prove that under root 5 minus under root 2 is irrational
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Arzoo12345:
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Hey there !!
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Let √5 – √2 be rational.
√5 – √2 = a / b where a and b are co-prime and (b≠0)
Squaring both sides,
(√5 – √2)² = ( a / b )²
=> 7 – 2√10 = a²/b²
=> 7 – (a²/b²) = 2√10
=> √10 = [ 7 – (a²/b²) ] / 2
Since a and b are integers so, [ 7 – (a²/b²) ] / 2 is a rational number.
So, √10 is also a rational number.
But this contradicts that the fact that √10 is irrational. Our consumption is false.
So, we conclude that √5 – √2 is an irrational number.
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____________________________________
Let √5 – √2 be rational.
√5 – √2 = a / b where a and b are co-prime and (b≠0)
Squaring both sides,
(√5 – √2)² = ( a / b )²
=> 7 – 2√10 = a²/b²
=> 7 – (a²/b²) = 2√10
=> √10 = [ 7 – (a²/b²) ] / 2
Since a and b are integers so, [ 7 – (a²/b²) ] / 2 is a rational number.
So, √10 is also a rational number.
But this contradicts that the fact that √10 is irrational. Our consumption is false.
So, we conclude that √5 – √2 is an irrational number.
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