Question

prove that √ 5+3 is an irrational number​

Answer 1

Answer:

ANSWER

Let us prove that 3

is irrational. The question will automatically follow.

We prove it by contradiction.

Assume 3

is a rational number.

Thus, 3=q/p

where p,q are co-prime integers.

Thus ⟹3q

2

=p

2

This means p

2

is a multiple of 3.

As p is an integer, p also must have a factor 3.

We can say p=3λ, where λ is a constant

Thus (3λ)

2

=3q

2

,

3λ

2

=q

2

Again, we see q

2

∣ 3 which means q ∣ 3.

What do we see?

Both p and q have a common factor 3!

This is not in agreement to our initial assumption that p and q must be co-prime.

So,

3

must be irrational.

Multiplying a rational number(in this case 5) with an irrational, makes the whole number irrational.

So 5

3

is irrational.