Tag: existence of irrational numbers

$\ \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 $

$\ \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 $

$\ \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 $

$\sqrt { 7 } =2.64575131106...\ The\quad decimal\quad representation\quad is\quad non\quad repeating\quad non\quad terminating.\ Hence,\quad \sqrt { 7 } is\quad an\quad irrational\quad number.\ \quad $

$\ { (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 $

$\ { (\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 $

$\ { \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 $

$\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 } \$

${ (2+\sqrt { 3 } ) }({ 2-\sqrt { 3 } ) }=4-3=1\ Hnece\quad rational.\ $

${ (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.\ $