prove that root 3 is an irrational number hence show that 7 + 2 root 3 is also an irrational number
Answers
Hey
√3 is irrational.
This can be proved by the method of contradiction. Let us assume that √3 is rational.
So we can find co-primes a and b such that √3 = a/b and b not equal to 0
that is b√3 = a
Squaring, (b√3)² = a²
or, a² = 3b² _________(1)
ie, a² is divisible by 3 or 3 divides a²
Therefore, 3 divides a also ie, 3 is a factor of a . Since 3 is a factor of a , we can find another integer c such that
a = 3c
Squaring, a² = (3c)² = 9c²
3b² = 9c² (from eq. (1) )
b² = 3c²
ie, b² is also a multiple of 3 or 3 divides b² and hence 3 divides b also
That is 3 is a common factor of a and b contradicting the fact that a and b are co-primes .
This contradiction arises due to the wrong assumption that √3 is rational. Hence √3 is irrational.
Now we have to prove that 7 + 2√3 is irrational. This can also be proved by the method of contradiction.
Let us assume that 7 + 2√3 is rational. Therefore we can find co-primes a and b such that 7 + 2√3 = a/b where b not equal to 0
2√3 = a/b - 7 = a-7b/b
√3 = a-7b/2b which is rational
ie , √3 is rational , which is a contradiction to the fact that √3 is irrational.
This contradiction arises due to the fact the wrong assumption that 7 + 2√3 is rational.
Hence it is proved that 7 + 2√3 is irrational.
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Hey
We know that √3 is irrational.This can be proved by the method of contradiction.
Let us assume that √3 is rational.
So we can find co-primes a and b such that √3 = a/b and b not equal to 0
that is = b√3 = a
Squaring, (b√3)² = a²
or, a² = 3b² _________(1)
ie, a² is divisible by 3 or 3 divides a²
Therefore, 3 divides a also ie, 3 is a factor of a . Since 3 is a factor of a , we can find another integer c such that
a = 3c
Squaring, a² = (3c)² = 9c²
3b² = 9c² (from eq. (1) )
b² = 3c²
ie, b² is also a multiple of 3 or 3 divides b² and hence 3 divides b also
That is 3 is a common factor of a and b contradicting the fact that a and b are co-primes .
This contradiction arises due to the wrong assumption that √3 is rational. Hence √3 is irrational.
Now we have to prove that 7 + 2√3 is irrational. This can also be proved by the method of contradiction.
Let us assume that 7 + 2√3 is rational. Therefore we can find co-primes a and b such that 7 + 2√3 = a/b where b not equal to 0
2√3 = a/b - 7 = a-7b/b
√3 = a-7b/2b which is rational
ie , √3 is rational , which is a contradiction to the fact that √3 is irrational.
This contradiction arises due to the fact the wrong assumption that
7 + 2√3 is rational.
Hence it is proved that 7 + 2√3 is irrational.
Hope it helps u , please mark it as brainliest..
