Question

Prove that 7+2√3 is irrational

Answer 1

Answer:

Let assume that 7-2root3 be rational and rational numbers are in the form of p/q form.

We know that root 3 is irrational and not p/q form. It contradicts the statement that it is irrational as it is p/q form.

Subtraction and multiplication of irrational number form irrational number so

7-2root3 is irrational.

explanation:

Answer 2

Answer:

Let us assume that 7 + 2√3 is rational number.

Then, we can find two co-prime integers a and b, such that

7 + 2√3 = a/b

2√3 = a/b - 7

2√3 = a - 7b/b

√3 = a - 7b/2b which is rational

Let us assume that √3 is rational.

Then, √3 = r/s, where r and s have a common factor other than 1.

Then divide by this common factor to get √3 =

a/b, where a and b are co-prime integers.

√3 = a/b

√3b = a

Squaring both sides.

(√3b)² = a²

3b² = a²

By the theorem if p divides a², p divides a.

Therefore, 3 divides a²

Therefore, 3 divides a

Let, a = 3c, for some integer c.

3b² = (3c)²

3b² = 9c²

b² = 3c²

By the theorem, if p divides a², p divides a. Here 3 divides b², hence 3 divides b

Therefore, a and b have a common factor other than 1.

This contradicts the fact that a and b have atleast 3 as a common factor.

This contradiction is arisen because of our incorrect assumption that √3 is rational.

Therefore, √3 is irrational.

Hence our assumption that 7 + 2√3 is irrational is wrong.

Therefore, 7 + 2√3 is irrational.

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