Show that √17 is an irrational number!
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Answers
Answer:
yes it is a irritational number
Step-by-step explanation:
Given :-
√17
To find :-
Show that √17 is an irrational number.
Solution :-
Given number = √17
Let us assume that √17 is a rational number
It must be in the form of p/q ,where p and q are integers and q≠0
Let √17 = a/b Where a and b are co-primes
On squaring both sides then
=>(√17)² = (a/b)²
=> 17 = a²/b²
=> 17b² = a²-------------(1)
=> b² = a²/17
17 divides a²
=> 17 divides a also
=> 17 is a factor of a -------------(2)
Put a = 17c in (1) then
=> 17b² = (17c²)
=> 17b² = 17×17×c²
=> b² = 17c²
=> 17c² = b²
=> c² = b²/17
17 divides b²
=> 17 divides b also
=> 17 is a factor of b -------------(2)
From (1)&(2)
17 is a common factor of a and b
But a and b are co primes which have only 1 is a common factor to them.
This contradicts to our assumption that is √17 is a rational number.
√17 is not a rational number.
√17 is an irrational number.
Hence, Proved.
Answer:-
√17 is an irrational number.
Used Method:-
Method of Contradiction (Indirect method)
Used formulae:-
Let a be any positive integer and P be a prime number,if P divides a² then p divides a also.