/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 Give an example of irrational nu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 56

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. $$a+b$$

Short Answer

Expert verified
Example (a): \( \sqrt{2} + (-\sqrt{2}) = 0 \) is rational. Example (b): \( \sqrt{2} + \sqrt{3} \) is irrational.

Step by step solution

01

Understand the Definitions

An irrational number is a number that cannot be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). A rational number, on the other hand, is a number that can be expressed in such a way.
02

Choose Irrational Numbers

Select two specific irrational numbers. A common example of irrational numbers are \( \sqrt{2} \) and \( \sqrt{3} \).
03

Find a Pair for a Rational Result

Consider the expression \( a + b \). If \( a = \sqrt{2} \) and \( b = -\sqrt{2} \), then their sum is \( a + b = \sqrt{2} - \sqrt{2} = 0 \). Zero is a rational number.
04

Verify the Rationality

Since \( 0 \) can be expressed as \( \frac{0}{1} \), which is in the form of a fraction, \( 0 \) is a rational number. Hence, adding the irrational numbers \( \sqrt{2} \) and \(-\sqrt{2}\) results in a rational number.
05

Find a Pair for an Irrational Result

Now we need example of an irrational result. If \( a = \sqrt{2} \) and \( b = \sqrt{3} \), then \( a + b = \sqrt{2} + \sqrt{3} \). The result is still irrational because it cannot be simplified to a fraction.
06

Verify the Irrationality

The sum \( \sqrt{2} + \sqrt{3} \) is known to be irrational because it cannot be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers.

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 those numbers that can be expressed as a fraction, where both the numerator and the denominator are integers. The denominator, importantly, should not be zero, as division by zero is undefined. This concept lets us describe a wide variety of numbers, including both positive and negative integers, zero, and fractions.
For instance:
  • The number 3 is a rational number because it can be written as \( \frac{3}{1} \).
  • The fraction \( \frac{-5}{6} \) is also rational because both -5 and 6 are integers.
  • The number 0, expressed as \( \frac{0}{1} \), is rational because 0 and 1 are integers.
Understanding rational numbers helps when evaluating expressions that involve different types of numbers, as it provides one of the foundational categories for classifying numbers.
Sum of Irrational Numbers
The sum of two irrational numbers can be either rational or irrational, depending on the specific numbers involved. It's a common misconception that adding two irrational numbers always results in an irrational sum, but this is not always the case.
Let's look at examples for clarity:
  • If you take \( a = \sqrt{2} \) and \( b = -\sqrt{2} \), their sum \( a + b = \sqrt{2} - \sqrt{2} = 0 \), which is a rational number.
  • Conversely, for \( a = \sqrt{2} \) and \( b = \sqrt{3} \), their sum \( a + b = \sqrt{2} + \sqrt{3} \) remains irrational, as it cannot be simplified to be expressed as a fraction of integers.
These examples illustrate that the properties of irrational numbers in arithmetic operations can vary, and it's crucial to analyze each case individually to determine if the outcome is rational or irrational.
Examples of Irrational Numbers
Irrational numbers cannot be expressed as a simple fraction, making them more complex compared to their rational counterparts. They have non-repeating, non-terminating decimal representations, which means their decimal form goes on forever without repeating. These numbers often arise in geometry and trigonometry.
Here are a few common examples:
  • \( \sqrt{2} \): Known as the square root of 2, it cannot be simplified into a fraction.
  • \( \sqrt{3} \): Similar to \( \sqrt{2} \), this is also irrational and arises frequently in mathematical calculations involving triangles.
  • \( \pi \): The constant used in calculations involving circles is famously irrational, as its decimal goes on infinitely without repeating.
  • \( e \): This is the base of the natural logarithm and is widely used in calculus.
Learning about these numbers expands our understanding beyond the rational number system, revealing how varied and expansive the set of real numbers truly is.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=x^{2}-2 x-2$$

In each of parts (a) through (d), first solve the equation for \(y\) so that you can enter it in your graphing utility. Then use the graphing utility to graph the equation in an appropriate viewing rectangle. In each case, the graph is a line. Given that the \(x\) - and \(y\) -intercepts are (in every case here) integers, read their values off the screen and write them down for easy reference when you get to part (e). (a) \(\frac{x}{2}+\frac{y}{3}=1\) (c) \(\frac{x}{6}+\frac{y}{5}=1\) (b) \(\frac{x}{-2}+\frac{y}{-3}=1\) (d) \(\frac{x}{-6}+\frac{y}{-5}=1\) (e) On the basis of your results in parts (a) through (d), describe, in general, the graph of the equation \(\frac{x}{a}+\frac{y}{b}=1,\) where \(a\) and \(b\) are nonzero constants.

Specify the center and radius of each circle. Also, determine whether the given point lies on the circle. $$(x+4)^{2}+(y+2)^{2}=20 ;(0,1)$$

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-5| \geq 2$$

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=x^{2}-2 x+1$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.