Question

show that root 3+root 5 is irrational

Answer 1

Ok. First, if we can show that sqrt(3) and sqrt(5) are both irrational, then this can be easily proven, right?

Answer 2

I'll be using reductio ad absurdum

Answer 3

Assume that sqrt(3) is a rational number
So, it can be written like this:\[\sqrt3=\frac{ p }{ q }\] where p and q are distinct natural numbers.

Answer 4

square both sides and get\[3=\frac{ p^2 }{ q^2 }\]
\[p^2=3q^2\]
So, p^2 is a multiple of 3.

Answer 5

That means that p has to be a multiple of 3 since p is a natural number.
So, we can say that \[p=3k\]
k is a natural number constant
substitute that in the last equation to get \[(3k)^2=3q^2\]
\[9k^2=3q^2\]
\[q^2=3k^2\]
So q^2 is also multiple of 3, which means that q is multiple of 3.
So, p and q have a common factor of 3.
But in the first place, we assumed that both of them are distinct, or without common factor.
So the assumption that sqrt(3) is wrong.

Answer 6

Correction: Assumption that sqrt(3) is rational number is wrong.
Similarly, we can prove that sqrt(5) is not rational number, either.
Becuase both of them are irrational number, sqrt(3)+sqrt(5) is irrational number.