The sum of a rational number and an irrational number will always be an irrational number.

Explanation:

• A rational number can be expressed as the fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.
• An irrational number cannot be expressed as a fraction; it has a non-terminating, non-repeating decimal expansion.

When you add a rational number $r$ to an irrational number $i$, the result cannot be expressed as a fraction (since the irrational "component" remains). Hence, the sum is irrational.

Example:

• Let $r = 3$ (rational) and $i = \sqrt{2}$ (irrational).
• $r + i = 3 + \sqrt{2}$, which is irrational because $\sqrt{2}$'s irrationality prevents the entire expression from becoming a fraction.

Would you like further clarification or examples?

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Related Questions:

1. What happens when two irrational numbers are added together?
2. Can the product of a rational and an irrational number ever be rational?
3. What is the sum of a rational number and a zero?
4. Why is $\pi + 2$ irrational but $\pi - \pi$ rational?
5. How can we prove that the sum of a rational and an irrational number is always irrational?

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Tip: To verify if a number is irrational, check whether it can be expressed as a fraction or if its decimal representation never terminates or repeats.