Tag: existence of irrational numbers

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Questions Related to existence of irrational numbers

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

State True or False.

$2+\sqrt{3}$ is an irrational number.

  1. True

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

$\ \sqrt { 3 } =1.73205080757......\ \sqrt { 3 } is\quad an\quad irrational\quad number,\quad since\quad it's\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ And\quad addition\quad of\quad a\quad rational\quad and\quad irrational\quad number\quad is\quad irrational.\ Hence,\quad 2+\sqrt { 3 } \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.

$\sqrt{3}+\sqrt{2}$ is an irrational number.

  1. True

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

$\ \sqrt { 3 } =1.73205080757......\ Also\quad \sqrt { 2 } =1.41421356237........\ \sqrt { 3 } and\quad \sqrt { 2 } are\quad irrational\quad numbers,\quad since\quad their\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ And\quad addition\quad of\quad two\quad irrational\quad numbers\quad is\quad irrational.\ Hence,\quad \sqrt { 2 } +\sqrt { 3 } \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.

$\sqrt{3}+\sqrt{5}$ is an irrational number.

  1. True

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

$\ \sqrt { 3 } =1.73205080757......\ Also\quad \sqrt { 5 } =2.2360679775........\ \sqrt { 3 } and\quad \sqrt { 5 } are\quad irrational\quad numbers,\quad since\quad their\quad decimal\quad representaion\quad is\quad non\quad terminating\quad non\quad repeating.\ And\quad addition\quad of\quad two\quad irrational\quad numbers\quad is\quad irrational.\ Hence,\quad \sqrt { 5 } +\sqrt { 3 } \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.

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

  1. True

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

$\ { (2-\sqrt { 2 } ) }(2+\sqrt { 2 } )=4-2=2\ \  { 2 } is\quad a\quad rational\quad number,\quad since\quad its\quad decimal\quad representaion\quad is\quad terminating.\ Hence,\quad { (2-\sqrt { 2 } ) }(2+\sqrt { 2 } )\quad is\quad a\quad rational\quad number.\ \quad $

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

State True or False.

$\sqrt{5}-2$ is an irrational number.

  1. True

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

$\ { (\sqrt { 5 } -2) }\ \sqrt { 5 } =2.2360679775........\ \ \sqrt { 5 } 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.\ Subtraction\quad of\quad rational\quad with\quad irrational\quad is\quad irrational.\ Hence,\quad { (\sqrt { 5 } -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.

$-\displaystyle\frac{2}{5}\sqrt{8}$ is an irrational number.

  1. True

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

$\ { \frac { -2 }{ 5 } \sqrt { 8 }  }={ \frac { -4 }{ 5 } \sqrt { 2 }  }=-0.8*\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 of\quad rational\quad with\quad irrational\quad is\quad irrational.\ Hence,\quad { (\frac { -2 }{ 5 } \sqrt { 8 } ) }\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
$\displaystyle\frac{(2+\sqrt{2})(3-\sqrt{5})}{(3+\sqrt{5})(2-\sqrt{2})}$ is Rational.

  1. True

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

$\displaystyle \frac { (2+\sqrt { 2 } )(3-\sqrt { 5 } ) }{ (3+\sqrt { 5 } )(2-\sqrt { 2 } ) } =\frac { { (2+\sqrt { 2 } ) }^{ 2 }{ (3-\sqrt { 5 } ) }^{ 2 } }{ (9-5)(4-2) } =\frac { (4+2+4\sqrt { 2 } )(9+5-6\sqrt { 5 } ) }{ 8 } \$


$\displaystyle =\frac { (6+4\sqrt { 2 } )(14-6\sqrt { 5 } ) }{ 8 } =\frac { (3+2\sqrt { 2 } )(7-3\sqrt { 5 } ) }{ 2 } =\frac { (21-9\sqrt { 5 } +14\sqrt { 2 } -6\sqrt { 10 } ) }{ 2 } \ The\quad above\quad given\quad expression\quad consists\quad of\quad an\quad algebraic\quad equation\quad in\quad numerator\quad \ consisting\quad of\quad irrational\quad terms,\quad hence\quad it\quad is\quad an\quad irrational\quad expression.\ $
The given statement is false.

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

State TRUE or FALSE 

${(2+\sqrt{3})}^{2}$ is Irrational

  1. True

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

${ (2+\sqrt { 3 } ) }^{ 2 }=4+3+4\sqrt { 3 } =7+4\sqrt { 3 } \ The\quad above\quad given\quad expression\quad consists\quad of\quad an\quad algebric\quad equation\quad \quad \ consisting\quad of\quad irrational\quad terms,\quad hence\quad it\quad is\quad an\quad irrational\quad expression.\ $