Question

root 2 upon 7 prove it is irrational number

Answer 1

Step-by-step explanation:

The proof that p–√ is irrational if p is prime would be a good starting point. That proof goes like this:

Assume there exist some integers a and b that are co-prime (do not share factors) such that

p–√=ab

Then we have that b2p=a2.

This shows us that a2 is divisible by p and since p is prime that means a is divisible by p, as well.

From our assumption that a and b do not share factors, we can deduce that p does not divide b.

So, on the LHS of the equation we have a number that can only be divided by p once and on the RHS we have a number that can be divided by p at least twice. But they are said to be equal, which is a contradiction.

Therefore, p–√ is irrational.

Now we know that 7–√ is irrational because 7 is prime. All there is left to prove is that the quotient of an irrational number and rational number is irrational.

I’ll leave that as an exercise for the reader.