/*! 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 14 Modify the argument in Theorem \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 14

Modify the argument in Theorem \(2.4 .7\) to show that there exists a positive real number \(y\) such that \(y^{2}=3\)

Short Answer

Expert verified
The positive real number \(y\) such that \(y^2 = 3\) is about 1.73205 (also known as the square root of 3) which can be found by continuous iteration of averaging.

Step by step solution

01

Establish Base Case

Start with a base case. We know that \(1^2 = 1\), which is less than 3, and \((2)^2 = 4\), which is greater than 3. Thus, the number we are looking for, \(y\), falls between 1 and 2.
02

Use Average

Using the average of these two numbers will get us closer to \(y\). Find the average \(1.5 = \frac{1+2}{2}\), and square the average to get \(2.25 = 1.5^2\), which is still less than 3.
03

Narrow the Range

Determine that our number is between 1.5 and 2. Take the average of the lower and upper bound \(1.75 = \frac{1.5+2}{2}\), and square it to get \(3.0625 = 1.75^2\), which is more than 3. Now we know that our number is between 1.5 and 1.75.
04

Continue Until Desired Accuracy

Following this process, we get closer and closer to the exact value of \(y\) with each iteration. If this process is repeated infinitely many times, we will arrive at a value of \(y\) which, when squared, equals exactly 3 due to the properties of our real number system.

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!

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

If \(S \subseteq \mathbb{R}\) is a nonempty bounded set, and \(I_{S}:=[\inf S\), sup \(S]\), show that \(S \subseteq I_{S}\). Moreover, if \(J\) is any closed bounded interval containing \(S\), show that \(I_{S} \subseteq J\).

Assuming the existence of roots, show that if \(c>1\), then \(c^{1 / m}n\).

If \(0 \leq a

(a) Give the first four digits in the binary representation of \(\frac{1}{3}\). (b) Give the complete binary representation of \(\frac{1}{3}\).

Let \(X\) and \(Y\) be nonempty sets and let \(h: X \times Y \rightarrow \mathbb{R}\) have bounded range in \(\mathbb{R}\). Let \(F: X \rightarrow \mathbb{R}\) and \(G: Y \rightarrow \mathbb{R}\) be defined by $$ F(x):=\sup \\{h(x, y): y \in Y\\}, \quad G(y):=\sup \\{h(x, y): x \in X\\} $$ Establish the Principle of the Iterated Suprema: $$ \sup \\{h(x, y): x \in X, y \in Y\\}=\sup \\{F(x): x \in X\\}=\sup \\{G(y): y \in Y\\} $$ We sometimes express this in symbols by $$ \sup _{x, y} h(x, y)=\sup _{x} \sup _{y} h(x, y)=\sup _{y} \sup _{x} h(x, y) . $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.