Mathematics
Number
A number is a mathematical concept used to quantify, count, and measure. It can represent quantities, positions, or identifiers. In mathematics, numbers can be classified into various types, such as natural numbers, integers, rational numbers, and real numbers, each with its own properties and characteristics.
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No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Number and Set Number A Number is a mathematical object used in counting and measuring. A notational symbol which represents a Number is called a numeral, but in common usage the word Number is used for both the abstract object and the symbol, as well as for the word for the Number. In addition to their use in counting and measuring, numerals are often used for labels (telephone Numbers), for ordering (serial Numbers), and for codes (e.g., ISBNs). In mathematics, the definition of Number has been extended over the years to include such Numbers as zero, negative Numbers, rational Numbers, irrational Numbers, and complex Numbers. Certain procedures which take one or more Numbers as input and produce a Number as output are called numerical operations. Unary operations take a single input Number and produce a single output Number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. More common are binary operations which take two input Numbers and produce a single output Number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic. Classification of Numbers Different types of Numbers are used in different cases. Numbers can be classified into sets, called Number systems. Number systems Natural (0), 1, 2, 3, 4, 5, 6, 7, ..., n Integers − n , ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 , ..., n Positive integers 1, 2, 3, 4, 5, ..., n Rational a ⁄ b where a and b are integers and b is not zero Real The limit of a convergent sequence of rational Numbers ________________________ WORLD TECHNOLOGIES ________________________ Complex a + bi where a and b are real Numbers and i is the square root of −1 Natural Numbers The most familiar Numbers are the natural Numbers or counting Numbers: one, two, three, and so on.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Number and Set Number A Number is a mathematical object used in counting and measuring. A notational symbol which represents a Number is called a numeral, but in common usage the word Number is used for both the abstract object and the symbol, as well as for the word for the Number. In addition to their use in counting and measuring, numerals are often used for labels (telephone Numbers), for ordering (serial Numbers), and for codes (e.g., ISBNs). In mathematics, the definition of Number has been extended over the years to include such Numbers as zero, negative Numbers, rational Numbers, irrational Numbers, and complex Numbers. Certain procedures which take one or more Numbers as input and produce a Number as output are called numerical operations. Unary operations take a single input Number and produce a single output Number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. More common are binary operations which take two input Numbers and produce a single output Number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic. Classification of Numbers Different types of Numbers are used in different cases. Numbers can be classified into sets, called Number systems. Number systems Natural (0), 1, 2, 3, 4, 5, 6, 7, ..., n Integers − n , ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ..., n Positive integers 1, 2, 3, 4, 5, ..., n Rational a ⁄ b where a and b are integers and b is not zero Real The limit of a convergent sequence of rational Numbers ________________________ WORLD TECHNOLOGIES ________________________ Complex a + bi where a and b are real Numbers and i is the square root of −1 Natural Numbers The most familiar Numbers are the natural Numbers or counting Numbers: one, two, three, and so on.
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
These two ideas are so strongly interwoven that we adopt the hypothesis as a definition: an infinite mathematical line is simply a representation of all of the points of R , and only those points. This is a very useful visualisation, but is of lit-tle use in questions concerning the explicit construction of the various real Numbers corresponding to different points on the line. Some real Numbers can be constructed rather simply: the quadratic equation x 2 = 2 provides us with the real Number which we call √ 2, and any rational Number has a very simple representation n/m , which is the solution x to the linear equa-tion mx = n . How far can we take these ideas? How many real Numbers can we construct starting from the integers and using only basic algebraic operations? We make a definition: • An algebraic Number is any real Number which is the solution to a polynomial equation with integer coefficients. Clearly all rational Numbers, and all Numbers √ n , where n is a natural Number, are algebraic Numbers. Furthermore, it is also possible to show, using a little theory of polynomials, that adding, subtracting, multiplying and dividing pairs of non-zero algebraic Numbers yields Numbers which are also algebraic. Therefore the set of all algebraic Numbers forms a field and may thus be considered as a well-defined Number system. Although it is a pleasant, mathematically clean notion that all of the real Numbers could be generated in this way from the integers, it turns out that the real Number Numbers 83 system is far more complex than simply the set of algebraic Numbers. There exist real Numbers which are not algebraic. Since such Numbers transcend the algebraic construction, they are called transcendental : • A transcendental Number is any real Number which is not algebraic. We have actually met one of these transcendental Numbers already: π .
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.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Types of Numbers 1. Natural Number Natural Numbers can be used for counting (one apple, two apples, three apples, ...) from top to bottom. In mathematics, natural Numbers are the ordinary counting Numbers 1, 2, 3, ... (some-times 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. ________________________ WORLD TECHNOLOGIES ________________________ 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 - PDFMathematics
A Second Start
- S. Page, J Berry, H Hampson(Authors)
- 2002(Publication Date)
- Woodhead Publishing(Publisher)
The individual items in a set are called the elements of the set. Sometimes all the elements may be similar as with the individual keys in a bunch. Sometimes it may only be membership of the set itself that the elements have in common. A plumber's toolbox might contain a pipe-wrench, a saw, some nails, a piece of solder and a ham sandwich! The objects still form a set. 10 Mathematics: A Second Start [Ch. 1 A subset is a set contained within some other set. For example, a mathematics class is a set. It will contain many subsets: The set of female students. The set of male students. The set of students (male and female) with blue eyes. The set of students who left their homework on the bus (again!). And so on. We will confine the rest of our discussion to what this book is all about: sets of Numbers. T H E NATURAL NumberS The simplest Numbers are the ones used for counting. Early civilisations all needed ways of keeping a check on Numbers: how much grain was stored for the winter, how many soldiers in the army, how many more days left until winter sets in. Thus people would represent Numbers by a notch on a stick, knots or beads on a cord or marks on a clay tablet. It was therefore natural to invent systems of marks and symbols to represent Numbers and the relationships between them. Some of these systems have disappeared because they were too restricted or clumsy to advance beyond simple counting. Try multiplying XVIII by CXLIV using the Roman system and you will see what we mean! (The answer, by the way, is MMDXCII.) The Numbers used for counting, i.e. 1,2,3,4... etc. are called natural Numbers. They are also called integers or whole Numbers. We are used to counting in the denary or decimal scale which was invented by Hindu mathematicians and extended by the Arabs. Zero was invented later and is needed when subtraction causes no result. The Number Line We can represent Numbers by using a line. The natural Numbers are placed in a line with equal distances between them. Early civilisations would have written their Number line like this; each natural Number is a point on the line. 1 2 3 4 5 etc. • · · · · The first thing we notice is that the natural Numbers have a particular order. Immediately we can begin to use statements like
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
(continued) 2 Chapter 1 ◆ Numerical Computation Number Type Definition Example Explanation Rational Numbers Includes integers and all other Numbers that can be expressed as the quotient of two integers. - - 1 2 , 3 5 , 57 23 , 98 99 , 7 A quotient is a Number formed by dividing one Number by another Number. Irrational Numbers Numbers that can- not be expressed as the quotient of two integers. π 2, 5, 7, 3 These are Numbers that, written as decimals, are nonterminating and nonrepeating. Real Numbers Rational and irrational Numbers combined together into one set of Numbers. Any Number other than a complex Number Rational and irrational Numbers are two separate Number types until we put them together into a single set of “real” Numbers. Complex Numbers Any imaginary Number (non-real) or a combination of a real Number and an imaginary Number Any imaginary Number like 4 - , and any combination like 4 9 - + - When we square a real Number, even a negative one, we get a positive Number. So, taking the square root of a negative Number doesn’t make sense as a real Number, and the result is an imaginary Number. Note that all of the Numbers we will work with in Chapter 1 are real Numbers. (Complex Numbers are covered in Chapter 21.) -1 +1 +2 +3 +4 +5 +6 +7 -2 -3 -4 -5 -6 -7 0 Positive Some positive Numbers are: Negative π √2 9 2 Some negative Numbers are: 2 3 - -5, , -2 , - √5 7 8 2 3 + +5, , +2 , + √5 7 8 5.6394 1.75 7 2 - -√30 FIGURE 1-1 Positional Number Systems A positional Number system is one which the position of a digit determines its value. Our decimal system is positional. Table 1-2 shows the values of positions in a decimal Number. Place Value Each position in a Number has a place value equal to the base of the Number system raised to the power of the position Number. The place values in the decimal Number system, as well as the place names, are shown in Table 1-2. TABLE 1-1 Number Types (continued)
eBook - PDFMathematics NQF2 SB
TVET FIRST
- M Van Rensburg, I Mapaling A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
10 Module 2 Understanding Numbers, their relationships and representation Module 2 Learning Outcomes This module will show you how to do the following: • Unit 2.1: Identify rational and irrational Numbers. • Unit 2.2: Round off rational and irrational Numbers to an appropriate degree of accuracy. • Unit 2.3: Convert rational Numbers between terminating and recurring decimals to the form: a __ b ; a , b ∈ ℤ ; b ≠ 0. • Unit 2.4: Apply the laws of exponents. • Unit 2.5: Rationalise fractions with surd denominators (binomial and monomial denominators), without using a calculator. • Unit 2.6: Add, subtract, multiply and divide simple surds. • Unit 2.7: Manipulate simple technical and non-technical formulae. • Unit 2.7: Solve an unknown variable in simple technical and non-technical formulae. • Unit 2.8: Identify and work with arithmetic sequences and series. Unit 2.1: Rational and irrational Numbers 2.1.1 Number types Table 2.1 summarises the different types of Numbers we use in Mathematics. Table 2.1: Types of Numbers Type of Numbers Examples Hints Even Numbers Even Numbers are divisible by 2 without a remainder, for example: {2; 4; 6; 8; 10; 12; 14 … } The last digit is 0, 2, 4, 6 or 8. Odd Numbers Odd Numbers are not divisible by 2 without a remainder, for example: {1; 3; 5; 7; 9; 11; 13; … } The last digit is 1, 3, 5, 7 or 9. Prime Numbers Prime Numbers are only divisible by 1 and by the Number itself without a remainder. Examples are: {2; 3; 5; 7; 11; 13; 17; 19; … } Prime Numbers have only two factors, itself and 1. Composite Numbers Composite Numbers are those Numbers which contain more than one factor, other than themselves and one. Examples are: {4; 6; 8; 9; 12; 14; 15; 16; 18; 20; 21; … } Composite Numbers are any Number that is not a prime Number. Natural Numbers (counting Numbers) ℕ Numbers we use for counting: ℕ = {1; 2; 3; 4; 5; … } It is natural to count on your 10 fingers.
eBook - PDFBeginning and Intermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Give an example of a Number that is used for counting. 1, 2, 3, etc. 2. Give an example of a Number that is used when dividing a pizza. 1 2 , 2 3 , etc. 3. Give an example of a Number that is used for measuring temperatures that are below zero. 2 3, 2 21 , etc. 4. What other types of Numbers can you think of? We will begin by discussing various sets of Numbers. List the Numbers in a set of real Numbers that are natural, whole, integers, rational, irrational, composite, prime, even, or odd. A set is a collection of objects. For example, the set 5 1, 2, 3, 4, 5 6 Read as “the set with elements 1, 2, 3, 4, and 5.” contains the Numbers 1, 2, 3, 4, and 5. The members , or elements , of a set are listed within braces 5 6 . Two basic sets of Numbers are the natural Numbers (often called the positive inte-gers ) and the whole Numbers . 1 Getting Ready 1.1 THE SET OF NATURAL NumberS (POSITIVE INTEGERS) 5 1, 2, 3, 4, 5, 6, 7, 8, 9 10, . . . 6 Copyright 2015 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. 4 CHAPTER 1 Real Numbers and Their Basic Properties The three dots in the previous definitions, called ellipses , indicate that each list of Numbers continues on forever. We can use whole Numbers to describe many real-life situations. For example, some cars might get 30 miles per gallon (mpg) of gas, and some students might pay $1,750 in tuition. Numbers that show a loss or a downward direction are called negative Numbers , and and a temperature of 20° below zero can be denoted as 2 20° . The negatives of the natural Numbers and the whole Numbers together form the set of integers .
eBook - PDF- Hugo D. Junghenn(Author)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 1 The Real Number System If the notion of limit is the cornerstone of analysis, then the real Number system is the bedrock. In this chapter we provide a description of the real Number system that is sufficiently detailed to allow a careful development of limit in the various forms that appear in this book. The real Number system is defined as a nonempty set R together with two algebraic operations, called addition and multiplication , and an ordering less than that collectively satisfy three sets of axioms: the algebraic or field axioms , the order axioms , and the completeness axiom . These are discussed in Sections 1.2–1.4. We begin, however, with a brief description of how the real Number system may be constructed from a more fundamental Number system. 1.1 From Natural Numbers to Real Numbers A rigorous construction of the real Number system starts with the set of natural Numbers (positive integers) N and then proceeds to the set of integers Z , the rational Number system Q , and, finally, the real Number system R . In this approach the natural Numbers are assumed to satisfy a set of axioms called the Peano Axioms . These are used to define the operations of addition and multiplication in N . Subtraction is introduced by enlarging the system of natural Numbers to Z , thereby allowing solutions of all equations of the form x + m = n , m, n ∈ Z . To obtain division, Z is enlarged to Q by forming all quotients m/n , where m, n ∈ Z , n = 0. In this system, one may solve all equations of the form ax + b = c , a = 0. The final step, the construction of R from Q , may be viewed as “filling in the gaps” of the rational Number line, these gaps corresponding to the so-called irrational Numbers . 1 For the details of this “bottom up” approach, the interested reader is referred to [ 7 ] or [ 10 ]. We shall instead take a “top down” approach, describing the real Number system axiomatically.
eBook - PDF- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
2 5 1.414213562373095 . . . 3 5 1.73205080756887 . . . p 5 3.14159265358979 . . . The set of real Numbers is composed of the rational Numbers along with the irrational Numbers. Every real Number is either a rational Number or an irrational Number. The following tree diagram summarizes the various classifications of the real Number system. Real Numbers Rational Numbers Irrational Numbers Integers Nonintegers 0 We can trace any real Number down through the diagram as follows: 7 is real, rational, an integer, and positive 2 2 3 is real, rational, noninteger, and negative 7 is real, irrational, and positive 0.38 is real, rational, noninteger, and positive Remark: We usually refer to the set of nonnegative integers, 5 0, 1, 2, 3, . . . 6 , as the set of whole Numbers , and we refer to the set of positive integers, 5 1, 2, 3, . . . 6 , as the set of natural Numbers . The set of whole Numbers differs from the set of natural Numbers by the inclusion of the Number zero. The concept of subset is convenient to discuss at this time. A set A is a subset of a set B if and only if every element of A is also an element of B . This is written as A # B and read as “ A is a subset of B .” For example, if A 5 5 1, 2, 3 6 and B 5 5 1, 2, 3, 5, 9 6 , then A # B because every element of A is also an element of B . The slash mark denotes nega-tion, so if A 5 5 1, 2, 5 6 and B 5 5 2, 4, 7 6 , we can say that A is not a subset of B by writ-ing A s B . Figure 1.3 represents the subset relationships for the set of real Numbers. Refer to Figure 1.3 as you study the following statements, which use subset vocabulary and sub-set symbolism. 1. The set of whole Numbers is a subset of the set of integers. 5 0, 1, 2, 3, . . . 6 # 5 . . . , 2 2, 2 1, 0, 1, 2, . . . 6 2. The set of integers is a subset of the set of rational Numbers. 5 . . . , 2 2, 2 1, 0, 1, 2, . . . 6 # 5 x 0 x is a rational Number 6 3. The set of rational Numbers is a subset of the set of real Numbers.
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