Tag: irrational numbers

Questions Related to irrational numbers

$\pi$ is _______

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. a rational number

  2. an integer

  3. an irrational number

  4. a whole number

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Correct Option: C
Explanation:
Sometimes we use $π = 22/7$ which is a popular approximation

$π = 3.14159265358...$

$22/7 = 3.142857142857...$

But $π$ and $22/7$ are close but not accurate.

Rational Numbers - $P/Q$ when $Q$ is not equal to $0$.

Let $x = 33.33333…. $——-(1)

$10x = 33.333333….. $——-(2)

Equation $(2) - (1)$

$9x = 30$

$x = 30/9$ which is in form of $P/Q$ and $x = 33.3333…$

The digit $‘3′$ is repeating itself and that’s why it can be written as $100/3.$

When it’s π, the value is $3.14159265358...$ The order of digits will not repeat itself in it but in $22/7 = 3.142857142857….$ you can see that $142857…$ is repeating itself that’s why $22/7$ is rational but $π$ is irrational.

So, option C is correct.

Which of the following number is irrational ?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\sqrt{16}-4$

  2. $(3-\sqrt{3}) (3+\sqrt{3})$

  3. $\sqrt{5}+3$

  4. $-\sqrt{25}$

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Correct Option: C
Explanation:

In the given options $\sqrt { 16 }$ and $\sqrt { 25 } $ are irrational numbers. Their real values are 4 and 5 respectively. So, option A and C are incorrect.

Option B can solved and its real value becomes 6. So it is also a rational number.
In option C, $\sqrt { 5 }$ is a irrational number. So, option C is a irrational number. 
So, correct answer is option C. 

Which one of the following is an irrational number ?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. 0.14

  2. 0.1416

  3. 0.14169452

  4. 0.4014001400014.....

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Correct Option: D
Explanation:

In the given options, only option D is non terminating non recurring decimal. 

So correct answer is option D.

A number is an irrational if and only if its decimal representation is :

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. non $-$ terminating

  2. non $-$ terminating and repeating

  3. non $-$ terminating and non $-$ repeating

  4. terminating

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Correct Option: C
Explanation:

Irrational numbers have decimal expansions that neither terminate nor repeating

So the correct answer is option C.

Which of the following is an irrational number ?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\sqrt{23}$

  2. $\sqrt{225}$

  3. $0.3796$

  4. $7.478$

Show answer
Correct Option: A
Explanation:

In the given options, 

$\sqrt { 225 }$ = 15. So, it is not a irrational number,
Option C and D are terminating decimals. So, they are also rational numbers.
$\sqrt{23}$ is a irrational number. 
So, option A is correct answer.  

$\pi$ is a(n) ________ while $\dfrac{22}{7}$ is rational.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Integer

  2. Whole Number

  3. Rational Number

  4. Irrational Number

Show answer
Correct Option: D
Explanation:

The value $\dfrac{22}7$ is a rational number, as it can be expressed in the form $\dfrac pq$. 

We consider it as an approximate value of $\pi$ because $\pi$ is close to $\dfrac{22}7$. 
But actually its value is $3.14159....$, which is neither terminating nor repeating. 
Thus, $\pi $ is irrational, but $\dfrac{22}7$ is rational.

$\sqrt{5}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$\sqrt5$ is irrational as it can never be expressed in the form a/b

Check whether following statement is true or false.
$7\sqrt{5}$ is a rational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

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Correct Option: B
Explanation:

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$7\sqrt5$ is irrational as it can never be expressed in the form a/b

$\dfrac{1}{\sqrt{2}}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$1/(\sqrt2)$ is irrational as it can never be expressed in the form a/b

$3+2\sqrt{5}$ a rational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Let's assume that $3+2\sqrt5$ is rational..... 

then 

$3+2\sqrt5 = p/q $

$\sqrt5 =( p-3q)/(2q) $ 

now take $p-3q$ to be P and $2q$ to be Q........where P and Q are integers 

which means, $\sqrt5= P/Q$...... 

But this contradicts the fact that $\sqrt5$ is rational 

So our assumption is wrong and $3+2\sqrt5$ is irrational.