Integers

12 Key excerpts on "Integers"

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  eBook - ePub

  Fostering Children's Mathematical Power

  An Investigative Approach To K-8 Mathematics Instruction

  • Arthur Baroody, Arthur J. Baroody, Jesse L.M. Wilkins, Ronald T. Coslick(Authors)
  • 1998(Publication Date)
  • Routledge

    (Publisher)

  Figure 8.5 ) and how to help children understand this type of number (Subunit 8•2•1). We then examine integer arithmetic (Subunit 8•2•2).

  Figure 8.5:  A Need for Negative Numbers

  8•2•1  The Concept of Integers Mathematics: Understanding Integers

  Integers, why another set of numbers?

  The Role of the Integers.  Sometimes it does not make sense to talk about numbers less than zero. For example, the attendance of class cannot drop below zero, even if a community is gripped by a plague or some other disaster. In such cases, the whole numbers do just fine, thank you. Sometimes it does make sense to talk about numbers less than zero. For example, it is cold out when the temperature drops below 0°C; it is very cold out when the temperature drops below 0°F. In the era of plastic credit, it is easier than ever to end up in debt. Bad football teams can end up with negative net yards gained. Some games, such as the television game show Jeopardy, entail penalty points and negative scores. In other games, such as the card game Hearts, a player is glad to run up a negative score, because the fewer points the better. In discussing elevation, one can go below as well as above sea level.

  Put differently, sometimes it is enough to know the size of a quantity; other times it is important to know a quantity’s size and its location. Whole numbers are fine for indicating a magnitude, but Integers are needed to indicate both magnitude and direction. With Integers, 0 is a reference point, and the (continued on page 8-23) sign of a number serves to indicate the number’s direction relative to 0. In brief, Integers are useful tools for situations when we need to know how big something is and where the bigness is.3

  Definition and Symbolism.  Concretely, the minus sign in front of a number, such as −2, “turns [it] upside down, like the reflection of trees … on a river” (Sawyer, 1971, p. 62). Mathematically, negative numbers are defined as the opposites of positive numbers. For example, −2 is defined as the solution to the expression 2 + □ =0. For example, +2 and −2 indicate the same magnitude but −2 goes in the opposite direction of +2. A number line provides a clear depiction of this interpretation. For example, +2 can be thought of as two units to the right of zero; −2, as two units to the left

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  Computer Algebra and Symbolic Computation

  Mathematical Methods

  • Joel S. Cohen(Author)
  • 2003(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  2 Integers, Rational Numbers, and Fields The chapter is concerned with the numerical objects that arise in computer algebra including the Integers, the rational numbers, and other classes of numerical expressions. In Section 2.1 we discuss the basic mathematical properties of the Integers and describe some algorithms that are important for computer algebra. Section 2.2 is concerned with the manipulation of rational numbers. We define a standard form for a rational number and describe an algorithm that evaluates involved arithmetic expressions with Integers and fractions to a rational number in standard form. In Section 2.3 we introduce the concept of a field, which is a mathematical system with axioms that describe in a general way the algebraic properties of the ratio-nal numbers and other classes of expressions that arise in computer algebra. We give a number of examples of fields and show that many transformations that are routinely used in the manipulation of mathematical expressions are logical consequences of the field axioms. 2.1 The Integers In this section we describe some mathematical and computational proper-ties of the Integers 17 18 Integers, Rational Numbers, and Fields The following theorem gives the basic division property of the Integers. 1 Theorem 2.1. For Integers a and b =£ 0, there are unique Integers q and r such that ) and ) The integer q is the quotient and is represented by the operator iquot(a, 6) (for integer quotient). The integer r is the remainder and is represented by irem(a, b). Example 2.2. In Theorem 2.1, the quotient and remainder are chosen so that r > 0.

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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 8.1 Integers 293 8.1 Integers Understand the definition of Integers, and graph Integers on an integer number line. Use an integer number line to order Integers. Find the absolute value of an integer. Recognize real-life problems that can be represented by Integers. Integers Definition of Integers Integers are the set of whole numbers and their opposites. Two numbers are opposites when they are the same distance from 0 on a number line. . . .−3, −2, −1, 0, 1, 2, 3, . . . Integers The integer number line is a measurement model that represents Integers. Integers to the right of zero on the integer number line are called positive Integers. Integers to the left of zero are called negative Integers. The integer zero is neither positive nor negative. -5 5 4 3 2 1 0 -1 -2 -3 Negative Integers -4 Positive Integers Opposites Zero is neither negative nor positive. Locating the point that represents an integer on the integer number line is called graphing an integer. EXAMPLE 1 Graphing Integers Graph the Integers −5, −1, 0, and 4. SOLUTION -6 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -5 -1 0 4 EXAMPLE 2 Graphing an Integer and Its Opposite Graph the integer 3 and its opposite. SOLUTION -6 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 Graph 3. -3 is the same distance from 0 as 3. You may hear students refer to the integer −5 as “minus five.” In mathematics, minus represents subtraction, whereas negative describes a number that is to the left of zero on the integer number line.

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  Mathematics for Elementary Teachers

  A Contemporary Approach

  • Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  For example, the subtraction problem 4 7 − has no answer when using whole numbers. Also, the equation x + = 7 4 has no whole-number solution. To remedy these situations, we introduce a new set of numbers, the Integers. Our approach here will be to introduce the Integers using a physical model. This model is related to a pro- cedure that was used in accounting. Numerals written in black ink represent amounts above zero (“in the black” is positive) and in red ink represent accounts below zero (“in the red” is negative). We will use the Integers to represent these situations. Algebraic Reasoning As can be seen here, the use of Integers allows one to solve a larger selection of equation types. Two examples of these equations are: 3 5 − = x and 8 2 + = x . Reflection from Research An understanding of Integers is crucial to an understanding of future work in algebra (Sheffield & Cruikshank, 1996). ADDITION AND SUBTRACTION Integers The set of Integers is the set I = − − − { } ..., , , , , , , , ... . 3 2 1 0 1 2 3 The numbers 1 2 3 , , ,... are called positive Integers and the numbers − − − 1 2 3 , , ,... are called negative Integers. Zero is neither a positive nor a negative integer. Children’s Literature www.wiley.com/college/musser See “50 Below Zero” by Robert Munsch. D E F I N I T I O N 8 . 1 In the introduction, temperature, elevation, stocks, and banking are presented as situa- tions where positive and negative numbers are used. Using one of these scenarios, write a word problem for each of the following expressions. − + − − − − − 30 14 30 14 30 14 ( ) In a set model, chips can be used to represent Integers. However, two colors of chips must be used, one color to represent positive Integers (black) and a second to represent negative Integers (red) (Figure 8.1). One black chip represents a credit of 1 and one red chip represents a debit of 1. Thus one black chip and one red chip cancel each other, or “make a zero” so they are called a zero pair [Figure 8.2(a)].

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  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 1 From Arithmetic to Algebra 1.1 Introduction

  Numbers are symbols used for counting and measuring. Hindu–Arabic numerals 0, 1, 2, 3, . . . . . ., 9 are grouped systematically in units, tens, hundreds, and so on, to solve problems containing numerical information. This is the subject of Arithmetic . It also involves an understanding of the structure of the number system and the facility to change numbers from one form to another; for example, the changing of fractions to decimals and vice versa. A detailed discussion about the Real Number System is given in Chapter 3. However, it would be instructive to recall some important subsets of real numbers , known to us.

  Numbers, which are used in counting, are called natural numbers or positive Integers . The set of natural numbers is denoted by

  1.2 The Set of Whole Numbers

  The set of natural numbers along with the number “0” makes the set of whole numbers, denoted by W . Thus,

  Note : “0” is a whole number but it is not a natural number.

  1.3 The Set of Integers

  All natural numbers , their negatives and zero when considered together, form the set of Integers denoted by Z . Thus,

  1.4 The Set of Rational Numbers

  The numbers of the form p /q where p and q are Integers , and the denominator q ≠ 0, form the set of rational numbers , denoted by Q .

  Examples: and so on, are all rational numbers.

  Remarks:

  a. Zero is a rational number , but division by zero is not defined. Thus, are meaningless expressions.

  b. All Integers are rational numbers , but the converse is not true.

  c. Positive rational numbers are called fractions .

  Let us discuss more about fractions.

  Generally, “fractions ” are used to represent the parts of a given quantity, under consideration. Thus, 3/7 tells us that a given quantity or an object is divided into seven equal parts and three parts are under consideration. A fraction is also used to express a ratio. Thus, 2:5 is also written as 2/5 and similarly 12:5 is written as 12/5. Since the ratio of two natural numbers can be greater than 1, all positive rational numbers are called fractions

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  Abstract Algebra

  An Inquiry Based Approach

  • Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom(Authors)
  • 2013(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  This is because any integer times zero is equal to zero; in other words, 0 · x = 0 for every integer x . (b) (67 − 11 + 925 − 81) + (81 + 11 − 925 − 67) (c) (125 − 982) + (982 − 43) + (43 − 620) + (620 − 79) + (79 − 125) (d) 75(147 − 229) + 229(75) − 147(75) 3 4 Investigation 1. The Integers: An Introduction Introduction Every journey has a beginning, and ours will begin with the Integers. For likely as long as you can remember, you have been using the Integers. When you first learned to count, your concept of number included only natural numbers , or what we might now refer to as positive Integers . The notions of zero and negative numbers came later on, just as they did throughout the historical development of the Integers. In fact, while the Integers may seem elementary to us now, it actually took mathematicians thousands of years to formally develop and understand them. This historical development was rife with controversy, and it led to serious philosophical and even theological debates. The daunting task of formally defining the Integers played a key role in the development of much of modern mathematics, and in particular the field of set theory. It might surprise you to learn the most common modern construction of the Integers is based entirely on sets and set operations. Such a rigorous development of the Integers is not necessary for our investigations, but we should at least define the terminology and notation that we will be using. ∗ Definition 1.2. • The set of natural numbers , denoted N , contains the counting numbers ( 1 , 2 , 3 , and so on); that is, N = { 1 , 2 , 3 ,... } . • The set of whole numbers , denoted W , contains the counting numbers and zero; that is, W = { 0 , 1 , 2 , 3 ,... } . • The set of Integers , denoted Z , contains the whole numbers and their opposites (or nega-tives); that is, Z = { ..., − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,...

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  An Open Door to Number Theory

  • Duff Campbell(Author)
  • 2018(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 1 The Integers, Z 1 Number systems When humans first started using numbers, they probably used the counting or natural numbers, N , first. These are the numbers in the set N = { 1 , 2 , 3 , . . . } . With these numbers one can do many useful things, like count cattle, levy taxes, or build pyramids and temples; also mathematical things like add and multiply, but we can’t always subtract. What is 5 − 5? We need a new number, call it zero , to mean nothing. Then we get the whole numbers, W = { 0 , 1 , 2 , 3 , . . . } . Now we still have trouble subtracting—what is 6 − 9? So we invent negative Integers and call the new system the Integers : Z = { . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . } . The symbol Z is from Zahlen, German for “number.” Now we can add, subtract, and multiply, but we can’t always divide: 6 ÷ 3 = 2, but 6 ÷ 5 = ? So we invent fractions, or rational numbers : Q = p q : p , q in Z , q = 0 . (Actually, this definition is incomplete, since, for instance, 3 6 is considered to be the same number as 1 2 , but we will ignore that complication for now.) Using Q , we can solve equations like 3 x − 7 = 0, but what about x 3 − 7 = 0? If we throw in all solutions to all polynomial equations (with integer coefficients), we get things like 3 √ 7, which are called roots, or more complicated numbers like 4 5 − √ 43 8 + √ 2 − 8 5 √ 3 + 2 . 1 2 Chapter 1. The Integers, Z Those are the algebraic numbers , denoted A or Q , which is where most algebraic number theory in fact occurs. But there are other ways of proceeding from Q . What is π ? It doesn’t solve a polynomial equation with integer coefficients, so it isn’t “an algebraic number”, but we can find (or approximate) it on the number line. We write π = 3 . 1415926 . . . to mean that the rational numbers 3, 3 . 1, 3 . 14, 3 . 141, etc., approach π as a limit. This is the basis for calculus and analysis, the idea of a limit.

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  Prealgebra 2e

  • Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
  • 2020(Publication Date)
  • Openstax

    (Publisher)

  Integers are counting numbers, their opposites, and zero ... –3, –2, –1, 0, 1, 2, 3 ... A negative number is less than zero. The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero. KEY CONCEPTS 3.1 Introduction to Integers • Opposite Notation ◦ −a means the opposite of the number a ◦ The notation −a is read the opposite of a. • Absolute Value Notation ◦ The absolute value of a number n is written as |n| . ◦ |n| ≥ 0 for all numbers. 3.2 Add Integers • Addition of Positive and Negative Integers 5 + 3 −5 + (−3) both positive, sum positive both negative, sum negative When the signs are the same, the counters would be all the same color, so add them. −5 + 3 5 + (−3) different signs, more negatives different signs, more positives Sum negative sum positive When the signs are different, some counters would make neutral pairs; subtract to see how many are left. 3.3 Subtract Integers • Subtraction of Integers Chapter 3 Integers 263 5 – 3 –5 – (–3) 2 –2 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. –5 – 3 5 – (–3) –8 8 5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives need neutral pairs need neutral pairs When there would not be enough of the counters to take away, add neutral pairs. Table 3.13 • Subtraction Property ◦ a − b = a + (−b) ◦ a − (−b) = a + b • Solve Application Problems ◦ Step 1. Identify what you are asked to find. ◦ Step 2. Write a phrase that gives the information to find it. ◦ Step 3. Translate the phrase to an expression. ◦ Step 4. Simplify the expression. ◦ Step 5. Answer the question with a complete sentence.

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  Elementary Algebra 2e

  • Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
  • 2020(Publication Date)
  • Openstax

    (Publisher)

  1.3 Add and Subtract Integers Learning Objectives By the end of this section, you will be able to: Use negatives and opposites Simplify: expressions with absolute value Add Integers Subtract Integers BE PREPARED : : 1.3 A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers. Use Negatives and Opposites Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than 0. The negative numbers are to the left of zero on the number line. See Figure 1.6. Figure 1.6 The number line shows the location of positive and negative numbers. The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number. Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative. Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See Figure 1.7. Figure 1.7 The numbers on a number line increase in value going from left to right and decrease in value going from right to left. MANIPULATIVE MATHEMATICS Doing the Manipulative Mathematics activity “Number Line-part 2” will help you develop a better understanding of Integers. Remember that we use the notation: a < b (read “a is less than b”) when a is to the left of b on the number line. a > b (read “a is greater than b”) when a is to the right of b on the number line. Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The Chapter 1 Foundations 41 numbers marked by points in Figure 1.8 are called the Integers.

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  Abstract Algebra

  A Comprehensive Introduction

  • John W. Lawrence, Frank A. Zorzitto(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 A Refresher on the Integers A good place to start might be the set of Integers: Z = {0, ±1, ±2, ±3, . . . }. We often encounter the notation N for the set of natural numbers. There seems to be no consensus on whether to include 0 as a natural number. Following the advice of Paul Halmos (1916–2006) in his seminal book Naive Set Theory, first published in 1960, we shall include 0 as a natural number. Thus N stands for the set {0, 1, 2, 3, . . . }. We will on occasion use the somewhat less common notation P for the set {1, 2, 3, . . . } of positive Integers. Often we shall prefer to say things like “let x be a positive integer,” instead of “let x ∈ P.” 1.1 Euclidean Division and the Greatest Common Divisor In the seventh book of Euclid we are told that any integer b, upon division by a positive integer a, yields a unique quotient q and a unique remainder r such that b = aq + r and 0 ≤ r < a. Indeed, q is the largest integer that is less than or equal to the fraction b/a, and r is simply b − aq. The procurement of q and r will be called Euclidean division. If the remainder r = 0, then b = aq. In this case we say that a divides b, and write a | b. For instance, 17 | 51 since 51 = 17 · 3. Conversely, a bit of reflection shows that if r = 0, then a  b. For example, the Euclidean division −91 = 17 · (−6) + 11 where 0 < 11 < 17, reveals that 17 does not divide −91. For visual clarity we may, on occasion just as above, use a dot to signify multiplication. When a | b, we say that b is a multiple of a, and also that a is a divisor or factor of b. Here are a few simple things to note. • If a | b and b | c, then a | c. • If a | b and a | c and x, y are any Integers, then a | bx + cy. • If b = 0 and a | b, then |a| ≤ |b|. • If a | b and b | a, then a = ±b. 1 2 A Refresher on the Integers The Greatest Common Divisor of Two Integers If a, b are Integers, an integer combination of a and b is any integer c built up as c = ax + by for some Integers x, y.

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  Introductory Algebra

  • Chris Nord(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  below sea level, but we would certainly not be at the same elevation as the summit of Mt. Everest! The elevation beneath the sea is different than above the sea.

  Is it reasonable to use the same number to describe these two very different types of numbers that are above or below zero? No, it isn’t.

  One limitation with the set of whole numbers is that there is no whole number that is less than zero. This not only restricts the usefulness of the set of whole numbers, it also introduces a lack of symmetry. Whole numbers are unbounded in one direction because there is no greatest whole number, but they are bounded in the other direction because 0 is the least of all the whole numbers.

  In this chapter, we expand our definition of number to include numbers that are the opposite of whole numbers. This new set of numbers, the Integers , is the set {…, -4, -3, -2, —1, 0, 1, 2, 3, 4, …}.

  2.1 Introduction to Integers

  Each positive integer corresponds with a negative integer. For example, the positive integer 7 corresponds to the negative integer -7. These corresponding Integers are called opposites of each other. When the subtraction symbol is placed in front of a positive integer in such a way that it cannot possibly indicate the operation of subtraction, it indicates that the integer is a negative integer.

  A. The Number Line

  Imagine a straight line that extends infinitely far in both directions. Evenly spaced ticks are marked out all along the line. Each tick corresponds to an integer. All of the positive Integers extend from the zero tick to the right in increasing order. All of the negative Integers extend from the zero tick to the left in decreasing order. This line is called the number line . It is a visual aid for understanding the set of Integers:

  Greater Than and Less Than

  Let a and b represent any two Integers. If a is to the left of b on the number line then we say “a is less than b.” Using math symbols, we write a < b

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  Set Theory: The Structure of Arithmetic

  • Norman T. Hamilton, Joseph Landin(Authors)
  • 2018(Publication Date)
  • Dover Publications

    (Publisher)

  3

  THE Integers AND THE RATIONAL NUMBERS

  3.1.INTRODUCTION

  The numbers we have constructed so far are clearly inadequate for many purposes. For example, if we wished, we could introduce the operation of subtraction in the set of natural numbers by means of the definition

  m – n is the natural number x, if there is any, such that m = n + x.

  The qualification “if there is any” is essential; for, if n > m, there is no such natural number. Thus, unrestricted subtraction cannot be carried out in the set of natural numbers.

  Similarly, the operation of division could be defined in the set of natural numbers by means of

  for n ≠ 0, m ÷ n is the natural number x, if there is any, such that n · x=m.

  Again the qualification is essential, since obvious y there are many instances in which such a natural number will not exist for given natural numbers n ≠ 0 and m.

  In order to extend the scope of our arithmetic, we shall, in the present chapter, construct two new systems of numbers, the Integers and the rationals. The system of Integers will admit unrestricted subtractions, and we shall prove that the Integers contain a “copy” of the natural numbers. Similarly, the system of rational numbers will admit unrestricted division, except by zero, as well as unrestricted subtraction, and it will be proved that the system of rational numbers contains a “copy” of the Integers. Thus, the set of Integers will constitute an “extension” of the natural numbers, and the rational numbers will be an “extension” of the Integers.

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