prove that root 3, is irrational
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Hey mate !!
Here is your solution
Let us suppose that root 3 is a rational number.
√3 = p/q , where p and q are co prime and q not equal to 0.
Squaring both the sides, we get ,
3 = p²/ q²
3q² = p² ........................( i )
=> p ² is divisible by 3.
So, p is also divisible by 3.
Let p = 3m for some integer m.
Substituting p = 3m in (i ) we, get,
3q² = (3m)² = 9m²
q² = 3m².
=> q² is divisible by 3.
So, q is also divisible by 3.
Since p and q both are divisible by 3, therefore 3 is a common factor of both p and q . But this contradicts our assumption that p and q are co prime .
This is bcoz of our contradictory assumption that √3 us a rational number.
Hence √3 is an irrational number.
Hope this will help you
Keep loving, keep smiling ☺☺
Here is your solution
Let us suppose that root 3 is a rational number.
√3 = p/q , where p and q are co prime and q not equal to 0.
Squaring both the sides, we get ,
3 = p²/ q²
3q² = p² ........................( i )
=> p ² is divisible by 3.
So, p is also divisible by 3.
Let p = 3m for some integer m.
Substituting p = 3m in (i ) we, get,
3q² = (3m)² = 9m²
q² = 3m².
=> q² is divisible by 3.
So, q is also divisible by 3.
Since p and q both are divisible by 3, therefore 3 is a common factor of both p and q . But this contradicts our assumption that p and q are co prime .
This is bcoz of our contradictory assumption that √3 us a rational number.
Hence √3 is an irrational number.
Hope this will help you
Keep loving, keep smiling ☺☺
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