Question

Prove √2 + √3 is an irrational number.
​

Answer 1

Let us assume that

2

+

3

is a rational number

Then. there exist coprime integers p, q,q



=0 such that

2

+

3

=

q

p

=>

q

p

−

3

=

2

Squaring on both sides, we get

=>(

q

p

−

3

)

2

=(

2

)

2

=>

q

2

p

2

−2

q

p

3

+(

3

)

2

=2

=>

q

2

p

2

−2

q

p

3

+3=2

=>

q

2

p

2

+1=2

q

p

3

=>

q

2

p

2

+q

2

×

2p

q

=

3

=>

2pq

p

2

+q

2

=

3