Mathematics
Types of Numbers
In mathematics, numbers can be classified into different types based on their properties and characteristics. Some common types of numbers include natural numbers (counting numbers), whole numbers (natural numbers and zero), integers (positive and negative whole numbers), rational numbers (fractions), and irrational numbers (non-repeating, non-terminating decimals). Each type of number has its own unique properties and uses in mathematical operations.
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eBook - PDF- Owen Bishop(Author)
- 2013(Publication Date)
- Newnes(Publisher)
Natural numbers, or counting numbers, are the numbers we use when answering the question *How many?'. They are always positive. For example, 1, 2, 3, 4, . . . 100, . . . and 2348, are all natural numbers. Whole numbers are the natural numbers together with zero. Integers comprise all the whole numbers and their negatives. For example, -14, -7 , 0, 3, and 12 are all integers. 4 UNDERSTAND ELECTRICAL AND ELECTRONICS MATHS Natural numbers Whole numbers -5 -4 -3 -2 -1 Integers Figure 1.1 Rational numbers comprise all the integers and all numbers that can be represented as ratios. A ratio is one integer divided by another integer, provided that the dividing integer is not zero. Examples of ratios are 2/5, 34/67, and 233/8. The ratios fill up most of the length between the integers on the number line. Ratios can also be expressed as decimal fractions. For example, 2/5 is a ratio and is also written as 0.4. The decimal form of 2/5 terminates with one decimal place (1 dp). In other words, it needs only one decimal place to express 2/5 exactly. In some cases the ratio produces a recurring decimal. For example 7/3 is 2.333 333 333 . . . indefinitely. This is usually written 2.3. Rational numbers include the natural numbers, whole numbers and integers because an integer such as 5, for example, can be represented by a ratio such as 20/4, or 5/1. Most values can be obtained by a suitable choice of numerator and divisor, so the rational numbers almost completely fill the number line. Figure 1.1 shows gaps among the rational numbers which indicate that there are some values that cannot be filled by ratios. The gaps are actually narrower than shown, and there are many more of them. Real numbers comprise the rational numbers and the irrational numbers. Irrational numbers are numbers that are written as non-recurring, non-terminating decimals. The sequence of digits after the decimal point continues indefinitely without ever repeating. Examples of irrational numbers include
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)
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 - 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)
Because the set of natural numbers and the set of whole numbers are included within the set of integers, these sets are called subsets of the set of integers. Integers cannot describe every real-life situation. For example, a student might study 3 1 2 hours, or a TV set might cost $217.37. To describe these situations, we need fractions, more formally called rational numbers. We cannot list the set of rational numbers as we have listed the previous sets in this section. Instead, we will use set-builder notation . This notation uses a variable (or vari-ables) to represent the elements in a set and a rule to determine the possible values of the variable. THE SET OF INTEGERS 5 . . . , 2 5, 2 4, 2 3, 2 2, 2 1, 0, 1, 2, 3, 4, 5, . . . 6 they are denoted with a 2 sign. For example, a debt of $1,500 can be denoted as 2 $1,500, 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. 1.1 Real Numbers and Their Graphs 5 Unless otherwise noted, all content on this page is © Cengage Learning. p < 3.141592654 Using a scientific calculator, press p . Using a graphing calculator, press 2nd ^ ( p ) ENTER . Read < as “is approximately equal to.” 2 < 1.414213562 Using a scientific calculator, press 2 . Using a graphing calculator, press 2nd x 2 ( ! ) 2 ENTER . If we combine the rational and the irrational numbers, we have the set of real numbers . The previous notation is read as “the set of all numbers x such that x is either a rational number or an irrational number.” Figure 1-1 illustrates how the various sets of numbers are interrelated.
eBook - PDFLearning and Teaching Mathematics using Simulations
Plus 2000 Examples from Physics
- Dieter Röss(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
If one applies the operations mentioned above to irrational numbers, then this does not lead to a more generalized number. Rational numbers constitute a countable set – they can be ordered in such a way that they constitute a countable sequence. The irrational numbers, on the other hand, do not constitute a countable set. In this sense, there are more irrational than rational numbers. 3.4.1 Algebraic numbers The need to introduce numbers that are not rational was recognized by the Pythagore-ans ( Pythagoras , 570–510 BC, mathematician and natural philosopher in the Greek colony Metapont in southern Italy) during their reflections on the calculation of right triangles with a hypotenuse c and legs a and b . In the domain of integers, there are only a few solutions to a right triangle, the Pythagorean triples, which are often used in homework problems: ( 3; 4; 5 I 6; 8; 10 ; 5; 12; 13 ; 8; 15; 17 ; 7; 24; 25 ; 9; 12; 15 ; 10; 24; 26 ; etc.) Theorem of Pythagoras: a 2 C b 2 D c 2 ! c D p a 2 C b 2 Example of an integer solution: c D p 3 2 C 4 2 D p 25 D 5 Example of a rational solution: c D s 3 2 2 C 2 2 D r 25 4 D 5 2 Example of an irrational solution: c D p 1 2 C 1 2 D p 2 Numbers that are generally obtained as the solutions of polynomial equations with rational coefficients, i.e. that are their roots, are designated as algebraic numbers. They include both rational and irrational numbers. 3.4.2 Transcendental numbers Irrational numbers that are not a root of a polynomial with rational coefficients are called transcendental numbers. Here transcendental simply means going beyond the rational numbers and does not have any mystical connotation whatsoever.
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.
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
It is therefore impossible to count the real numbers. We have 12 Note that there is some ambiguity in this process of which we must be aware; for example the number 0 . 49999 . . . actually equals 0.5, as is seen by applying the fundamen-tal axiom to the sequence of rational numbers 4 . 9 , 4 . 99 . 4 . 999 . 4 . 9999 , . . . . To prove the uncountability result properly requires us to take this technical point into consideration. 82 A Mathematical Bridge thus at last managed to go beyond ∞ and have unearthed a new type of infinite number: the number of real numbers. We call this uncountably infinite number the continuum C = | R | . It is well worth stressing at this point that C is larger than ∞ from the point of view of infinities . There are hugely more reals than there are rationals. In fact, by any reasonable definition, virtually every real number is irrational because we can show that between any two real numbers there are uncountably many irrational numbers, yet only countably many rationals. Thus the fundamental axiom opens a numerical floodgate, drowning the rational numbers in a sea of irrational ones. 2.2.3 Algebraic and transcendental numbers The real numbers are frequently pictured in a geometrical context as cor-responding to all of the points on an idealised mathematical line. Pick any point on the line and it will correspond to a real number; pick any real number and it will correspond to a point on the line. 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.
eBook - PDF- Karl Smith(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
These are the entries of the second diagonal. The answers of 2 in. and ! 2 in . are independent from the number of stairs. 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 5.6 Groups, Fields, and Real Numbers 217 You are familiar with the rational numbers (fractions, for example) and the irrational numbers ( p or square roots of certain numbers, for example), and now we wish to consider the most general set of numbers to be used in elementary mathematics. This set consists of the annexation of the irrational numbers to the set of rational numbers and is called the set of real numbers. Definition of Real Numbers Groups, Fields, and Real Numbers 5.6 Real Numbers The set of real numbers, denoted by R , is defined as the union of the set of rationals and the set of irrationals. The set of real numbers is the most common set used in elementary math. Decimal Representation Let’s consider the decimal representation of a real number. If a number is rational , then its decimal representation is either terminating or repeating. Example 1 Convert fractional form to decimal form Find the decimal representation of each of the given rational numbers. a. 1 4 b. 5 8 c. 58 10 d. 2 3 e. 1 6 f. 5 11 g. 1 7 Solution a. 1 4 5 0.25 1 4 4 This is a terminating decimal. b. 5 8 5 0.625 5 4 8 This is a terminating decimal. c. 58 10 5 5.8 58 4 10 This is a terminating decimal. d. 2 3 5 0.666 . . . 2 4 3 Display: . 6666666667 This is a repeating decimal.
eBook - PDF- John B. Conway(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
In others it is the properties of the rational numbers. We are going to start somewhere in between these two. We will definitely assume you know the natural numbers. There are, however, some properties of N that I think you will readily 1.2 The Real Numbers 9 accept but may not have seen explicitly stated. Here is one that is a first cousin of the fact that there is no largest integer. No proof is given. 1.2.5. Lemma. If m, n ∈ N, then there is a natural number N such that Nn > m The definition of Q has already been given in (1.1.2). It’s an algebraically defined entity and we are certainly going to assume the reader is knowledgeable of all its algebraic properties. There are some non-algebraic properties of Q that we need and that some readers may not have been exposed to. These will involve the order structure of Q. Here is one that is the version of the preceding lemma for Q. 1.2.6. Proposition. If x and are positive rational numbers, then there is an n in N with n > x. Proof. Put x = a/b and = c/d with a, b, c, d in N. If n ∈ N, then − x n = c d − a nb = nbc − ad nbd By the preceding lemma we can choose n such that nbc > ad . For that value of n, n > x. In the Quick Approach above we defined the concept of a set of real numbers that is bounded above or below. The same concept applies to subsets of Q, which are, of course, subsets of R. But here we want to underline that we must choose the upper bound to be a rational number. This is not so important until we also discuss the concept of the supremum and infimum of subsets of Q. Here for a bounded subset of Q to have a supremum it must be rational. There are subsets of Q that are bounded above but do not have a supremum in Q. To do this we present two lemmas. 1.2.7. Lemma. There is no rational number x with x 2 = 2. Proof. In fact if there were such a rational number, we could write it as x = n/m, where n, m ∈ Z and n and m have no common divisor other than ±1.
eBook - PDF- John Newsome Crossley(Author)
- 1987(Publication Date)
- World Scientific(Publisher)
Part 3 Real Numbers It is a relatively small step to go from the natural numbers to rational numbers, that is, fractions of the form p/q. All that is required is to think in terms of new units which are q times as big as before. Then measuring (the old) p by the new units gives p/q in the new scale. This is what is to be found in Euclid (see Book VII). Euclid writes of one number measuring another, by which he means that one is an integral multiple of the other (Euclid VII.5). However, the Greeks discovered that although one could measure one length or magnitude by another, there were also easily constructed lengths, e.g. the diagonal of the unit square, which could not be measured exactly. The diagonal of a square is incommensurable with the side of the square. They discovered irrational numbers. We shall study the background to this discovery in chapter V. 1 Now the Greeks were quite clear that no infinite processes were involved in showing that certain numbers are irrational. In the nineteenth century, people began to think about all the points on the line and this notion of all necessarily involved consideration of the infinite. In chapter VI we shall see how this was treated. We shall also point out that a new axiom, which is mentioned very infrequently these days, is required. This axiom allows us to identify the geometric line and the number line; an identification which the Greeks never made. Finally we shall see how the formalization of our intuition of the line has developed quite recently and see that the way is open for further developments. 109 Chapter V Irrationals 1. Introduction The discovery of irrational numbers by the Greeks has long, though perhaps erroneously, been regarded as causing a great crisis in Greek mathematics.
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