prove that 4 minus square root of 3 is an irrational number

 

Asked by koushalvishal109 | 28th Feb, 2019, 02:16: PM

Expert Answer:

If 4-√3 is a rational number, then we can write  4-√3 = a/b  , where a and b are integers.
 
4-√3 = a/b
 
4 = ( a/b) +√3
 
4b = a + √3b
 
By squaring both sides, 16b2 = ( a +√3b )2 = a2 + 3b2 + 2√3ab
 
(13b2 - a2)/(2ab)  = p/q  = √3
 
where p = (13b2 - a2) and q = 2ab  are integers.
 
But right side of above equation is a irrartional number  that can not be equal to p/q if p and q are integers
 
we get this contradiction due to the assumption that 4-√3 is a rational number.
 
Hence 4-√3 is an irrational number

Answered by Thiyagarajan K | 28th Feb, 2019, 03:03: PM

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