Question
Question: Prove that \(3 + 2\sqrt 5 \) is an irrational number....
Prove that 3+25 is an irrational number.
Solution
Hint:In order to prove that the number is irrational, we start off by considering it to be rational and disproving that it does not hold the property of a rational number i.e by contradiction method.
Complete step-by-step answer:
If a number that can be expressed in the form of qp is called rational number. Here p and q are integers,
and q is not equal to 0.
Let us assume 3+25 is a rational number, it can be expressed as
3+25= qp
⟹5=2qp - 3q
The RHS 2qp - 3qlooks like a rational number but the LHS of the equation5 is not a rational number.
Hence 3+25 is not a rational number, i.e. it is an irrational number.
Note – In order to solve questions of this type the key is to know the definitions of rational and irrational numbers. An irrational number is a number that cannot be expressed as a fraction for any integers and Irrational numbers have decimal expansions that neither terminate nor become periodic.Contradiction method is a common proof technique that is based on a very simple principle: something that leads to a contradiction can not be true, and if so, the opposite must be true.