Daniel Rosenthal David Rosenthal - A Readable Introduction to Real Mathematics
Here you can read online Daniel Rosenthal David Rosenthal - A Readable Introduction to Real Mathematics full text of the book (entire story) in english for free. Download pdf and epub, get meaning, cover and reviews about this ebook. year: 2016, publisher: Springer International Publishing, genre: Children. Description of the work, (preface) as well as reviews are available. Best literature library LitArk.com created for fans of good reading and offers a wide selection of genres:
Romance novel
Science fiction
Adventure
Detective
Science
History
Home and family
Prose
Art
Politics
Computer
Non-fiction
Religion
Business
Children
Humor
Choose a favorite category and find really read worthwhile books. Enjoy immersion in the world of imagination, feel the emotions of the characters or learn something new for yourself, make an fascinating discovery.
- Book:A Readable Introduction to Real Mathematics
- Author:
- Publisher:Springer International Publishing
- Genre:
- Year:2016
- Rating:4 / 5
- Favourites:Add to favourites
- Your mark:
A Readable Introduction to Real Mathematics: summary, description and annotation
We offer to read an annotation, description, summary or preface (depends on what the author of the book "A Readable Introduction to Real Mathematics" wrote himself). If you haven't found the necessary information about the book — write in the comments, we will try to find it.
The book is an introduction to real mathematics and is carefully written in a precise but readable and engaging style and is tightly organised into eight short core chapters and four longer standalone extension chapters.Designed for an undergraduate course or for independent study, this text presents sophisticated mathematical ideas in an elementary and friendly fashion. The fundamental purpose of this book is to engage the reader and to teach a real understanding of mathematical thinking while conveying the beauty and elegance of mathematics. The text focuses on teaching the understanding of mathematical proofs. The material covered has applications both to mathematics and to other subjects. The book contains a large number of exercises of varying difficulty, designed to help reinforce basic concepts and to motivate and challenge the reader. The sole prerequisite for understanding the text is basic high school algebra; some trigonometry is needed for Chapters 9 and 12.Topics covered include: *Mathematical induction *Modular arithmetic *The fundamental theorem of arithmetic *Fermats little theorem *RSA encryption *The Euclidean algorithm *Rational and irrational numbers *Complex numbers *Cardinality *Euclidean plane geometry *Constructability (including a proof that an angle of 60 degrees cannot be trisected with a straightedge and compass). This textbook is suitable for a wide variety of courses and for a broad range of students in the fields of education, liberal arts, physical sciences and mathematics. Students at the senior high school level who like mathematics will also be able to further their understanding of mathematical thinking by reading this book.
Daniel Rosenthal David Rosenthal: author's other books
Who wrote A Readable Introduction to Real Mathematics? Find out the surname, the name of the author of the book and a list of all author's works by series.
Daniel Rosenthal David Rosenthal
Michael Hoy
Joel David Hamkins
Reeping David
Kenneth Falconer
Daniel W. Cunningham
Rautenberg
Franco Vivaldi
Donald Bindner
A Readable Introduction to Real Mathematics — read online for free the complete book (whole text) full work
Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "A Readable Introduction to Real Mathematics" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.
Light
Font size:
↓
↑
Reset
Interval:
↓
↑
Bookmark:
Make
Springer International Publishing Switzerland 2014
Daniel Rosenthal , David Rosenthal and Peter Rosenthal A Readable Introduction to Real Mathematics Undergraduate Texts in Mathematics 10.1007/978-3-319-05654-8_11. Introduction to the Natural Numbers
Daniel Rosenthal 1, David Rosenthal 2 and Peter Rosenthal 1
(1)
Department of Mathematics, University of Toronto, Toronto, ON, Canada
(2)
Department of Mathematics and Computer Science, St. Johns University, Queens, NY, USA
We assume basic knowledge about the numbers that we count with; that is, the numbers 1, 2, 3, 4, 5, 6, and so on. These are called the natural numbers , and the set consisting of all of them is usually denoted by 
. They do seem to be very natural, in the sense that they arose very early on in virtually all societies. There are many other names for these numbers, such as the positive integers and the positive whole numbers . Although the natural numbers are very familiar, we will see that they have many interesting properties beyond the obvious ones. Moreover, there are many questions about the natural numbers to which nobody knows the answer. Some of these questions can be stated very simply, as we shall see, although their solution has eluded the thousands of mathematicians who have attempted to solve them.
. They do seem to be very natural, in the sense that they arose very early on in virtually all societies. There are many other names for these numbers, such as the positive integers and the positive whole numbers . Although the natural numbers are very familiar, we will see that they have many interesting properties beyond the obvious ones. Moreover, there are many questions about the natural numbers to which nobody knows the answer. Some of these questions can be stated very simply, as we shall see, although their solution has eluded the thousands of mathematicians who have attempted to solve them.We assume familiarity with the two basic operations on the natural numbers, addition and multiplication. The sum of two numbers will be indicated using the plus sign +. Multiplication will be indicated by putting a dot in the middle of the line between the numbers, or by simply writing the symbols for the numbers next to each other, or sometimes by enclosing them in parentheses. For example, the product of 3 and 2 could be denoted 3 2 or (3)(2). The product of the natural numbers represented by the symbols m and n could be denoted mn , or m n , or ( m )( n ).
We also, of course, need the number 0. Moreover, we require the negative whole numbers as well. For each natural number n there is a corresponding negative number n such that 
. Altogether, the collection of positive and negative numbers and 0 is called the integers . It is often denoted by 
.
. Altogether, the collection of positive and negative numbers and 0 is called the integers . It is often denoted by .We assume that you know how to add two negative integers and also how to add a negative integer to a positive integer. Multiplication appears to be a bit more mysterious. Most people feel comfortable with the fact that, for m and n natural numbers, the product of m and ( n ) is mn . What some people find more mysterious is the fact that 
for natural numbers m and n ; that is, the product of two negative integers is a positive integer. There are various possible explanations that can be provided for this, one of which is the following. Using the usual rules of arithmetic: 
Adding mn to both sides of this equation gives 
or 
Thus, 
so, 
Therefore, the fact that 
is implied by the other standard rules of arithmetic.
for natural numbers m and n ; that is, the product of two negative integers is a positive integer. There are various possible explanations that can be provided for this, one of which is the following. Using the usual rules of arithmetic: is implied by the other standard rules of arithmetic.1.1 Prime Numbers
One of the important concepts we will study is divisibility . For example, 12 is divisible by 3, which means that there is a natural number (in this case, 4) such that the product of 3 and that natural number is 12. That is, 12=3 4. In general, we say that the integer m is divisible by the integer n if there is an integer q such that m = nq . There are many other terms that are used to describe such a relationship. For example, if m = nq , we may say that n and q are divisors of m and that each of n and q divides m . The terminology q is the quotient when m is divided by n is also used when n is different from 0. In this situation, n and q are also sometimes called factors of m ; the process of writing an integer as a product of two or more integers is called factoring the integer.
The number 1 is a divisor of every natural number since, for each natural number m , m =1 m . Also, every natural number m is a divisor of itself, since m = m 1.
The number 1 is the only natural number that has only one natural number divisor, namely itself. Every other natural number has at least two divisors, itself and 1. The natural numbers that have exactly two natural number divisors are called prime numbers . That is, a prime number is a natural number greater than 1 whose only natural number divisors are 1 and the number itself. We do not consider the number 1 to be a prime; the first prime number is 2. The primes continue: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on.
And so on? Is there a largest prime? Or does the sequence of primes continue without end? There is, of course, no largest natural number. For if n is any natural number, then n + 1 is a natural number and n + 1 is bigger than n . It is not so easy to determine if there is a largest prime number or not. If p is a prime, then p + 1 is almost never a prime. Of course, if p =2, then 
and p and p + 1 are both primes. However, 2 is the only prime number p for which p + 1 is prime. This can be proven as follows. First note that, since every even number is divisible by 2, 2 itself is the only even prime number. Therefore, if p is a prime other than 2, then p is odd and p + 1 is an even number larger than 2 and is thus not prime.
and p and p + 1 are both primes. However, 2 is the only prime number p for which p + 1 is prime. This can be proven as follows. First note that, since every even number is divisible by 2, 2 itself is the only even prime number. Therefore, if p is a prime other than 2, then p is odd and p + 1 is an even number larger than 2 and is thus not prime.Is it nonetheless true that, given any prime number p , there is a prime number larger than p ? Although we cannot get a larger prime by simply adding 1 to a given prime, there may be some other way of producing a prime larger than any given one. We will answer this question after learning a little more about primes.
A natural number, other than 1, that is not prime is said to be composite . (The number 1 is special and is neither prime nor composite.) For example, 4, 68, 129, and 2010 are composites. Thus, a composite number is a natural number other than 1 that has a divisor in addition to itself and 1.
To determine if a number is prime, what potential factors must be checked to eliminate the possibility that there are factors other than the number and 1? If m = n q , it is not possible that n and q are both larger than the square root of m , for if two natural numbers are both larger than the square root of m , then their product is larger than m . It follows that a natural number (other than 1) that is not prime has at least one divisor that is larger than 1 and is no larger than the square root of that natural number. Thus, to check whether or not a natural number m is prime, you need not check whether every natural number less than m divides m . It suffices to check if m has a divisor that is larger than 1 and no larger than the square root of m . If it has such a divisor, it is composite; if it has no such divisor, it is prime.
Light
Font size:
↓
↑
Reset
Interval:
↓
↑
Bookmark:
Make
Similar books «A Readable Introduction to Real Mathematics»
Look at similar books to A Readable Introduction to Real Mathematics. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.
Michael Hoy
Joel David Hamkins
Reeping David
Kenneth Falconer
Daniel W. Cunningham
Rautenberg
Franco Vivaldi
Donald Bindner
Reviews about «A Readable Introduction to Real Mathematics»
Discussion, reviews of the book A Readable Introduction to Real Mathematics and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.