Grade 1210 grade maths
Prove that 3 + 2√5 + is irrational Prove that 3 + 2√5 + is irrational
Prove that 3 + 2√5 + is irrational
Prove that 3 + 2√5 + is irrational
1 Answer
Harshit Singh
Dear Student
Let us assume 3 + 2√5is rational.
Then we can find co-prime x and y (y≠ 0) such that 3 + 2√5= x/y
Rearranging, we get,
2√5= x/y -3
√5= 1/2( x/y -3)
Since, x and y are integers, thus,
1/2( x/y -3)is rational number
Therefore,√5is also a rational number. But this contradicts the fact that√5is irrational.
So, we conclude that 3 + 2√5is irrational.
Thanks
Let us assume 3 + 2√5is rational.
Then we can find co-prime x and y (y≠ 0) such that 3 + 2√5= x/y
Rearranging, we get,
2√5= x/y -3
√5= 1/2( x/y -3)
Since, x and y are integers, thus,
1/2( x/y -3)is rational number
Therefore,√5is also a rational number. But this contradicts the fact that√5is irrational.
So, we conclude that 3 + 2√5is irrational.
Thanks
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