Question

prove that 7+√2 is an irrational number​

Answer 1

Answer:

Can you prove that 7+root 2 is an irrational number?

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This can be proven by contradiction. Assume that 7+2–√ is a rational number. Then by definition there must be integers m and n such that

7+2–√=mn

That would mean that

2–√=mn−7=m−7nn

which, in turn, would mean that 2–√ was rational. However, it’s well known that 2–√ is irrational (see , for instance, for many proofs of this), so our original assumption—that 7+2–√ is rational—was incorrect. ■

Note that this proof is easily generalized to establish the following result: the sum of a rational number and an irrational number is irrational.

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Let us assume that7+2–√is a rational number.

Then it can be written in the form of pq

for some integers p and q where q≠0

7+2–√=pq

⟹2–√=pq−7

⟹2–√=p−7qq

Right hand side is a rational number. Hence left hand side must be a

rational number. But we know that2–√is irrational.

∴Our assumption is wrong.

7+2–√can not be a rational number. Hence it is irrational.