Question

Prove that \[\left( {4 - 5\sqrt 2 } \right)\] is an irrational number.

Answer 1

Hint:
Here, we will prove the given number by using the method of Contradiction. We will assume that the given number is a rational number. We will express it as a rational number and simplify it further to prove that the given number is an irrational number. An irrational number is a number that cannot be expressed as a fraction or the ratio of two integers. An irrational number can neither be terminating or recurring.

Complete step by step solution:
To prove that the number \[\left( {4 - 5\sqrt 2 } \right)\] is an irrational number, we have to prove that the number \[\left( {4 - 5\sqrt 2 } \right)\] is not a rational number.
A rational number is a number which can be expressed as a fraction or the ratio of two integers as \[\dfrac{p}{q}\].
Now, we will consider \[\left( {4 - 5\sqrt 2 } \right)\] is a rational number.
Expressing the number as rational number, we get
\[ \Rightarrow 4 - 5\sqrt 2 = \dfrac{p}{q}\]
Subtracting 4 on both sides, we get
\[ \Rightarrow - 5\sqrt 2 = \dfrac{p}{q} - 4\]
By taking L.C.M on RHS, we get
\[ \Rightarrow - 5\sqrt 2 = \dfrac{p}{q} - 4 \times \dfrac{q}{q}\]
\[ \Rightarrow - 5\sqrt 2 = \dfrac{p}{q} - \dfrac{{4q}}{q}\]
Subtracting the like terms, we get
\[ \Rightarrow - 5\sqrt 2 = \dfrac{{p - 4q}}{q}\]
By rewriting the equation, we get
\[ \Rightarrow \sqrt 2 = \dfrac{{p - 4q}}{{ - 5q}}\]
We know that a surd is always an irrational number.
So, \[\sqrt 2 \] is an irrational number whereas \[\dfrac{{p - 4q}}{{ - 5q}}\] is a rational number.
Hence, our assumption that \[4 - 5\sqrt 2 \] is a rational number is wrong.

Therefore, \[\left( {4 - 5\sqrt 2 } \right)\] is an irrational number.
Hence proved.

Note:
We know that addition of two irrational numbers may or may not be irrational. Also, subtraction of two irrational numbers may or may not be irrational, but the difference of a rational number and an irrational number is always irrational. Surd is defined as the number which cannot be simplified to find the square root. Every rational number is not a surd but every irrational number is a surd. We are using the concept of contradiction. The contradiction is a method of proving the statement true by showing the assumption to be false.