Prove that 7-√3 is irrational. - Mathematics

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Prove that 7-`sqrt3` is irrational.

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Method I:

Let `7-sqrt3` is rational number

∴ `7-sqrt3=p/q`   (p, q are integers, q ≠ 0)

∴ `7 -p/q=sqrt3`

⇒ `sqrt3=(7q-p)/q`

Here p,q are integers.

∴ `(7q-p)/q` is also integer.

∴ LHS =`sqrt3` is also integer but this sqrt3 is contradiction that is irrational so our assumption is wrong that  `7-sqrt3` is rational

∴  `7-sqrt3` is irrational proved.

Method II:

Let `7-sqrt3` is rational

we know sum or difference of two rationals is also rational.

∴ `7-7-sqrt3 = sqrt3 = `

but this is contradiction that `sqrt3` is irrational

`:.7-sqrt3` is irrational proved.

Concept: Proofs of Irrationality
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