Question

bet ( sqrt{2} ) rational
[
begin{array}{c}
operatorname{sos} sqrt{2}=frac{a}{b}
a=sqrt{2} b
a^{2}=2 b^{2}
end{array}
]
Hona 2 is factos of ( a^{2} ) ( S_{0} )
2 is also factos of a
2
( frac{3+4}{2^{2}} ) side
So, let ( a=2 c ) as kes a has 2 factos.
[
begin{aligned}
operatorname{Moc} 2 &=frac{a^{2}}{b^{2}}
2 &=frac{4 c^{2}}{b^{2}}
b^{2} &=2 c^{2}
operatorname{sos} & b^{2}=text { has } 2 text { as factor }
end{aligned}
]
So, b also has 2 factos Fhus ( a, b ) has 2 common factor hemee ( frac{a}{b} ) is not rational. Flemce ( sqrt{2} ) is livational.

# Prove that 2 is not a rational number.

Solution