Question

Prove that 3 + 2√5 is irrational.​

Answer 1

Let 3 + 2√5 be a rational number.

Then the co-primes x and y of the given rational number where (y ≠ 0) is such that:

3 + 2√5 = x/y

Rearranging, we get,

2√5 = (x/y) – 3

√5 = 1/2[(x/y) – 3]

Since x and y are integers, thus, 1/2[(x/y) – 3] is a rational number.

Therefore, √5 is also a rational number. But this confronts the fact that √5 is irrational.

Thus, our assumption that 3 + 2√5 is a rational number is wrong.

Hence, 3 + 2√5 is irrational.

Answer 2

Answer:

Prove 3+2

5

is irrational.

→ let take that 3+2

5

is rational number

→ so, we can write this answer as

⇒3+2

5

=

b

a

Here a & b use two coprime number and b



=0.

⇒2

5

=

b

a

−3

⇒2

5

=

b

a−3b

∴

5

=

2b

a−3b

Here a and b are integer so

2b

a−3b

is a rational number so

5

should be rational number but

5

is a irrational number so it is contradict

Hence 3+2

5

is irrational.

I hope it help you...