Question

1. which list shows only irrational numbers?

a ( 29.7, pi, \frac{77}{99}, sqrt{1} )
b ( \frac{13}{13}, 3^2, 7.overline{7}, -0.2 )
c ( 12.0overline{8}, -6, 3\frac{14}{15}, sqrt{2} )
d ( pi, sqrt{5}, \frac{pi}{4}, sqrt{13} )

Explanation:

Step1: Recall irrational number definition

Irrational numbers are non - repeating, non - terminating decimals and cannot be expressed as a fraction of two integers. Rational numbers can be expressed as a fraction, are terminating or repeating decimals, or are perfect square roots.

Step2: Analyze Option A

• \(29.7\) is a terminating decimal, so it is rational.
• \(\frac{77}{99}\) is a fraction of two integers, so it is rational.
• \(\sqrt{1} = 1\), which is an integer (a rational number). Only \(\pi\) is irrational here. So Option A is incorrect.

Step3: Analyze Option B

• \(\frac{13}{13}=1\) (rational), \(3^{2}=9\) (rational), \(7.\overline{7}\) is a repeating decimal (rational), \(- 0.2\) is a terminating decimal (rational). All numbers in this option are rational. So Option B is incorrect.

Step4: Analyze Option C

• \(12.0\overline{8}\) is a repeating decimal (rational), \(-6\) is an integer (rational), \(3\frac{14}{15}=\frac{3\times15 + 14}{15}=\frac{59}{15}\) (a fraction, rational). Only \(\sqrt{2}\) is irrational here. So Option C is incorrect.

Step5: Analyze Option D

• \(\pi\) is a non - repeating, non - terminating decimal (irrational).
• \(\sqrt{5}\): Since \(5\) is not a perfect square, \(\sqrt{5}\) is a non - repeating, non - terminating decimal (irrational).
• \(\frac{\pi}{4}\): Since \(\pi\) is irrational and dividing an irrational number by a non - zero rational number (\(4\)) gives an irrational number.
• \(\sqrt{13}\): Since \(13\) is not a perfect square, \(\sqrt{13}\) is a non - repeating, non - terminating decimal (irrational). All numbers in this option are irrational.

Answer:

D. \(\boldsymbol{\pi,\sqrt{5},\frac{\pi}{4},\sqrt{13}}\)