# Prove that 7-√3 is irrational. - Mathematics

Sum

Prove that 7-sqrt3 is irrational.

#### Solution

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.

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