Rational numbers

Correct answer: B. Rational number

Explanation: In \(1.232323\ldots\), the block \(23\) repeats indefinitely, so it is a recurring decimal. Every recurring decimal can be expressed as a fraction; in fact, \(1.232323\ldots=\frac{122}{99}\), so it is rational. It is not irrational because an infinite decimal is irrational only when it is non-terminating and non-recurring. Exam tip: terminating and recurring decimals are always rational.

Correct answer: B. Rational number

Explanation: The block 45 repeats after the decimal point, so this is a recurring decimal. Every recurring decimal is rational; in fact, \(3.454545\ldots=3+\frac{45}{99}=\frac{38}{11}\). It is not an integer because \(\frac{38}{11}\) is not a whole number. Exam tip: Both terminating and recurring decimals are rational numbers.

Correct answer: C. \((\sqrt{13})^2\)

Explanation: In option C, \((\sqrt{13})^2=13\), and 13 is an integer, so it is rational. In the other options, an irrational square root is added to or subtracted from a rational number, so the result remains irrational. Exam tip: squaring the square root of a number removes the radical and gives the number itself.