Nootan solutions for मैथमैटिक्स [अंग्रेजी] कक्षा ९ आईसीएसई chapter 1 - Rational and Irrational Numbers [Latest edition]

Insert one rational number between the following rational numbers:

3 and 8

Insert one rational number between the following rational numbers:

–2 and 5

Insert one rational number between the following rational numbers:

`1/4` and `3/5`

Insert one rational number between the following rational numbers:

`-2/5` and `-3/2`

Insert three rational numbers between the following rational numbers and write them in descending order:

`3/8` and `-1/4`

Insert three rational numbers between the following rational numbers and write them in descending order:

`6/11` and `8/5`

Insert five rational numbers between the following rational numbers:

`1/2` and `5/14`

Insert five rational numbers between the following rational numbers:

`3/4` and `17/8`

Insert two rational numbers between the following rational numbers:

`3/5` and `5/6`

Insert two rational numbers between the following rational numbers:

`-1/3` and `-1/4`

Insert four rational numbers between the following rational numbers:

`-2/5` and `1/3`

Insert four rational numbers between the following rational numbers:

`1/2` and `4/9`

Insert 9 rational numbers between the following rational numbers:

`3/10` and `7/5`

Insert 9 rational numbers between the following rational numbers:

`3/2` and 2

Prove that `sqrt(3)` is an irrational number.

Prove that `sqrt(5)` is an irrational number. 

Prove that `sqrt(11)` is an irrational number.

Prove that `1/sqrt(2)` is an irrational number.

Prove that `(5 + sqrt(3))` is an irrational number.

Prove that `6sqrt(2)` is an irrational number.

Prove that `3 + 2sqrt(5)` is an irrational number.

Prove that `4 - 3sqrt(5)` is an irrational number.

Prove that `sqrt(3) + sqrt(5)` is an irrational number.

Using suitable examples, show that the

1. sum of two irrational numbers may be rational.
2. difference of two irrational numbers may be rational.
3. product of two irrational numbers may be rational.
4. quotient of two irrational numbers may be rational.

Represent the following terminating decimal in the form of a rational number `bb(("in fraction form" p/q)`:

0.36

Represent the following terminating decimal in the form of a rational number `bb(("in fraction form" p/q)`:

0.875

Represent the following terminating decimal in the form of a rational number `bb(("in fraction form" p/q)`:

3.125

Represent the following terminating decimal in the form of a rational number `bb(("in fraction form" p/q)`:

42.75

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`21/50`

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`3/11`

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`2 1/5`

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`1/7`

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`12 5/6`

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`17/400`

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`3/8`

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`11/48`

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`7/200`

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`13/350`

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`19/3750`

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`47/225`

Write the decimal expansion of those numbers from the following that have terminating decimals.

`17/343`

Write the decimal expansion of those numbers from the following that have terminating decimals.

`11/4000`

Write the decimal expansion of those numbers from the following that have terminating decimals.

`37/400`

Write the decimal expansion of those numbers from the following that have terminating decimals.

`15/160`

Write the decimal expansion of those numbers from the following that have terminating decimals.

`77/175`

Write the decimal expansion of those numbers from the following that have terminating decimals.

`7/375`

Write the decimal expansion of `1/7`. Hence, write the decimal expansions of `2/7, 3/7, 4/7, 5/7` and `6/7`.

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

`0.bar(6)`

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

`0.bar(24)`

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

1.46

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

`0.2bar(73)`

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

`3.2bar(35)`

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

`11.bar(53)`

Express `1.26 + 0.bar(35)` in the form of a rational number `("in fraction form" p/q)`.

The following real numbers have decimal expansions as given below. State whether they are rational or irrational. If they are rational, express them in the form `p/q`, where p and q are co-prime integers and q ≠ 0 and then what can you say about the prime factors of q?

15.764

The following real numbers have decimal expansions as given below. State whether they are rational or irrational. If they are rational, express them in the form `p/q`, where p and q are co-prime integers and q ≠ 0 and then what can you say about the prime factors of q?

`18.bar(32)`

The following real numbers have decimal expansions as given below. State whether they are rational or irrational. If they are rational, express them in the form `p/q`, where p and q are co-prime integers and q ≠ 0 and then what can you say about the prime factors of q?

1.585585558...

The following real numbers have decimal expansions as given below. State whether they are rational or irrational. If they are rational, express them in the form `p/q`, where p and q are co-prime integers and q ≠ 0 and then what can you say about the prime factors of q?

1.636363

Insert an irrational number between the following:

4 and 5

Insert an irrational number between the following:

`1/3` and `1/2`

Insert an irrational number between the following:

0.3 and 0.5

Insert an irrational number between the following:

`-1/3` and `-4/5`

Insert two irrational numbers between the following: 

4 and 6

Insert two irrational numbers between the following: 

`1/2` and `13/4`

Find a rational number between the following:

`sqrt(2)` and `sqrt(3)`

Find a rational number between the following:

`sqrt(18)` and `sqrt(19)`

Find an irrational number between `sqrt(6)` and `sqrt(10)`.

Find an irrational number between 3 and `sqrt(15)`.

Find two irrational numbers between `4sqrt(2)` and `3sqrt(3)`.

Represent `sqrt(10)` on the number line.

Represent `sqrt(3)` on the number line.

Represent `sqrt(6)` on the number line.

State with reason, whether the following is a surd and which is not?

`sqrt(12)`

State with reason, whether the following is a surd and which is not?

`6sqrt(16)`

State with reason, whether the following is a surd and which is not?

`2sqrt(3) xx 6sqrt(27)`

State with reason, whether the following is a surd and which is not?

`root(3)(8) xx root(3)(32)`

State with reason, whether the following is a surd and which is not?

`3sqrt(2) - 7`

Simplify:

`(sqrt(2) + 1)^2`

Simplify:

`(sqrt(3) - sqrt(2))^2`

Simplify:

`(sqrt(7) - sqrt(3))(sqrt(7) + sqrt(3))`

Simplify:

`(4 + sqrt(3))(4 - sqrt(3))`

Simplify:

`(1 + sqrt(3))(sqrt(2) - 1)`

Simplify:

`(6 - sqrt(5))^2`

Show that `root(3)(2)` is an irrational number.

Show that `root(3)(5)` is an irrational number.

Show that `root(3)(4)` is an irrational number.

Show that `root(3)(12)` is an irrational number.

If `sqrt(2) = 1.414`, find the value of the following:

`sqrt(200) + sqrt(50) - 5sqrt(32)`

If `sqrt(2) = 1.414`, find the value of the following:

`3sqrt(72) + 6sqrt(128) - sqrt(288)`

If `sqrt(2) = 1.414, sqrt(3) = 1.732`, find the value of the following:

`5sqrt(12) + sqrt(98) - sqrt(108)`

If `sqrt(2) = 1.414, sqrt(3) = 1.732`, find the value of the following:

`3sqrt(243) + 6sqrt(72) - 3sqrt(8)`

Simplify the following:

`sqrt(7 + 4sqrt(3))`

Simplify the following:

`sqrt(3 - 2sqrt(2))`

Simplify the following:

`sqrt(10 + 2sqrt(21))`

Simplify the following:

`sqrt(7 - 2sqrt(10))`

Which is greater?

`3sqrt(5)` and `6sqrt(2)`

Which is greater?

`6sqrt(5)` and `5sqrt(6)`

Arrange in ascending order.

`2sqrt(7), 3sqrt(2), 4sqrt(3)`

Arrange in ascending order.

`7sqrt(2), 3sqrt(11), 5sqrt(3)`

Compare.

`root(3)(3)` and `root(4)(5)`

Compare.

`root(4)(7)` and `root(3)(6)`

Rationalise the denominator of the following: 

`1/sqrt(2)`

Rationalise the denominator of the following:

`sqrt(3)/sqrt(7)`

Rationalise the denominator of the following:

`3/(4 + sqrt(7))`

Rationalise the denominator of the following:

`1/(sqrt(6) - sqrt(5))`

Rationalise the denominator of the following:

`(sqrt(2) + 1)/(sqrt(2) - 1)`

Rationalise the denominator of the following:

`(3 + sqrt(2))/(3 - sqrt(2))`

Rationalise the denominator of the following:

`(7 - 4sqrt(3))/(7 + 4sqrt(3))`

Rationalise the denominator of the following:

`(3sqrt(5) - 4sqrt(2))/(3sqrt(5) + 4sqrt(2))`

Simplify the following:

`1/(sqrt(2) + 1) + 1/(sqrt(3) + sqrt(2)) + 3/(sqrt(6) + sqrt(3)) - 2/(sqrt(6) + 2)`

Simplify the following:

`1/(sqrt(2) + sqrt(3)) + 1/(sqrt(3) + sqrt(4)) + 1/(sqrt(5) + sqrt(6)) + 1/(sqrt(6) + sqrt(7))`

Rationalise the denominator of the following and hence find their value by using `sqrt(2) = 1.414` and `sqrt(3) = 1.732`:

`7/(3 - sqrt(2))`

Rationalise the denominator of the following and hence find their value by using `sqrt(2) = 1.414` and `sqrt(3) = 1.732`:

`13/(4 + sqrt(3))`

Rationalise the denominator of the following and hence find their value by using `sqrt(2) = 1.414` and `sqrt(3) = 1.732`:

`5/(3sqrt(3) - 4sqrt(2))`

If a and b are rational numbers, find the value of a and b:

`a + bsqrt(2) = (4 + 3sqrt(2))/(4 - 3sqrt(2))`

If a and b are rational numbers, find the value of a and b:

`a + bsqrt(6) = (2sqrt(3) - sqrt(2))/(2sqrt(3) + sqrt(2))`

If a and b are rational numbers, find the value of a and b:

`a - bsqrt(3) = (4sqrt(3) - 7)/(4sqrt(3) + 7)`

If a and b are rational numbers, find the value of a and b:

`a + bsqrt(5) = (4 + 2sqrt(5))/(sqrt(5) + 1) - (4 - 2sqrt(5))/(sqrt(5) - 1)`

If `x = sqrt(2) + 1`, find `1/x`.

If `x = sqrt(2) + 1`, find `x + 1/x`.

If `x = sqrt(2) + 1`, find `x - 1/x`.

If `x = sqrt(2) + 1`, find `x^2 + 1/x^2`

If `x = 2 + sqrt(3)`, find `1/x`.

If `x = 2 + sqrt(3)`, find `x + 1/x`.

If `x = 2 + sqrt(3)`, find `x - 1/x`.

If `x = 2 + sqrt(3)`, find `x^2 + 1/x^2`.

If `x = (3 + sqrt(2))/(3 - sqrt(2)), y = (3 - sqrt(2))/(3 + sqrt(2))`, find x².

If `x = (3 + sqrt(2))/(3 - sqrt(2)), y = (3 - sqrt(2))/(3 + sqrt(2))`, find y².

If `x = (3 + sqrt(2))/(3 - sqrt(2)), y = (3 - sqrt(2))/(3 + sqrt(2))`, find xy.

If `x = (3 + sqrt(2))/(3 - sqrt(2)), y = (3 - sqrt(2))/(3 + sqrt(2))`, find x² + y² + xy.

If `x = 7 + 4sqrt(3)`, find the value of `sqrt(x) - 1/sqrt(x)`.

Prove that `1/(sqrt(3) + 1` is an irrational number.

π is ______.

The rationalising factor of `sqrt(5)` is ______.

The decimal representation of an irrational number is ______.

Which of the following is an irrational number?

The rationalising factor of `sqrt(2) + 1` is ______.

A rational number between `sqrt(2)` and `sqrt(3)` is ______.

Which of the following numbers has non-terminating repeating decimal expansion?

The decimal expansion of the rational number `17/160` will terminate after ______ decimal places.

The decimal expansion of `15/32` is ______.

`(sqrt(3) + sqrt(5))(sqrt(5) - sqrt(3))` is equal to ______.

If `x = sqrt(2) + 1` then `x + 1/x` is equal to ______.

`sqrt(8) xx sqrt(64)` is equal to ______.

Which of the following is an irrational number?

The simplest form of the rational number `63/84` is ______.

`sqrt(32) + sqrt(50)` is equal to ______.

Every irrational number is ______.

Assertion: `sqrt(2)` is an irrational number.

Reason: The sum of a rational and an irrational number is irrational.

Assertion: `3/(sqrt(6) + sqrt(3)) = sqrt(6) - sqrt(3)`

Reason: The rationalising factor of `sqrt(6) + sqrt(3)` is `sqrt(6) - sqrt(3)`.

Assertion: `7/120` is terminating repeating decimal expansion.

Reason: π is an irrational number.

Assertion: 0.7 is a rational number between 0.5 and 0.9.

Reason: A rational number between a and b is `(a + b)/3`.

(i) `sqrt3` is a rational number.

(ii) The rationalising factor of `sqrt3 + 1` is `sqrt3 - 1`.

(i) `(sqrt(3) + 1)/(sqrt(3) - 1) = 2 + sqrt(3)`.

(ii) `22/7` is an irrational number.

(i) A rational number between `1/2` and `1/5` is `1/10`.

(ii) `(sqrt(7) + 1)/(sqrt(7) - 1)` can be written as `(sqrt(7) + 1)^2`.

(i) If m is prime then `sqrt(m)` is an irrational number.

(ii) There are infinitely many irrational numbers between any two rational numbers.