Question

name :
date :
8.ns.a.1: classify numbers as rational or irrational

1. which expression represents an irrational number?

1.) √2 2.) 3/4 3.) 0.17 4.) 0

1. which is an irrational number?

1.) 10 2.) π 3.) -1/3 4.)√9

1. which is a rational number?

1.) √7 2.) π 3.) 5√9 4.) 6√2

1. which number is rational?

1.) π 2.) 5/4 3.) √8 4.) √(3/2)

1. which number is irrational and explain why is it an irrational number?

√(4/9), √20, √121

1. which number is rational and explain why is it rational?

√(99)/11, √164, √196

1. which number is irrational?

1.) √9 2.) √8 3.) 0.3333 4.) 2/3

1. which number is irrational?

1.) 5/4 2.) 0.3 3.) √121 4.) π

1. which is an irrational number?

1.) √9 2.) 3.14 3.) √3 4.) 3/4

1. the number 0.14114111411114... is

1.) rational 2.) irrational
3.) integer 4.)whole
© beulah jonnalagadda 2022

Explanation:

Step1: Recall definitions

A rational number can be expressed as \(\frac{p}{q}\) where \(p,q\) are integers and \(q eq0\), or as a terminating/repeating decimal. An irrational number is non - repeating, non - terminating and cannot be expressed as \(\frac{p}{q}\). Square roots of perfect squares are rational, square roots of non - perfect squares are irrational (except when simplified to a rational form).

Step2: Solve Question 1

- For \(\sqrt{2}\): 2 is not a perfect square, so \(\sqrt{2}\) is irrational.
- For \(\frac{3}{4}\): It is in the form \(\frac{p}{q}\) (\(p = 3,q = 4\)), so rational.
- For \(0.17\): Terminating decimal, so rational.
- For \(0\): \(0=\frac{0}{1}\), so rational.
So the answer for question 1 is 1.) \(\sqrt{2}\)

Step3: Solve Question 2

- For \(10\): \(10=\frac{10}{1}\), rational.
- For \(\pi\): \(\pi\approx3.1415926\cdots\) is non - repeating, non - terminating, so irrational.
- For \(-\frac{1}{3}\): In the form \(\frac{p}{q}\) (\(p=- 1,q = 3\)), rational.
- For \(\sqrt{9}\): \(\sqrt{9}=3=\frac{3}{1}\), rational.
So the answer for question 2 is 2.) \(\pi\)

Step4: Solve Question 3

- For \(\sqrt{7}\): 7 is not a perfect square, irrational.
- For \(\pi\): Irrational.
- For \(5\sqrt{9}\): \(\sqrt{9}=3\), so \(5\sqrt{9}=5\times3 = 15=\frac{15}{1}\), rational.
- For \(6\sqrt{2}\): \(\sqrt{2}\) is irrational, so \(6\sqrt{2}\) is irrational.
So the answer for question 3 is 3.) \(5\sqrt{9}\)

Step5: Solve Question 4

- For \(\pi\): Irrational.
- For \(\frac{5}{4}\): In the form \(\frac{p}{q}\) (\(p = 5,q = 4\)), rational.
- For \(\sqrt{8}\): 8 is not a perfect square (\(\sqrt{8}=2\sqrt{2}\)), irrational.
- For \(\sqrt{\frac{3}{2}}\): \(\frac{3}{2}\) is not a perfect square, so \(\sqrt{\frac{3}{2}}\) is irrational.
So the answer for question 4 is 2.) \(\frac{5}{4}\)

Step6: Solve Question 5

- \(\sqrt{\frac{4}{9}}=\frac{2}{3}\) (rational, since \(\frac{4}{9}\) is a perfect square of \(\frac{2}{3}\)).
- \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\), 5 is not a perfect square, so \(\sqrt{20}\) is irrational (because it cannot be expressed as \(\frac{p}{q}\) and is non - repeating, non - terminating).
- \(\sqrt{121}=11=\frac{11}{1}\) (rational, since 121 is a perfect square).
So the irrational number is \(\sqrt{20}\) because it simplifies to \(2\sqrt{5}\) and \(\sqrt{5}\) is irrational (non - perfect square square root), so \(\sqrt{20}\) cannot be written as a ratio of two integers and has a non - repeating, non - terminating decimal expansion.

Step7: Solve Question 6

- \(\frac{\sqrt{99}}{11}=\frac{\sqrt{9\times11}}{11}=\frac{3\sqrt{11}}{11}\), \(\sqrt{11}\) is irrational, so \(\frac{\sqrt{99}}{11}\) is irrational.
- \(\sqrt{164}=\sqrt{4\times41}=2\sqrt{41}\), \(\sqrt{41}\) is irrational, so \(\sqrt{164}\) is irrational.
- \(\sqrt{196}=14=\frac{14}{1}\) (rational, since 196 is a perfect square).
So the rational number is \(\sqrt{196}\) because it equals 14, which can be written as \(\frac{14}{1}\) (a ratio of two integers).

Step8: Solve Question 7

- \(\sqrt{9}=3=\frac{3}{1}\), rational.
- \(\sqrt{8}=2\sqrt{2}\), \(\sqrt{2}\) is irrational, so \(\sqrt{8}\) is irrational.
- \(0.3333\): Terminating decimal, rational.
- \(\frac{2}{3}\): In the form \(\frac{p}{q}\), rational.
So the answer for question 7 is 2.) \(\sqrt{8}\)

Step9: Solve Question 8

- \(\frac{5}{4}\): In the form \(\frac{p}{q}\), rational.
- \(0.\overline{3}=\frac{1}{3}\) (repeating decimal, rational).
- \(\sqrt{121}=11=\frac{11}{1}\), rational.
- \(\pi\): Irrational (non - repeating, non - terminating).
So the answer for question 8 is 4.) \(\pi\)

Step10: Solve Question 9

- \(\sqrt{9}=3=\frac{3}{1}\), rational.
- \(3.14\): Terminating decimal, rational.
- \(\sqrt{3}\): 3 is not a perfect square, so \(\sqrt{3}\) is irrational.
- \(\frac{3}{4}\): In the form \(\frac{p}{q}\), rational.
So the answer for question 9 is 3.) \(\sqrt{3}\)

Step11: Solve Question 10

The number \(0.14114111411114\cdots\) is non - repeating and non - terminating. So it is irrational. So the answer is 2.) irrational

Answer:

1. 1.) \(\boldsymbol{\sqrt{2}}\)
2. 2.) \(\boldsymbol{\pi}\)
3. 3.) \(\boldsymbol{5\sqrt{9}}\)
4. 2.) \(\boldsymbol{\frac{5}{4}}\)
5. \(\boldsymbol{\sqrt{20}}\) (because \(\sqrt{20}=2\sqrt{5}\), \(\sqrt{5}\) is irrational, so \(\sqrt{20}\) cannot be expressed as \(\frac{p}{q}\) and has a non - repeating, non - terminating decimal expansion)
6. \(\boldsymbol{\sqrt{196}}\) (because \(\sqrt{196} = 14=\frac{14}{1}\), a ratio of two integers)
7. 2.) \(\boldsymbol{\sqrt{8}}\)
8. 4.) \(\boldsymbol{\pi}\)
9. 3.) \(\boldsymbol{\sqrt{3}}\)
10. 2.) \(\boldsymbol{irrational}\)