Problem 78 For Exercises $75-80,$ conside... [FREE SOLUTION]

Chapter 1: Problem 78

For Exercises $75-80,$ consider the following list: $$\begin{array}{llll} {18,} & {-4.7,} & {0,} & {-\frac{5}{9},} & {\pi,} & {\sqrt{17}, \quad 2, \overline{16},} & {-37} \end{array}$$ List all irrational numbers.

Short Answer

Expert verified

The irrational numbers are $\pi$ and $\sqrt{17}$.

Step by step solution

- Understand Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b eq 0$. They have non-repeating, non-terminating decimal expansions.

- Identify Candidates

Examine each number in the list and determine if it is irrational based on its decimal expansion or roots.

- Evaluate Each Number

Check each number one by one: 18 (rational), $-4.7$ (rational), 0 (rational), \[ -\frac{5}{9} \] (rational), $\pi$ (irrational), $\sqrt{17}$ (irrational), \2.\overline{16}\ (rational), $-37$ (rational).

- List Irrational Numbers

The irrational numbers identified in the previous step are $\pi$ and $\sqrt{17}$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational numbers

Rational numbers are numbers that can be written as a fraction. This fraction form is $\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is not zero. For example, numbers like 3, -1, and $\frac{3}{4}$ are rational. They either terminate or repeat in decimal form.
For example:

3 can be written as $\frac{3}{1}$
-1 can be written as $\frac{-1}{1}$
0.75 can be written as $\frac{3}{4}$

Rational numbers include integers, fractions, and some decimals. In the exercise, 18, -4.7, 0, $\frac{-5}{9}$, 2.\overline{16}, and -37 are rational because they can be written as fractions or have repeating decimal forms.

fraction representation

Fraction representation of rational numbers is crucial for identifying them easily. A rational number can always be expressed with two integers. For instance, $\frac{1}{2}$ is a simple fraction where 1 is the numerator and 2 is the denominator. Here are some examples for clarity:

18 can be represented as $\frac{18}{1}$
-4.7 can be written as $\frac{-47}{10}$
$-37$ is simply $\frac{-37}{1}$

Recognizing this form helps us understand that these are all rational. If a number can be written this way, it means it fits within rational numbers.

non-terminating decimals

Non-terminating decimals are decimals that do not end. They go on forever. If these decimals do not repeat, they are irrational. For example, $ \pi $ and $ \sqrt{17} $ are irrational because their decimal forms are non-terminating and non-repeating.
However, some non-terminating decimals repeat, like 2.\overline{16}. These are rational because they can be expressed as fractions. For example:

2.\overline{16} = $\frac{216}{99}$

Knowing if decimals repeat or not helps us decide if a number is rational or irrational.

roots

Roots, specifically square roots, can be rational or irrational. A root is irrational if it is not a perfect square. For example, $ \sqrt{17} $ is not a perfect square and is irrational. But, $ \sqrt{16} $ equals 4, a rational number.
To summarize:

$ \sqrt{17} $ is irrational because 17 isn't a perfect square.
$ \pi $ is irrational, known as transcendental.

Remembering which numbers give perfect squares can help identify irrational roots easily.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

For Exercises \(75-80,\) consider the following list: $$\begin{array}{llll} {18,} & {-4.7,} & {0,} & {-\frac{5}{9},} & {\pi,} & {\sqrt{17}, \quad 2, \overline{16},} & {-37} \end{array}$$ List all irrational numbers.

Short Answer

Step by step solution

- Understand Irrational Numbers

- Identify Candidates

- Evaluate Each Number

- List Irrational Numbers

Key Concepts

rational numbers

fraction representation

non-terminating decimals

roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Discrete Mathematics

Probability and Statistics

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.