Question
A number is irrational if and only if its decimal representation is:
(a) non-terminating
(b) non-terminating and repeating
(c) non-terminating and non-repeating
(d) terminating
Solution
An irrational number cannot be expressed in the form \( \frac{p}{q} \).
Its decimal expansion has two important properties:
- It is non-terminating (never ends)
- It is non-repeating (no recurring pattern)
Examples:
\[ \sqrt{2} = 1.4142135\ldots \]
\[ \pi = 3.1415926\ldots \]
Thus, irrational numbers have decimal expansions that neither terminate nor repeat.
Final Answer
✔ Correct option: (c) non-terminating and non-repeating