Mathematics
Natural Numbers
Natural numbers are the set of positive integers starting from 1 and continuing indefinitely. They are used for counting and ordering objects. In mathematical notation, natural numbers are often represented by the symbol "N" and are fundamental in various mathematical operations and theories.
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11 Key excerpts on "Natural Numbers"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 1 Natural Number Natural Numbers can be used for counting (one apple, two apples, three apples, ...) from top to bottom. ________________________ WORLD TECHNOLOGIES ________________________ In mathematics, Natural Numbers are the ordinary counting numbers 1, 2, 3, ... (sometimes zero is also included). Since the development of set theory by Georg Cantor, it has become customary to view such numbers as a set. There are two conventions for the set of Natural Numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century. Natural Numbers have two main purposes: counting (there are 6 coins on the table) and ordering (this is the 3rd largest city in the country). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. A more recent notion is that of a nominal number, which is used only for naming. Properties of the Natural Numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics. History of Natural Numbers and the status of zero The Natural Numbers had their origins in the words used to count things, beginning with the number 1. The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622.
eBook - PDFAdvanced Calculus, An Introduction To Mathematical Analysis
An Introduction to Mathematical Analysis
- Samuel Zaidman(Author)
- 1997(Publication Date)
- World Scientific(Publisher)
Chapter I NUMBERS Numbers are the basic building bricks of analysis; everything else is based on them. In this Chapter we first present an informal discussion of Natural Numbers and negative integers. The subsequent discussion of rational and then of real numbers is much more elaborate and gives a real flavour of what the analysis is all about. 1.1 The Natural Numbers We indicate in the following lines, a familiar discussion about the so called Natural Numbers. These are indicated by symbols 1,2,3,... and originate in counting finite collections of objects; the Natural Numbers are also called positive integers. In the realm of these (mathematical) objects one considers two (algebraic) operations: addition (+) and multiplication (•). To any given pair of Natural Numbers each of these operations associates another natural number, in such a way that the following properties are true: m + n = n + ra, m • n = n -m (the commutative laws) (m + n ) + p = m + (n + p), (m -n) • p = m • (n • p) (the associative laws) m • (n + p) = m • n + m • p (the distributive laws) m -1 = m (1 is an identity with respect to multiplication) We note also the following trichotomy law: l 2 Advanced Calculus Given any Natural Numbers m and n, one and only one of the following possibilities occurs: (i) m = n; (ii) 777, = n + x for some natural number x; (iii) n = m -f 7/ for some natural number y. If (ii) holds we write m > n or n < m and we say that m is larger or greater than n and that n is smaller or less than m. If either (i) or (ii) holds, we write m > n or n < m and say that m is larger or equal to n and that n is less than or equal to m. Let us denote with N the set (collection) of all Natural Numbers. A fundamental result that we consider valid in N is the so called principle of mathematical induction. By this we understand the following property: (a) Let M be a subset of N such that: 1 G M and if n G M then n + 1 is also in M.
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- 2014(Publication Date)
- Learning Press(Publisher)
Notation Mathematicians use N or (an N in blackboard bold, displayed as ℕ in Unicode) to refer to the set of all Natural Numbers. This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null . To be unambiguous about whether zero is included or not, sometimes an index 0 is added in the former case, and a superscript * or subscript 1 is added in the latter case: (Sometimes, an index or superscript + is added to signify positive. However, this is often used for nonnegative in other cases, as R + = [0,∞) and Z + = { 0, 1, 2,... }, at least in European literature. The notation *, however, is standard for nonzero, or rather, invertible elements.) Some authors who exclude zero from the naturals use the terms Natural Numbers with zero , whole numbers , or counting numbers , denoted W , for the set of nonnegative inte-gers. Others use the notation P for the positive integers if there is no danger of confusing this with the prime numbers. Set theorists often denote the set of all Natural Numbers including zero by a lower-case Greek letter omega : ω. This stems from the identification of an ordinal number with the set of ordinals that are smaller. Algebraic properties The addition and multiplication operations on Natural Numbers have several algebraic properties: • Closure under addition and multiplication: for all Natural Numbers a and b , both a + b and a × b are Natural Numbers. • Associativity: for all Natural Numbers a , b , and c , a + ( b + c ) = ( a + b ) + c and a × ( b × c ) = ( a × b ) × c . • Commutativity: for all Natural Numbers a and b , a + b = b + a and a × b = b × a . • Existence of identity elements: for every natural number a , a + 0 = a and a × 1 = a . • Distributivity of multiplication over addition for all Natural Numbers a , b , and c , a × ( b + c ) = ( a × b ) + ( a × c )
eBook - PDFMathematics with Understanding
The Commonwealth and International Library: Mathematical Topics, Volume 1
- Harold Fletcher, Arnold A. Howell, C. Plumtpon(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
C H A P T E R 6 Natural Numbers AND ADDITION THE Natural Numbers are those we use when we make a count of the elements of any set. This counting is a mapping operation in which one set of objects is placed in one-to-one correspondence with another set. The sets, (a ball, a bat, a wicket), (Tom, Dick, Harry), (a bicycle, a car, a bus), have one property in common, which is the fact that they can be mapped one-to-one exactly. They are equivalent sets. This property is defined by the number of elements in each set, and such a number is called a cardinal number. Sets can be arranged by their cardinality. The cardinal numbers obtained in this way are called the counting numbers, but for mathemati-cal purposes they are called the set of Natural Numbers, and are represent-ed by the numerals {1,2,3,4, . . . } . The condition governing this set of Natural Numbers is that each element represents a class of equivalent sets. It is through experiences of three pens, three dolls, three cars and so on that a child eventually understands the abstract notion of three. The difficulty about number is that it is abstract and therefore not easy to illustrate. It is quite easy to produce a set of three objects, but to point out this set is not to point out the number three. We record how many elements this set contains by writing a numer-al, 3. We are not pointing out the number 3, for only after vast experience of equivalent sets of three objects does the abstract idea of a whole class of three, that is, threeness, become part of a person's abstract thought of number. As an analogy the idea of blueness only becomes part of us after many experiences of objects which we describe as blue. An out-standing property of the Natural Numbers might be described as size, resulting in a definite order. This means that there is a first number, 1, 111 112 MATHEMATICS WITH UNDERSTANDING — BOOK 1 and that each number has one, and only one, definite successor and so the set is an infinite series.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
The oldest and most classical set-theoretic definition of the Natural Numbers is the definition commonly ascribed to Frege and Russell under which each concrete natural number n is defined as the set of all sets with n elements. This may appear circular, but can be made rigorous with care. Define 0 as {{ }} (clearly the set of all sets with 0 elements) and define S ( A ) (for any set A ) as { x ∪ { y } | x ∈ A ∧ y ∉ x }. Then 0 will be the set of all sets with 0 elements, 1 = S (0) will be the set of all sets with 1 element, 2 = S (1) will be the set of all sets with 2 elements, and so forth. The set of all Natural Numbers can be defined as the intersection of all sets containing 0 as an element and closed under S (that is, if the set contains an element n , it also contains S ( n )). One could also define finite independently of the notion of natural number, and then define Natural Numbers as equivalence classes of finite sets under the equivalence relation of equipollence. This definition does not work in the usual systems of axiomatic set theory because the collections involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New Foundations (and in related systems known to be relatively consistent) and in some systems of type theory.
eBook - PDFSet Theory
A First Course
- Daniel W. Cunningham(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
4 The Natural Numbers In this chapter, it shall be shown that number theory can be embedded within set theory. One consequence of this embedding is that all of the theorems in number theory can be proven from the axioms of set theory. In order to show that number theory is, in fact, a special branch of set theory, we must first represent each natural number as a set. How can one define the Natural Numbers in set theory? To answer this question, we next consider a set theoretic construction that makes sense for any set. Definition 4.0.1. For each set x , the successor x + is the set that is obtained by adjoining x to the elements of x , namely, x + = x ∪ { x } . We note the following three properties concerning the successor of a set x : 1. a ∈ x + iff ( a ∈ x ∨ a = x ). 2. x ∈ x + . 3. x ⊆ x + . Using the successor operation, we can now construct, in set theory, the first few Natural Numbers as follows: 0 = ∅ . 1 = 0 + = 0 ∪ { 0 } = { 0 } . 2 = 1 + = 1 ∪ { 1 } = { 0 , 1 } . 3 = 2 + = 2 ∪ { 2 } = { 0 , 1 , 2 } . 4 = 3 + = 3 ∪ { 3 } = { 0 , 1 , 2 , 3 } . We note some interesting properties that these “Natural Numbers” possess (see the above bulleted list): 1. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ 5 ∈ · · · . 2. 0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ 4 ⊆ 5 ⊆ · · · . 84 The Natural Numbers 4.1 Inductive Sets • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • The existence of an infinite set is crucial in modern set theory and mathematics. Zermelo realized that one cannot prove that an infinite set exists and thus found a fairly simple way to assert the existence of an infinite set. Infinity Axiom . There is a set A that contains the empty set as an element and whenever x ∈ A , then x + ∈ A . Using the infinity axiom, we shall prove that there is a set consisting of only the Natural Numbers; this set is denoted by ω . First, we define a property of a set that will ensure that each natural number belongs to such a set.
eBook - PDFBackgrounds Of Arithmetic And Geometry: An Introduction
An Introduction
- Dan Branzei, Radu Miron(Authors)
- 1995(Publication Date)
- World Scientific(Publisher)
CHAPTER D ARITHMETIC Contemporary Mathematics uses two methods for its edification: the constructive method and the axiomatic one. The first method, the constructive one, consists in defining the notions involved in mathematical theories in a direct, effective manner, as it has been done in the set theory in the preceding chapter. We will apply this idea to the construction of some set of numbers. The axiomatic method will be used in the other chapters. §l.The set N of Natural Numbers The problems discussed in the previous chapter regarding cardinal numbers find their first application in defining the set N of Natural Numbers. The basic idea, in constructing this set, has been suggested by Frege and Russel, and that is why it is called the Frege -Russell theory of Natural Numbers. By using the void set 0 , whose uniqueness has already been remarked, let us consider the sequence: (1.1) 0 , { 0 } , { 0 , { 0 } } , { 0 , { 0 } , { 0 , {0}}}, ... in which each term, beginning with the second one, is the set whose elements are all the preceding terms in the sequence. Definition 1.1 We call Natural Numbers the cardinal numbers of the sets in the sequence (I. I). We note the Natural Numbers with 0, 1, 2, ... and their set with N, respectively. From the way of constructing the sequence (1.1) it results n = {0, 1, 2, ..., n -1 } We recall that the cardinal n sets are called finite, those of cardinal No = N being called countable. On the basis of the definition of order relation between cardinal numbers it follows that 0 < n < No. Therefore: the set N is not finite and any element n £ N is smaller than ffi. It is obvious that the order relation on N, determined by the order between (1.1) 0 , { 0 } , { 0 , { 0 } } , { 0 , { 0 } , { 0 , {0}}}, ... n = {0, 1, 2, ..., n -1 } Arithmetic 19 cardinal numbers is equivalent to the relation of natural order given by the succession in the sequence (1.1).
eBook - PDFLearning and Teaching Mathematics using Simulations
Plus 2000 Examples from Physics
- Dieter Röss(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
The simulation in Figure 3.1 visualizes the sharp threshold that nature imposes for spontaneously grasping the number of elements of a set. In this simulation, points 3.2 Whole numbers 15 are shown in a random arrangement that can be spontaneously grasped as a group. The number changes with a frequency that can be specified between 1 and a max-imum number. You can establish experimentally where your own grasping thresh-old lies. The description pages of the simulation contain further details and hints for experiments. Even numbers are a multiple of the number 2; a prime number cannot be decom-posed into a product of Natural Numbers, excluding 1. The lower limit of the Natural Numbers is the unity 1. This number had a close to mystical meaning for number theoreticians of antiquity, as the symbol for the unity of the computable and the cosmos. It also has a special meaning in modern arithmetic as that number which, when multiplied with another number, produces the same number again. There is, however, no upper limit of the Natural Numbers: for each number there exists an even larger number. As a token for this boundlessness, the notion of infinity developed, with the symbol 1 , which does not represent a number in the usual sense. Already, the preplatonic natural philosophers ( Plato himself lived from 427–347 BC) worked on the question of the infinite divisibility of matter (If one divides a sand grain infinitely often, is it then still sand?) and time (if one adds to a given time interval infinitely often half of itself, will that take infinitely long?) Zenon of Elea (490–430 BC) showed in his astute paradoxes, Achilles and the tor-toise and the arrows , 11 that the ideas of movement and number theory at the time were in contradiction to each other. Subtraction is the logical inversion of addition: for Natural Numbers it is only permissible if the number to subtract is smaller than the original number by at least 1.
eBook - PDF- John Newsome Crossley(Author)
- 1987(Publication Date)
- World Scientific(Publisher)
In our view the approaches just described mark a big change in the study of the Natural Numbers. Hitherto the Natural Numbers had been taken for granted, used in counting objects in the world and much theoretical work had been done with a collection of numbers which were, to put it a little crudely, just as sound and real as solid bodies studied in mechanics. Dedekind's and Peano's moves to characterize a structure or system of Natural Numbers immediately take us into the abstract realm of infinity (as Dedekind himself shows). Here we are dealing with something different, for there is no possibility of verifying whether any infinite collection of 49 The Natural Numbers objects exists in the universe. At this point we may say we shift from applied mathematics to pure mathematics : from the study of the world by the use of mathematics to the study of the world of mathematics itself. One cannot argue that Dedekind's Natural Numbers are identical with the numbers we use in everyday life, for the former have a precise definition, the latter none. It is, of course, perfectly true that no-one has (Vet?) found any discrepancy between results obtained by using the Dedekind-Peano axioms and any obtained by (market-place) calculation. With Dedekind the system of Natural Numbers is characterized by a starting point (the axiom that says 1 is a natural number) and a process of going from one number to the (uniquely determined) next number, together with saying that this is the only way to obtain a number. Here there is no clear need for any idea of counting any longer, but only a purely abstract construction. There is, however, a difficulty. For in order to show that at least one system with these properties exists, Dedekind turns to a proof which is remarkably different from the technical proofs which precede it in his book [1912]. He writes: 66. Theorem. There exist infinite systems.
eBook - PDF- Karl Smith(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
If Integers 5.3 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 5.3 Integers 195 this new number is annexed to the set of Natural Numbers, a set called the set of whole numbers is formed: W 5 {0, 1, 2, 3, 4, … } This one annexation to the existing numbers satisfied society’s needs for several thousand years. However, as society evolved, the need for bookkeeping advanced, and eventually the need to answer this: Can we annex new numbers to the set W so that it is possible to carry out all subtractions? The numbers that need to be annexed are the opposites of the Natural Numbers. The opposite of 3, which is denoted by 2 3, is the number that when added to 3 gives 0. If we add these opposites to the set W , we have the following set: Z 5 { … , 2 3, 2 2, 2 1, 0, 1, 2, 3, … } This set is known as the set of integers. It is customary to refer to certain subsets of Z as follows: 1. Positive integers: {1, 2, 3, 4, … } 2. Zero: {0} 3. Negative integers: { 2 1, 2 2, 2 3, … } Now with this new enlarged set of numbers, are we able to carry out all possible additions, subtractions, and multiplications? Before we answer this question, let’s review the process by which we operate within the set of integers. It is assumed that you have had an algebra course, so the following summary is intended only as a review. You might recall that the process for describing the operations with integers requires the notion of absolute value , which represents the distance of a number from the origin when plotted on a number line.
eBook - PDF- Lev D. Beklemishev(Author)
- 2000(Publication Date)
- Elsevier Science(Publisher)
It is this amplification which liberates subtraction from the restrictions to which its application is subjected in the system of Natural Numbers. Consequently, we can perform within the system of integers every 98 EXTENSION OF THE SYSTEM OF Natural Numbers operation which was possible within the system of Natural Numbers. In addition, we can perform within the system of integers certain operations which were impossible within the system of Natural Numbers. Example 1. Show that the above relation R between ordered pairs of Natural Numbers is reflexive, symmetric and transitive. Example 2. Prove the associative law for the addition of integers. Example 3. Discuss the introduction of the usual order and of the usual multiplication into the system of integers, and prove the most important laws concerning these notions. E x am p l e 4. We can give the following alternative to the above introduction of the integers. Let 1 be the set of all Natural Numbers a such that either ~(a)=l or n,,(a) = 1. For the elements of 1 we define addition +1 and multi-plication . I' as follows. (i) a+1b will be the number c in 1 such that: nl(a) +n,(b) + n,,(c) = n.(a) +n.(b) +nl(c). (ij) a' 1b will be the number d in 1 such that: nl(a) ·n,(b)+n.(a) . n.(b)+n.(d)=n,(a) ·n.(b)+n.(a) ·n,(b)+~(d). Complete the treatment of the integers along these lines. -Hint: ct. Section 53. 39. THE SYSTEM OF RATIONAL NUMBERS It is well-known that the system of rational numbers constitutes likewise an extension of the system of integers; we amplify the system of integers to liberate ourselves from the restrictions to which the application of division is subjected within the system of integers. The passage from the system of integers to the system of rational numbers can be effected as follows. Let S be the set of all ordered pairs (a, b) of integers a and b with b of O. Then we can define in S, a relation R as follows: R[(a, b), (c, d)], if and only if a·d=b·c.
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