show that 7+2 under root3 is irrational no
YASHIKAJINDAL:
let 7+2under root 3 be an rational number . Then it must satisfy the condition of rational number that is 7+2 under root 3=p/q or 2 under root 3 = p/q-7
p-7q÷2q =under root 3 . Hence under root 3 is an irrational number and p-7q÷2q is an rational number . Irrational number is not equal to Rational .Thus our supposition is wrong that 7+2under root 3 is a rational number . Hence 7+2under root 3 is an irrational number
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Let, if possible, 7 + 2 under root 3 be a rational no. This will give rise to 2 unique integers a and b such that, 7+2 under root 3, where a and b are co primes and b is not equal to 0.
Now,
7+2 under root 3 = a/b
2 under root 3 = a+7b÷b
Under root 3 = a+7b÷2b
Here, as b, a, 7 and 2 are all integers, a+7b÷2b is real.
This implies that under root 3 is also a rational no.
However, we know that under root 3 is an irrational no.
This is a contradiction.
Therefore, our assumption was wrong.
Therefore, 7+2 under root 3 is an irrational no.
Hence proved.
Now,
7+2 under root 3 = a/b
2 under root 3 = a+7b÷b
Under root 3 = a+7b÷2b
Here, as b, a, 7 and 2 are all integers, a+7b÷2b is real.
This implies that under root 3 is also a rational no.
However, we know that under root 3 is an irrational no.
This is a contradiction.
Therefore, our assumption was wrong.
Therefore, 7+2 under root 3 is an irrational no.
Hence proved.
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