Which of these decimal numbers is RATIONAL?

Class 9Number SystemsNumber Systems

Step-by-Step Solution

A repeating block means rational: 0.123123… =

\frac{123}{999} = \frac{41}{333}

. The other three never settle into a repeating block, so they are non-terminating non-recurring and hence irrational.

Answer: D) 0.123123123… (the block 123 repeats)

**Rational ↔ terminating OR non-terminating recurring decimal.**
**Irrational ↔ non-terminating non-recurring decimal.**

Example:

\frac{1}{4} = 0.25

(terminating → rational).
Example:

\frac{1}{3} = 0.333

… (recurring → rational).
Example:

2 = 1.41421356

… (non-terminating non-recurring → irrational).

**When does

\frac{p}{q}

terminate?** Exactly when the denominator (in simplest form) has only 2's and 5's as prime factors. So

\frac{1}{8} (= 1/ 2^{3})

terminates: 0.125. But

\frac{1}{6} (= 1/ (2 \times 3))

doesn't: 0.1666…

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