decimal is a rational number. It is the decimal form of the fraction . Rational numbers are in the form of is not equal to zero

Rational numbers are those numbers which can be expressed in the form of where are integers. So, 11 and 13 are respectively and they are integers too.! … Rational numbers are in the form of is not equal to zero.

(iv)

can be written as, , where is any non-zero integer. Hence, 0 is a rational number.

(v) : Irrational

is non terminating recurring, so it is a rational number.

Every whole number is a rational number, because any whole number can be written as a fraction. For example, can be written as .

(vii) : Irrational

is an irrational number. Hence is irrational.

Proof: Lets first assume to be rational.

, where is not equal to .

Here, are co-primes whose HCF is .

( squaring both sides )…

… … … … … (i)

Here, divides also a ( because, if a prime number divides the square of a positive integer, then it divides the integer itself )

Now, let ( squaring both sides )…

… … … … … (ii)

Substituting Equation (i) in Equation (ii),

divides as well as .

Conclusion:

Here, are both divisible by . But our assumption that their HCF is is being contradicted.

Therefore, our assumption that is rational is wrong. Thus, it is irrational.

(viii) : Irrational

Proof: Let is a rational number.

Squaring both sides,

Irrational number.

.

(ix)

Rational numbers are those numbers which can be expressed in the

(x) : Irrational

The basic answer to this question is that is irrational because it represents the ratio of the circumference of a circle to its diameter and that ratio is irrational.

is irrational. It cannot be represented as a ratio of integers (with non zero denominator). is just an approximation of . It is used to simplify the problems and achieve a result as less deviated as possible.

(xi) Rational

(xii) : Rational

It is in a repeating decimal and hence can be represented in the form of

(xiii) : Irrational

.

All unresolved roots are known as an irrational number since is an unresolved root which is also known as surds, so it is an irrational number.

(xiv)

(xv)

Question / Answer 19: State whether True or False:

(i) Every real number is either rational or irrational: True

(ii) Every real number can be represented on a number line: True

(iii) There exists and integer which is not a rational number: False

(iv) There exist a point on a number line which do not represent any real number: False

(v) An infinite number of rational numbers can be inserted between any two rational numbers: True

(vi) The multiplicative inverse of any rational number a is 1/a : False

Question / Answer 20: Fill in the blanks

(i) 0 is a rational number that is its own additive inverse.

(ii) 0 is a rational number that does not have a multiplicative inverse.

(iii) 1 and -1 are two rational numbers that are equal to their own reciprocal.

(iv) The product of a rational number with its reciprocal is 1.

(v) The reciprocal of a negative number is negative.

(vii) Number of irrational numbers between any two rational numbers is infinite.

Question 21: Arrange in ascending order

(i) (ii)

Answer:

(i)

First take everything within under root sign. That way we can compare the numbers easily.

The numbers would then be

Now arrange in ascending order

Or

(ii)

First take everything within under root sign. That way we can compare the numbers easily.

The numbers would then be

Now arrange in ascending order

Or

Question 22: Write the rationalizing factors of the following:

(i) (ii) (iii)

(iv) (v) (vi)

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 23: Rationalize the denominator of each of the following:

Answer:

Question 24: Insert 5 rational numbers between:

(i) and (ii) and (iii)

Answer:

(i) and

(ii) and

First take everything within the root sign. So we need to find rational numbers between and

Hence the numbers are

(iii)

We can make the numbers as square root. So we need to find rational numbers between and

Hence the numbers are

Question & Answer 25: State True or False:

(i) : False

(ii) : True

(iii) : True

(iv) is a rational number: True (the value is which is a rational number)

(v) is irrational number: False (the value is which is a rational number)

(vi) is a rational number: True (the value is 6 which is a rational number)