Let's understand why $\sqrt{3} - \sqrt{5}$ is not a rational number.

Step 1: Definition of Rational Numbers

A rational number is any number that can be expressed in the form $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ .

Step 2: Understanding the Expression

The expression $\sqrt{3} - \sqrt{5}$ is a combination of two irrational numbers. Both $\sqrt{3}$ and $\sqrt{5}$ are irrational because they cannot be expressed as a simple fraction.

Step 3: Proof by Contradiction

Let us assume, for the sake of contradiction, that $\sqrt{3} - \sqrt{5}$ is rational.

Therefore, let $\sqrt{3} - \sqrt{5} = r$ , where $r$ is a rational number.

Rearranging the equation, we get:

$\sqrt{3} = r + \sqrt{5}$

Squaring both sides:

${(\sqrt{3})}^{2} = {(r + \sqrt{5})}^{2}$

$3 = r^{2} + 2 r \sqrt{5} + 5$

Rearranging terms:

$2 r \sqrt{5} = 3 - 5 - r^{2}$

$2 r \sqrt{5} = - 2 - r^{2}$

Therefore,

$\sqrt{5} = \frac{- 2 - r^{2}}{2 r}$

Step 4: The Contradiction

The left side of the equation, $\sqrt{5}$ , is irrational. However, the right side is a rational expression (a fraction of rational numbers, since $r$ is rational). This is a contradiction because a rational number cannot equal an irrational number.

Conclusion

Our initial assumption that $\sqrt{3} - \sqrt{5}$ is rational must be false. Therefore, it is an irrational number.

Related Topics & Formulae

Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal expansion is non-terminating and non-repeating. Examples include $\sqrt{2}$ , $π$ , and $e$ .

Properties: The sum or difference of two irrational numbers is not necessarily irrational. However, if the irrational parts are not additive inverses of each other (like in this case), the result often remains irrational.

Proof Technique: Proof by contradiction is a powerful method to show that a number is irrational. You assume it is rational and then show this leads to an impossible situation.