Question

. Prove that 5 – √3 is irrational number ​

Answer 1

Answer:

Let us assume that 5 - √3 is a rational number in the form of p/ q where p and q are coprimes and q ≠ 0.

5 - √3 = p /q

Add √3 to both sides.

5 - √3 + √3 = p /q + √3

5 = p/ q + √3

Subtract both sides p/ q.

5 - p/ q = √3

(5q - p)/ q = √3

Since we already know that √3 is an irrational number.

Thus, a rational number can not be equal to an irrational number

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Answer 2

To prove :- 5 – √3 is irrational number

proof:-

Let us assume on the contrary that 5− √3 is

rational. Then, there exist prime positive integers a and b such that

5− √3= a/b

⇒ 5− a/b = √3

⇒ √3 is rational [∵a,b are integers∴

5b−a/b is a rational number]

This contradicts the fact that √3 is irrational.

So, our assumption is incorrect. Hence, 5−√3 is

Hence, 5−√3 is an irrational number.

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