REAL NUMBERS | Irrational Numbers | Part -3

Sum of a Rational Number and an Irrational Number

In mathematics, understanding the interaction between different types of numbers is essential. An interesting characteristic in number theory involves the sum of a rational number and an irrational number.

What Are Rational and Irrational Numbers?

• Rational numbers: Numbers that can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and q ≠ 0. Examples: \( \frac{2}{3}, -5, 0.75 \).
• Irrational numbers: Numbers that cannot be written as \( \frac{p}{q} \), and have non-terminating, non-repeating decimal expansions. Examples: \( \pi, \sqrt{2}, \sqrt{5} \).

Theorem

The sum of a rational number and an irrational number is always an irrational number.

Proof

Let \( r \) be a rational number and \( i \) be an irrational number. Suppose their sum \( s = r + i \) is rational.

Now, eliminate ( r ) by subtracting it from both sides: \[ s - r = (r + i) - r = i \]

This implies that \( i \) is rational (since the difference of two rational numbers is rational), which contradicts the fact that \( i \) is irrational. Therefore, the assumption is false, and the sum must be irrational.

Example

Let \( r = 1 \) and \( i = \sqrt{2} \). Then: \[ r + i = 1 + \sqrt{2} \] Since this value cannot be written as a rational number, it is classified as irrational.

Is the Sum of Two Irrational Numbers Always Irrational?

No, the sum of two irrational numbers is not always irrational.

Explanation

Sometimes, the sum of two irrational numbers can be rational.

Example 1 (Irrational + Irrational = Rational)

Let \( i_1 = \sqrt{2} \) and \( i_2 = -\sqrt{2} \). Both are irrational.

Their sum: \[ i_1 + i_2 = \sqrt{2} + (-\sqrt{2}) = 0 \] which is rational.

Example 2 (Irrational + Irrational = Irrational)

Let \( i_1 = \sqrt{2} \), \( i_2 = \sqrt{3} \). Then: \[ i_1 + i_2 = \sqrt{2} + \sqrt{3} \] This number is irrational since it cannot be written as a ratio of two integers.

• Adding a rational number to an irrational number always results in an irrational number.
• The result of adding two irrational numbers can be either rational or irrational, based on their specific values.