numbers 2. Show that if a and b are positive integers, then there is a smallest positive integer of the form a-bk, k = Z. -Prove that both the sum and th
numbers 2. Show that if a and b are positive integers, then there is a smallest positive integer of the form a-bk, k = Z. -Prove that both the sum and th
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.1: Basic Assumptions
Problem 40WE
Related questions
Question
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In “Elementary Number Theory & Its applications” by Kenneth H. Rosen, consider the following definitions in chapter 1.1, then exercise 2:
The Well-Ordering Property: Every non-empty set of positive integers has a least element
The Greatest Integer Function:
The greatest integer in a real number x, denoted by [x], is the largest integer less than or equal to x. That is, [x] is the integer satisfying:
[x]
Transcribed Image Text:1.1
0
44141
2
0
:
Figure 1.1 Listing the rational numbers.
The initial terms of the sequence are 0/1 = 0, 1/1 = 1, -1/1 = -1, 1/2, 1/3, -1/2,
2/1=2,-2/1=-2, -1/3, 1/4, and so on.) We leave it to the reader to fill in the details,
to see that this procedure lists all rational numbers as the terms of a sequence.
We have shown that the set of rational numbers is countable, but we have not given an
example of an uncountable set. Such an example is provided by the set of real numbers,
as shown in Exercise 45.
EXERCISES
1. Determine whether each of the following sets is well ordered. Either give a proof using the
well-ordering property of the set of positive integers, or give an example of a subset of the
set that has no smallest element.
a) the set of integers greater than 3
b) the set of even positive integers
c) the set of positive rational numbers
d) the set of positive rational numbers that can be written in the form a/2, where a is a
positive integer
e) the set of nonnegative rational numbers
2. Show that if a and b are positive integers, then there is a smallest positive integer of the form
a-bk, k € Z.
3. Prove that both the sum and the product of two rational numbers are rational.
4. Prove or disprove each of the following statements.
a) The sum of a rational and an irrational number is irrational.
b) The sum of two irrational numbers is irrational.
![To begin, we will introduce several different types of numbers. The integers are the
numbers in the set
{..., -3, -2, -1, 0, 1, 2, 3, ...).
The integers play center stage in the study of number theory. One property of the positive
integers deserves special mention.
The Well-Ordering Property Every nonempty set of positive integers has a least
element.
The well-ordering property may seem obvious, but it is the basic principle that allows
us to prove many results about sets of integers, as we will see in Section 1.3.
The well-ordering property can be taken as one of the axioms defining the set of
positive integers or it may be derived from a set of axioms in which it is not included.
(See Appendix A for axioms for the set of integers.) We say that the set of positive
integers is well ordered. However, the set of all integers (positive, negative, and zero)
is not well ordered, as there are sets of integers without a smallest element, such as the
set of negative integers, the set of even integers less than 100, and the set of all integers
itself.
Another important class of numbers in the study of number theory is the set of
numbers that can be written as a ratio of integers.
Definition. The real number r is rational if there are integers p and q, with q #0,
such that r = p/q. If r is not rational, it is said to be irrational.
Example 1.1. The numbers -22/7, 0=0/1, 2/17, and 1111/41 are rational numbers.
Note that every integer n is a rational number, because n = n/1. Examples of irrational
numbers are √2, 7, and e. We can use the well-ordering property of the set of positive
integers to show that √2 is irrational. The proof that we provide, although quite clever,
is not the simplest proof that √2 is irrational. You may prefer the proof that we will give
in Chapter 4, which depends on concepts developed in that chapter. (The proof that e is
irrational is left as Exercise 44. We refer the reader to [HaWr08] for a proof that is
irrational. It is not easy.)
Theorem 1.1. √√2 is irrational.
Proof. Suppose that √2 were rational. Then there would exist positive integers a and b
such that √2= a/b. Consequently, the set S= (k√√2|k and k√2 are positive integers
is a nonempty set of positive integers (it is nonempty because a = b√2 is a member
of S). Therefore, by the well-ordering property, S has a smallest element, say, s=1√2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9acf228d-ec9e-49d8-a78a-8ceb13ba5105%2Fa274f764-9193-4d6a-bb39-ab613f8ea54a%2Fa8z84vd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To begin, we will introduce several different types of numbers. The integers are the
numbers in the set
{..., -3, -2, -1, 0, 1, 2, 3, ...).
The integers play center stage in the study of number theory. One property of the positive
integers deserves special mention.
The Well-Ordering Property Every nonempty set of positive integers has a least
element.
The well-ordering property may seem obvious, but it is the basic principle that allows
us to prove many results about sets of integers, as we will see in Section 1.3.
The well-ordering property can be taken as one of the axioms defining the set of
positive integers or it may be derived from a set of axioms in which it is not included.
(See Appendix A for axioms for the set of integers.) We say that the set of positive
integers is well ordered. However, the set of all integers (positive, negative, and zero)
is not well ordered, as there are sets of integers without a smallest element, such as the
set of negative integers, the set of even integers less than 100, and the set of all integers
itself.
Another important class of numbers in the study of number theory is the set of
numbers that can be written as a ratio of integers.
Definition. The real number r is rational if there are integers p and q, with q #0,
such that r = p/q. If r is not rational, it is said to be irrational.
Example 1.1. The numbers -22/7, 0=0/1, 2/17, and 1111/41 are rational numbers.
Note that every integer n is a rational number, because n = n/1. Examples of irrational
numbers are √2, 7, and e. We can use the well-ordering property of the set of positive
integers to show that √2 is irrational. The proof that we provide, although quite clever,
is not the simplest proof that √2 is irrational. You may prefer the proof that we will give
in Chapter 4, which depends on concepts developed in that chapter. (The proof that e is
irrational is left as Exercise 44. We refer the reader to [HaWr08] for a proof that is
irrational. It is not easy.)
Theorem 1.1. √√2 is irrational.
Proof. Suppose that √2 were rational. Then there would exist positive integers a and b
such that √2= a/b. Consequently, the set S= (k√√2|k and k√2 are positive integers
is a nonempty set of positive integers (it is nonempty because a = b√2 is a member
of S). Therefore, by the well-ordering property, S has a smallest element, say, s=1√2.
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