Guide The Real Number System in an Algebraic Setting

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Royal Festival Hall. Unit interval. Relations, less than and greater than. The set of natural numbers. The set of whole numbers. The set of integers. The set of rational numbers. That is, a ratio or.

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The set of real numbers , denoted R ,. The absolute value of a real number a is its distance from the origin. Two prime numbers that differ by 2 are called a pair of twin primes. It has been conjectured that there are infinitely many pairs of twin primes, but this has never been proved.

Real Number System

Find the first pair of twin primes greater than In , C. Goldbach , Russian conjectured that every even number greater than 2 is the sum of two primes. The Goldbach conjecture has never been proved. Write each of 20, 30, 40, 50, and 60 as the sum of two primes in all possible ways. If a, b, and c are natural numbers, if a and c are relatively prime, and if b and c are relatively prime, prove that ab and c are relatively prime.

If a and c are relatively prime and n is a natural number, prove that a n and c are relatively prime. If a, b, and c are natural numbers, if a and b are relatively prime, if a c, and if b c, prove that ab c. An example is the standard method of long division cf. One of the most useful algorithms of arithmetic — called the Euclidean Algorithm — consists essentially of repeated application of the Fundamental Theorem of Euclid.

See the preface for a discussion of the role of starring in this book. Euclidean Algorithm. Apply the Euclidean Algorithm to the numbers and This number d has a more remarkable property than merely being the greatest member of C. This property is incorporated into the following definition, and proved to exist in Theorem I below. If a and b are any two natural numbers, their GCD exists and is unique.

To prove ii , we let c be an arbitrary common divisor of a and b. This completes the proof. The following two theorems are easily established, and their proofs are left to the reader. Any two natural numbers a and b possess common multiples as well as common divisors — for example, their product ab is a multiple of each.

As might be expected, among the common multiples of a and b is one of special significance, called the least common multiple or sometimes the lowest common multiple , according to the definition : Definition II. A natural number e is called the least common multiple, or LCM for short, of the natural numbers a and b if and only if it has the following two properties: i a e and b e; ii whenever c is a natural number and a c and b c, then e c.

If a and b are any two natural numbers, their LCM exists and is unique. Therefore e c, as we wished to show. A useful by-product of the Euclidean Algorithm is a proof of the fact that the GCD of two natural numbers a and b can be expressed in a simple "linear" fashion in terms of a and b: Theorem V. We start by proving a lemma. If a and b are natural numbers, then any natural number that can be expressed in the form ma — nb where m and n are natural numbers can also be expressed in the form rb — sa where r and s are natural numbers , and conversely.

Proof of Lemma. Proof of Theorem II, continued. By the Fundamental Theorem of Mathematical Induction the formula 3 is obtained. The only part of the proof remaining to be estab- lished is that 4 is a sufficient condition that a and b be relatively prime. This is true since any common divisor of a and b, in the presence of equation 4 , must also be a divisor of the right-hand member 1. If one examines the proof of the Lemma of Theorem V, and the solutions of Examples 3 and 4, it should be clear that there are infinitely many possibilities for the values of m, n, r, and s in Theorem V.

We have made no effort to find all solutions, or even to show that the solutions obtained in Examples 3 and 4 are the smallest possible. Such questions as these, relating to solutions involving natural numbers for equations in more than one unknown, are a part of one of the oldest and most fascinating chapters of number theory, called Diophantine Equations, after the Greek mathematician Diophantus circa a.

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For further reading, cf. Prove that the LCM of two natural numbers is their product if and only if they are relatively prime. State and prove a general principle of which this is a particular illustration. These are called distributive laws. State and prove corresponding commutative laws. Use the representations of Exercise 25 to prove that ad, bd, cd — d a, b, c. State and prove distributive laws for three numbers similar to those of Exercise Formulate and prove a few general theorems of your own design.

Integers and Rational Numbers The set of all integers is denoted J '. If x is an integer, then exactly one of the three statements listed in the preceding definition is true. A number is a negative integer if and only if its negative is a positive integer, or, equivalently, a natural number statement Hi. On the other hand, by the definition of an integer, for any integer x at least one of the three must hold. If an integer x is positive, alternatives ii and Hi are eliminated and x must belong to Jf.

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