Prove the sum of <msqrt>

Lucille Cummings

Lucille Cummings

Answered question

2022-06-24

Prove the sum of 3 + 4 3 is Irrational number

Answer & Explanation

Daniel Valdez

Daniel Valdez

Beginner2022-06-25Added 19 answers

Let x = 3 + 4 3 so 0 = ( x 3 ) 3 4 = x 3 3 x 2 3 + 9 x 3 3 4. If x Q , 3 = x 3 + 9 x 4 3 ( x 2 + 1 ) Q , a contradiction.
Dania Mueller

Dania Mueller

Beginner2022-06-26Added 6 answers

Every number of the form α = a + b 3 with a , b Q is a root of a quadratic polynomial because ( a + b 3 ) 2 = a 2 + 3 b 2 + 2 a b 3 and so α is a solution of
0 = ( X 3 4 ) ( X + 2 a ) ( X 2 2 a X + a 2 3 b 2 ) = ( ) X + ( )
and therefore rational - contradiction. (Verify that a least one of the "(…)" is non-zero!)

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