Tag: properties of irrational numbers

Questions Related to properties of irrational numbers

The value of  $\displaystyle \pi $ upto $50$ decimal places is $:\:314159265358979323846264338327950288419716939937510$
Which are the least occurring digits?
maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: A
Explanation:

We will be considering the digits only after the decimal point.


Digit after decimal point frequency
0 2
1 5
2 5
3 8
4 4
5 5
6 4
7 4
8 5
9 8
Total 50

The maximum occurring digits are 3 and 9. 
The least occurring digit is 0.

Which of the following is irrational?

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

  1. $\dfrac {22}{7}$

  2. $3.141592$

  3. $2.78181818$

  4. $0.123223222322223.......$

Show answer
Correct Option: D
Explanation:

An irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.
Among all the options only $(D)$ $0.123223222322223$...... is non terminating and non repeating decimal.Therefore, it is a irrational number.

$\sqrt 7$ is

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

  1. A rational number

  2. An irrational number

  3. Not a real number

  4. Terminating decimal

Show answer
Correct Option: B
Explanation:

Rational numbers are those numbers which can be expressed in the form $ \dfrac {p}{q} $, where p and q are integers and $ q \neq 0 $
Numbers which are not rational numbers are called irrational numbers.
Since, $ \sqrt {7} $ cannot be written in $ \dfrac {p}{q} $, where $p$ and $q$ are integers and $ q \neq 0 $; it is an irrational number.

State whether the following statement are true or false? Justify your answers.

Every irrational number is a real number.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Real number consists of collection of rationals and irrationals.

Hence, every irrational number is also a real number.

Example-2 is also real.

State whether the following statement are true or false? Justify your answers.

Every real number is an irrational 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:

The statement is false since real numbers consists of both rational and irrational numbers. $5,65,8/9...$ are all real numbers which are rational.

Classify the following numbers as rational or irrational : $2-\sqrt{5}$

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

  1. Irrational number

  2. Rational number

  3. Less Data

  4. None of the above

Show answer
Correct Option: A
Explanation:

$2$ is rational

$\sqrt 5 =2.035.........$ which is non terminating and non repeating hence irrational number.
We know that rational- irrational= irrational number.
Hence $2-\sqrt 5= irrational \,  number$
Hence, option A is the correct answer.

Decimal representation of an irrational number is always

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

  1. Terminating

  2. Terminating, Repeating

  3. Non-Terminating, Repeating

  4. Non-Terminating, Non-Repeating

Show answer
Correct Option: D
Explanation:

Decimal representation of an irrational number is always non terminating non repeating.

 For example,$\sqrt{2}$ $=1.41421356237309504880168872420969807856967187537694807317667973799...$

Are the square roots of all positive integers irrational?

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:

No. Not all square roots of integers are irrational. Examples are $\sqrt{4}=2$, $\sqrt{1}=1$, $\sqrt{9}=3$, etc.

Which among the following is true?

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

  1. There is no rational number between two irrational numbers.

  2. If ${x}^{2}=0.4$,then x is a rational number.

  3. The only real numbers are rational numbers.

  4. The reciprocal of an irrational number is irrational.

Show answer
Correct Option: D

The decimal expansion of the number $\sqrt{2}$ is 

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

  1. A finite decimal

  2. 1.4121

  3. Non-Terminating, Recurring

  4. Non-Terminating, Non-Recurring

Show answer
Correct Option: D
Explanation:

$\sqrt{2} = 1.4142136...$
The decimal expansion of the number $\sqrt{2}$ is Non-terminating non recurring