Mathematics
Sum of Natural Numbers
The sum of natural numbers refers to the result of adding all the positive integers from 1 to a given number. This can be calculated using the formula n(n+1)/2, where n is the given number. The concept is fundamental in mathematics and has applications in various areas such as arithmetic progression and series.
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8 Key excerpts on "Sum of Natural Numbers"
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.
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)
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. We record like this: Natural numbers { 1 , 2 , 3, 4 , 5, . . . } Fundamental Operations for Natural Numbers Addition The union of two sets A and Β will be a set which contains either the members of set A or Β or both set A and B. A - {3. 4. 5. 6} Β - {6. 7. 8} AÖB - {3, 4 , 5, 6, 7,8} n{A) = 4 n(B) -3 η(ΑΌΒ) = 6 We observe 4 + 3 ^ 6 . The element 6 is a member of both sets and is counted twice when the sets are numerated separately but once when considering the union of the sets. Study these examples. A = {3, 4 , 5, 6} B = {4, 5, 6} ADB-={3, 4 , 5, 6} n{A) = 4 n{B) -3 n(AÖB) = 4 but 4 + 3 ^ 4 The sets are not disjoint. There are common members, {4, 5, 6}. A - {3, 4 , 5, 6} Β = {5, 6} AUB = {3, 4 , 5, 6} /ι(Λ) -4 « ( 5 ) -2 / 7 ( , 4 U £ ) : = 4 but 4 + 2 ^ 4 NATURAL NUMBERS AND ADDITION The sets are not disjoint. There are common members, {5, 6}. A = {3, 4, 5, 6} Β = {7, 8} A{JB={3, 4, 5, 6, 7, 8} n(A) =4 n(B) =2 n(Af]B)=6 and 4 + 2 -6 The sets are disjoint. There are no common members. If we consider defining addition in terms of the union of sets we can only do this if the members of the sets are non-overlapping. There must be no common members, the intersection of the sets is an empty set. When we meet sets which have no common members we refer to the sets as disjoint. Thesefore, addition can be defined as the union of disjoint sets or sets which have no common members. The cardinal number of the union of disjoint sets will be the sum of the cardinal numbers of each set. Given any two natural numbers, we can add them, the solution being known as the sum of the two natural numbers.
eBook - ePub- Karel Hrbacek, Thomas Jech(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
n elements.This idea still has a fundamental deficiency. We have defined 0, 1, 2, 3, 4, and 5 and could easily define 17 and—not so easily—324. But no list of such definitions tells us what a natural number is in general. We need a statement of the form: A set n is a natural number if ... . We cannot just say that a set n is a natural number if its elements are all the smaller natural numbers, because such a “definition” would involve the very concept being defined.Let us observe the construction of the first few numbers again. We defined 2 = {0,1}. To get 3, we had to adjoin a third element to 2, namely, 2 itself:3 = 2 ∪ { 2 } = { 0 , 1 } ∪ { 2 } .Similarly,4 =3 ∪ { 3 } = { 0 , 1 , 2 } ∪ { 3 } .5 =4 ∪ { 4 } , etc .Given a natural number n , we get the “next” number by adjoining one more element to n, namely, n itself. The procedure works even for 1 and 2: 1 = 0 U {0}, 2 = 1 U {1}, but, of course, not for 0, the least natural number.These considerations suggest the following.1.1 DefinitionThe successor of a set x is the set S (x ) = x U {x }.Intuitively, the successor S (n ) of a natural number n is the “one bigger” number n+1 . We use the more suggestive notation n + 1 for S (n ) in what follows. We later define addition of natural numbers (using the notion of successor) in such a way that n + l indeed equals the sum of n and 1. Until then, it is just a notation, and no properties of addition are assumed or implied by it.We can now summarize the intuitive understanding of natural numbers as follows:- 0 is a natural number.
- If n is a natural number, then its successor n + 1 is also a natural number.
- All natural numbers are obtained by application of (a) and (b), i.e., by starting with 0 and repeatedly applying the successor operation: 0, 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, .. . etc.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
Sometimes the set of natural numbers with 0 included is called the set of whole numbers or counting numbers . 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 integers. 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. Constructions based on set theory A standard construction A standard construction in set theory, a special case of the von Neumann ordinal construction, is to define the natural numbers as follows: We set 0 := { }, the empty set, and define for every set a . S ( a ) is the successor of a , and S is called the successor function. ________________________ WORLD TECHNOLOGIES ________________________ If the axiom of infinity holds, then the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function.
eBook - PDF- Karl Smith(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
. . } This set of numbers is called the set of counting numbers or natural numbers. Let’s assume that you understand what the numbers in this set represent, and you understand the operation of addition, 1 . That is, we assume, without definition, knowledge of the operation of addition of natural numbers. 9 9 8 8 4 4 3 3 6 6 7 7 1 1 2 2 5 5 out of order There are a few self-evident properties of addition for this set of natural numbers. They are called “self-evident” because they almost seem too obvious to be stated explic-itly. For example, if you jump into the air, you expect to come back down. That assump-tion is well founded in experience and is also based on an assumption that jumping has certain undeniable properties. But astronauts have found that some very basic assump-tions are valid on earth and false in space. Recognizing these assumptions (properties, axioms, laws, or postulates) is important. Closure Property When we add or multiply any two natural numbers, we know that we obtain a natural number. This “knowing” is an assumption based on experiences (inductive reasoning), but we have actually experienced only a small number of cases for all the possible sums and products of numbers. The scientist—and the mathematician in particular—is very skeptical about making assumptions too quickly. The assumption that the sum or product of two natural numbers is a natural number is given the name closure and is referred to as the closure property. The property is phrased in terms of sets and operations. Think of a set as a “box”; there is a label on the box—say, addition. If all additions of numbers in the box have answers that are already in the box, then we say the set is closed for addition. If there is at least one answer that is not contained in the box, then the set is said to be not closed for that operation.
eBook - PDFNumbers and Functions
Steps into Analysis
- R. P. Burn(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
PART I Numbers 1 Mathematical induction Preliminary reading: Rosenbaum, Polya ch. 7 . Concurrent reading: Sominskii. Further reading: Thurston chs 1 and 2 , Ledermann and Weir. The set of all whole numbers { 1 , 2 , 3 , . . . } is often denoted by N . We will usually call N the set of natural numbers , and, sometimes, the set of positive integers . Note that N excludes 0. 1 1 2 + 2 2 + 3 2 + . . . + n 2 = n ( n + 1 )( 2 n + 1 ) 6 . Is this proposition true or false? Test it when n = 1, n = 2 and n = 3. How many values of n should you test if you want to be sure it is true for all natural numbers n ? If we write f ( n ) = n ( n + 1 )( 2 n + 1 ) 6 , show that f ( n ) + ( n + 1 ) 2 = f ( n + 1 ). Now suppose that the proposition with which we started holds for some particular value of n : add ( n + 1 ) 2 to both sides of the equation and deduce that the proposition holds for the next value of n . Since we have already established that the proposition holds for n = 1 , 2, and 3, the argument we have just formulated shows that it must hold for n = 4, and then by the same argument for n = 5, and so on. 2 When n is a natural number, 6 n − 5 n + 4 is divisible by 5. Check this proposition for n = 1 , 2 and 3. By examining the difference between this number and 6 n + 1 − 5 ( n + 1 ) + 4, show that if the proposition holds for one value of n , it holds for the succeeding value of n . When you have done this you have 3 4 1 Mathematical induction established the two components of the proof of the proposition by mathematical induction. 3 Prove the following propositions by induction (some have easy alternative proofs which do not use induction): (i) ( Triangular numbers ) 1 + 2 + 3 + . . . + n = 1 2 n ( n + 1 ) , (ii) a + ( a + d ) + ( a + 2 d ) + . . . + ( a + ( n − 1 ) d ) = 1 2 n ( 2 a + ( n − 1 ) d ) , (iii) 1 3 + 2 3 + 3 3 + . . . + n 3 = [ 1 2 n ( n + 1 ) ] 2 , (iv) 1 · 2 + 2 · 3 + 3 · 4 + . . . + n ( n + 1 ) = 1 3 n ( n + 1 )( n + 2 ) , (v) 1 + 2 + 4 + .
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 - 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.
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