le.9.11 rove that V3 is irrational.

1.	le.9.11 rove that V3 is irrational.
Answer» Let us assume that √3 is a rational number. then, as we know a rational number should be in the form of p/q where p and q are co- prime number. So,√3 = p/q { where p and q are co-prime}√3q = p Now, by squaring both the side we get,(√3q)² = p²3q² = p² ........ ( i ) So,if 3 is the factor of p²then, 3 is also a factor of p ..... ( ii ) => Let p = 3m { where m is any integer } squaring both sidesp² = (3m)²p² = 9m² putting the value of p² in equation ( i )3q² = p² 3q² = 9m²q² = 3m² So,if 3 is factor of q²then, 3 is also factor of q Since, 3 is factor of p & q bothSo, our assumption that p & q are co-prime is wrong hence, √3 is an irrational number

Answer»

Let us assume that √3 is a rational number.

then, as we know a rational number should be in the form of p/q where p and q are co- prime number.

So,√3 = p/q { where p and q are co-prime}√3q = p

Now, by squaring both the side

we get,(√3q)² = p²3q² = p² ........ ( i )

So,if 3 is the factor of p²then, 3 is also a factor of p ..... ( ii )

=> Let p = 3m { where m is any integer }

squaring both sidesp² = (3m)²p² = 9m²

putting the value of p² in equation ( i )3q² = p² 3q² = 9m²q² = 3m²

So,if 3 is factor of q²then, 3 is also factor of q

Since, 3 is factor of p & q bothSo, our assumption that p & q are co-prime is wrong

hence, √3 is an irrational number

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