Tag: existence of irrational numbers

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

$\sqrt 7$ is

  1. A rational number

  2. An irrational number

  3. Not a real number

  4. Terminating decimal
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

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

  1. Irrational number

  2. Rational number

  3. Less Data

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

Decimal representation of an irrational number is always

  1. Terminating

  2. Terminating, Repeating

  3. Non-Terminating, Repeating

  4. Non-Terminating, Non-Repeating
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

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

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

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

Which among the following is true?

  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.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Option D is correct because the reciprocal of an irrational number (like 1/sqrt(2)) is also irrational. Option A is false as there are infinitely many rational numbers between any two irrational numbers. Option B is false because x = sqrt(0.4) is irrational. Option C is false because irrational numbers are also real numbers.

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

State True or False.

${(\sqrt{2}-2)}^{2}$ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\ { (\sqrt { 2 } -2) }^{ 2 }=2+4-4\sqrt { 2 } =6-4\sqrt { 2 } \ \sqrt { 2 } =1.41421356237........\ \ \sqrt { 2 } is\quad an\quad irrational\quad number,\quad since\quad its\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ Multiplication\quad and\quad subtration\quad of\quad rational\quad with\quad irrational\quad is\quad irrational.\ Hence,\quad { (\sqrt { 2 } -2) }^{ 2 }\quad is\quad an\quad irrational\quad number.\ \quad $

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

State True or False.

(√2+√3)2(2√+3√)2 is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\ { (\sqrt { 2 } +\sqrt { 3 } ) }^{ 2 }=2+3+2\sqrt { 6 } =5+2\sqrt { 6 } \ \sqrt { 6 } =2.44948974278........\ \ \sqrt { 6 } is\quad an\quad irrational\quad number,\quad since\quad its\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ Multiplication\quad ans\quad addition\quad of\quad rational\quad with\quad irrational\quad is\quad irrational.\ Hence,\quad { (\sqrt { 2 } +\sqrt { 3 } ) }^{ 2 }\quad is\quad an\quad irrational\quad number.\ \quad $