Question

Show that 2+√3 is an irrational number where√3 is irrational no

Answer 1

we know that sum product subtraction and multiplication of rational and irrational no is always irrational

Answer 2

Answer:

First we prove that √2 is irrational.

for that we assume that  √2  is rational. And as rational number can be written as $\frac{p}{q}$   form.

where we assume that p and q have no common factor. Squaring in both side. After that  2= $\frac{p^{2} }{q^{2} }$

which implies that

$p^{2} = 2q^{2}$

thus $p^{2}$  is even. The only way this can be true is that p itself is even.

But then $p^{2}$  is actually divisible by 4. Hence $q^{2}$  and therefore q must be even.

So p and q are both even which is a contradiction to our assumption that they have no common factors. The square root of 2 cannot be rational.

and we similarly prove for √3

​And as we know that sum of two irrational no. is always irrational.