Mathematics
Rational Numbers and Fractions
Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. Fractions are a way of representing rational numbers, with the numerator indicating the number of equal parts being considered and the denominator representing the total number of parts in a whole. Rational numbers and fractions are fundamental concepts in mathematics, used in various calculations and comparisons.
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10 Key excerpts on "Rational Numbers and Fractions"
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
There are two directions we can take. First, we could take all the fractions together with their opposites. This would give us a new collection of numbers, namely the fractions and numbers such as − − − 2 3 5 7 11 2 , , . A second approach would be to take the integers and form all possible “fractions” where the numerators are integers and the denominators are nonzero integers. We adopt this second approach, in which a rational number will be defined to be a ratio of integers. The set of rational numbers defined in this way will include the opposites of the fractions. NCTM Standard All students should understand the meaning and effects of arith- metic operations with fractions, decimals, and integers. Common Core – Grade 7 Understand every quotient of integers (with nonzero divisor) is a rational number. THE RATIONAL NUMBERS Rational Numbers The set of is the set Q a b a b b = ≠ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ | . and are integers, 0 D E F I N I T I O N 9 . 1 Examples of rational numbers are 2 3 5 7 4 9 0 1 7 9 , , , . − − − − , and Mixed numbers such as − = − − = − = 3 1 4 13 4 5 2 7 37 7 1 3 7 3 , , and 2 are also rational numbers, since they can The fraction 2 3 can be thought of as the number 0.6 which lies on the number line between 0 and 1. It can also be thought of as one whole broken into 3 pieces where 1 piece is 1 3 and 2 pieces is 2 3 . How are the symbols − 2 3 and − 2 3 related to each other or to either of the meanings described here? –2 –1 0 Two pieces One whole 1 2 2 3 1 3 342 Chapter 9 Rational Numbers, Real Numbers, and Algebra be expressed in the form a b , where a and b are integers, b ≠ 0. Notice that every fraction is a rational number; for example, in the case when a ≥ 0 and b > 0 in a b . Also, every integer is a rational number, for example, in the case when b = 1 in a b . Thus we can extend our diagram in Figure 9.1 to include the set of rational numbers (Figure 9.2).
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
fraction when I mean non-negative rational numbers.Rational Numbers
Although many people mistakenly use the terms fractions and rational numbers synonymously, they are very different number sets. Other important distinctions related to the rational numbers are made in the following examples.• All rational numbers may be written in fraction form.3 4,4 3(,usually written as)2 32.1 4.1(, andusually written as)21 411 21 4are all fractions and rational numbers.(usually written as)2 1• Not all numbers written in fraction form are rational.is not a rational number although it is written in fraction form.π 2• Each fraction does not correspond to a different rational number.There is not a different rational number for each of the three fractions. Just as one and the same woman might be addressed as Mrs. Jones, Mom, Mother, Maggie, Dear, Aunty Meg, and Margaret, these fractions are different numerals designating the same rational number. A single rational number underlies all of the equivalent forms of a fraction.2 3,6 9, and10 15• Rational numbers may be written as fractions, but they may be written in other forms as well.Terminating decimals are rational numbers. Non-terminating, repeating decimals are rational numbers. Percentages are rational numbers. Non-terminating, non-repeating decimals are not rational numbers. Ratios and rates are rational numbers.Fractions as Numbers
When we speak of a fraction as a number, we are really referring to the underlying rational number. Understanding a fraction as a number entails realizing, for example, thatrefers to the same relative amount in each of the following pictures. There is but one rational number underlying all of these relative amounts. Whether we call it1 41 4,4 16,3 8, or2 8
eBook - ePubTeaching Mathematics in Primary Schools
Principles for effective practice
- Robyn Jorgensen, Shelley Dole, Kevin Larkin(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
CHAPTER 11RATIONAL NUMBERTopics that fall under the heading of rational number include fractions, decimals, ratio and proportion, rate and per cent. These topics are linked mathematically, but conceptually they are subtly different. In this chapter, key ideas associated with the topics of fractions, decimals, ratio, rate, proportion and per cent and their interlinked nature are presented. Approaches for enhancing students' knowledge of these topics are described.Common and decimal fractionsFractional numbers can be represented in fraction form (e.g.¼) and in decimal form (e.g. 0.25), and the terms 'common fraction' and 'decimal fraction', respectively, are used to distinguish the two symbolic representations. The word fraction is frequently applied to numbers in both fraction form and decimal form, yet there are subtle conceptual differences between common fractions and decimal fractions. Common fraction understanding is based on the part-whole concept. Decimal fraction understanding stems from a combination of an understanding of common fractions, and whole number and place value knowledge. For simplification, in this chapter common fractions are referred to as 'fractions', and decimal fractions as 'decimals'.Whole number and rational number connectionsWhole number understanding provides the foundation for understanding of rational numbers. Particular rational number topics provide a foundation as well as a link to other rational number topics. Decimal understanding is connected to both fraction and whole number knowledge. Ratio and proportion understanding links to fractions, as well as to multiplicative thinking developed through the study of whole numbers. Rate links to ratio. Per cent links to decimals and fractions, and to ratio and proportion. The interconnected nature of rational number topics to each other and to whole number is depicted in the accompanying flowchart. New Zealand's Number Framework (Ministry of Education, New Zealand 2008) includes nine global stages of number knowledge and strategy understanding that encompass the development of rational number knowledge.
eBook - PDF- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
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. Unless otherwise noted, all content on this page is © Cengage Learning. 1.1 • Sets, Real Numbers, and Numerical Expressions 15 We define a rational number as follows: Definition 1.2 Rational Numbers A rational number is any number that can be written in the form a b , where a and b are integers, and b does not equal zero. We can easily recognize that each of the following numbers fits the definition of a ra-tional number. 2 3 4 2 3 15 4 and 1 2 5 However, numbers such as 2 4, 0, 0.3 , and 6 1 2 are also rational numbers. All of these numbers could be written in the form a b as follows. 2 4 can be written as 2 4 1 or 4 2 1 0 can be written as 0 1 5 0 2 5 0 3 5 . . . 0.3 can be written as 3 10 6 1 2 can be written as 13 2 We can also define a rational number in terms of decimal representation. We classify decimals as terminating, repeating, or nonrepeating. Type Definition Examples Rational numbers Terminating A terminating decimal ends. 0.3, 0.46, 0.6234, 1.25 Yes Repeating A repeating decimal has a block of digits that repeats indefinitely. 0.66666 . . . 0.141414 . . . 0.694694694 . . . 0.23171717 . . . Yes Nonrepeating A nonrepeating decimal does not have a block of digits that repeats indefinitely and does not terminate. 3.1415926535 . . . 1.414213562 . . . 0.276314583 . . . No A repeating decimal has a block of digits that can be any number of digits and may or may not begin immediately after the decimal point. A small horizontal bar (overbar) is commonly used to indicate the repeat block. Thus 0.6666 . . . is written as 0.6 , and 0.2317171717 . . . is written as 0.2317 . In terms of decimals, we define a rational number as a number that has a terminating or a repeating decimal representation.
eBook - PDF- Keith Hirst(Author)
- 1994(Publication Date)
- Butterworth-Heinemann(Publisher)
As in the previous section we shall only discuss addition, leaving the details for the other arithmetic operations to the exercises. The Rational Numbers 45 There are some important principles here, relating to the fact that we have defined rational numbers as equivalence classes of fractions. Our task is not simply to state how to add fractions, but how to define addition of the equivalence classes. The method is to take a representative fraction from two equivalence classes, add them, and then define the sum of the classes to be the class containing the sum of the two fractions. There is a problem with this, namely that we have to be certain that if we were to choose different fractions we would always end up in the same equivalence class. The next result shows that this is indeed the case. • Proposition 2 Suppose we have two rational numbers x and y and that we are given two representative fractions from each, so that and We then have a2 d2 + b2 C2 bidz • PROOF The equivalence of the given fractions tells us that a 1b2 = a2bl and that c 1d2 = C2dl. We multiply the first equation by dld2 and the second by blb2. We then add them and factorize. This procedure gives alb2dl d2 = azbidida c1d2blb2 = C2 dlblb2 alb2dl d2 + c1d2b1 b2 = a2bl dl d2 + cidibibi (a1dl + bl CI)b 2d2 = (a2 d2 + b2 C2)b 1dl. From Definition 4 this equation tells us that • This kind of situation occurs in many places in algebra when we wish to combine equivalence classes in some way. The phraseology commonly used is to say, when we have proved such a result, that the operation (in this case addition) is 'well-defined' . We introduced this section by saying that the integers could be embedded in the rationals. This means that there is a subset of the rational numbers which behaves in every way, both arithmetically and algebraically, like the integers. In the language of abstract algebra this subset would be said to be isomorphic to the integers.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Most people who have degrees in mathematics were students who could learn mathematics on their own. This doesn’t mean that you must learn it all on your own, or that you study alone, or that you don't ask questions. It means that you know your resources, both internal and external, and you can count on those resources when you need them. Attaining this goal gives you independence and puts you in control of your success in any math class you take. © clu/iStockPhoto This is the last chapter in which we will mention study skills. You know by now what works best for you and what you have to do to achieve your goals for this course. From now on, it is simply a matter of sticking with the things that work for you and avoiding the things that do not. It seems simple, but as with anything that takes effort, it is up to you to see that you maintain the skills that get you where you want to be in the course. 415 6.1 Learning Objectives In this section, we will learn how to: 1. Evaluate a rational expression. 2. Determine when a rational expression is undefined. 3. Reduce a rational expression to lowest terms. 4. Reduce a rational expression containing factors that are opposites. Evaluating and Reducing Rational Expressions Introduction We will begin this section with the definition of a rational expression. Recall from Chapter 1 that a rational number is any number that can be expressed as the ratio of two integers: Rational numbers = a __ b a and b are integers, b ≠ 0 We define a rational expression in a similar fashion. A rational expression is any expression that can be written in the form P __ Q where P and Q are polynomials and Q ≠ 0. rational expression DEFINITION Some examples of rational expressions are 2 x − 3 ______ x + 5 x 2 − 5 x − 6 _________ x 2 − 1 a − b _____ b − a Evaluating Rational Expressions To evaluate a rational expression means to find its value when any variables in the expression are replaced by specific numbers.
eBook - PDFThe Algebra Teacher's Guide to Reteaching Essential Concepts and Skills
150 Mini-Lessons for Correcting Common Mistakes
- Judith A. Muschla, Gary Robert Muschla, Erin Muschla(Authors)
- 2011(Publication Date)
- Jossey-Bass(Publisher)
Negative numbers can make the process even more confusing. 1. Explain that a rational number is a number that can be expressed as a quotient of integers. Numbers such as 1 4 , − 5 2 , and 8 are examples of rational numbers. 2. Explain that adding positive and negative rational numbers involves many of the same steps that are necessary for adding positive fractions. Depending on the abilities of your students, you may find it helpful to review the steps for finding a common denominator, writing equiv-alent fractions, adding the numerators, and simplifying the results. 3. Review the steps for adding rational numbers and the example on the worksheet with your students. Emphasize that to add negative rational numbers, students must also do the following: • Write all mixed numbers as improper fractions. • Write the denominators of the fractions as positive numbers. • Add the numerators, following the rules for adding integers. EXTRA HELP: Pay close attention to negative signs as you add rational numbers. It is a good idea to write negative signs before the numerators. ANSWER KEY: (1) − 19 20 (2) 27 56 (3) 2 5 24 (4) − 2 5 14 (5) − 4 11 20 (6) − 1 8 (7) 1 17 28 (8) − 6 19 30 (9) − 15 56 (10) − 1 7 10 ------------------------------------------------------------------------------------------(Challenge) Answers may vary. 1 and − 1 2 is one solution. ------------------------------------------------------------------------------------------62 T H E A L G E B R A T E A C H E R ’ S G U I D E Name Date WORKSHEET 2.8: ADDING RATIONAL NUMBERS -------------------------------------------------------------------------------------Follow the steps below to add positive and negative rational numbers: 1. Express any mixed or whole numbers as improper fractions. 2. Rewrite any negative fractions so that the denominator is positive. 3. Find the least common denominator and write equivalent fractions. 4. Add the numerators, following the rules for adding integers.
- Richard Lehrer, Leona Schauble(Authors)
- 2023(Publication Date)
- Routledge(Publisher)
7 MEASUREMENT MODELS OF ARITHMETIC OPERATIONS AND RATIONAL NUMBER DOI: 10.4324/9781003287476-7 Fractions are pervasive in K-5 mathematics education, but children and teachers in the elementary grades often find fractions challenging to construct and interpret and, further, find it difficult to make sense of arithmetic operations on fractions. These difficulties stem, at least in part, from the fact that what students and teachers have learned about whole numbers and arithmetic operations on whole numbers can be misleading when applied to fractions (Depaepe et al., 2015 ; Ni & Zhou, 2005 ; Siegler & Lortie-Forgues, 2017). Moreover, instruction about fractions in elementary mathematics too often primarily emphasizes how to carry out procedural rules for solving problems, rather than providing a conceptual frame for situating fraction knowledge within a wider system of mathematical knowledge (e.g., Smith, 2002). Historically, fractions arose in response to the need to refine measures of magnitudes by fracturing units (Davydov & Tsvetkovich, 1991), subdividing them into ever-finer gradations to meet particular requirements of different situations of measure. For students, the need to refine measurements first arises when they confront challenges of measuring magnitudes of length, area, angle, and volume that are not divisible by whole-number units. As described in Chapters 3 – 6, subdividing units challenges students to conceptualize relations between magnitudes of a subdivided unit and its parent unit, to measure magnitudes in subdivided units, and to engage in additive and multiplicative comparisons of these measures. In the current chapter, we lift away from this embedding of rational number within the strands of measure to illuminate how children’s investigations of measurement can support their understandings of fractions and arithmetic operations involving fractions
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Understand the completeness of the real number line. Become familiar with some famous irrational numbers. Classifying Real Numbers In Section 7.3, you learned that any number written as a — b , where a and b are integers and b ≠ 0, can be written as a terminating decimal or a repeating decimal. 1 — 4 = 0.25 Terminating decimal − 1 — 3 = −0.333. . . Repeating decimal There are other types of numbers whose decimal representations do not terminate or repeat. √ — 2 = 1.41421. . . Nonrepeating decimal Together, these three types of decimals help make up the set of real numbers. Definition of the Set of Real Numbers The set of real numbers is the set of all numbers that can be written in decimal form, including those numbers that require an infinite decimal expansion. If the decimal form terminates or repeats, then the real number is a rational number. If the decimal form neither terminates nor repeats, then the real number is an irrational number. Recall that a rational number is a number that can be written as the ratio of two integers. This implies that an irrational number cannot be written as the ratio of two integers. EXAMPLE 1 Showing that √ — 2 Is Irrational Use proof by contradiction to show that √ — 2 is irrational. SOLUTION To begin, assume that √ — 2 is rational and can be written as the ratio of two integers (in simplest form). √ — 2 = a — b Assumption: √ — 2 can be written as a rational number a — b . 2 = a 2 — b 2 Square each side. 2b 2 = a 2 Multiply both sides by b 2 . If 2b 2 = a 2 , then the prime factorizations of 2b 2 and a 2 are the same by the Fundamental Theorem of Arithmetic. So the factor 2 must appear the same number of times in each prime factorization of 2b 2 and a 2 . Because squares have prime factors that occur in pairs, a 2 and b 2 have an even number of 2s. But for 2b 2 , there is another factor of 2, so there is an odd number of 2s in the prime factorization of 2b 2 .
eBook - PDF- Inder K Rana(Author)
- 1998(Publication Date)
- WSPC(Publisher)
4.5 Geometric representation 147 Consider a horizontal line and mark two distinct points O and A on the line, A on the right of O. Choose the segments OA as unit of length. Let O represent the integer zero and A represent the integer 1. Mark points on the line spaced at unit distance apart on the right of A and let them represent positive integers 2,3,..., re-spectively. Similarly, let points on the line spaced at unit distance apart on the left of O represent the negative integers — 1, — 2,..., respectively. O A n + f -4 -3 -2 -1 0 1 2 3 4 ^ n + 1 Figure 4.5 : Geometric representation of rationals. p Let r 6 Q, r > 0. Let r = - + n, where n,p, q £ IN are unique num-9 P bers such that 0 < - < 1. We divide the segment n to n + 1 into q q equal parts. Let the point corresponding to the right end-point of the p th part be the point on the line which represents r. Nega-tive rationals can be represented similarly. Thus for every rational there is a point on the line and distinct rationals get represented by distinct points. As was noted by Hippasus of Metapontum in 500 B.C., there are points on the line which do not represent any ratio-nal (see section 4.1). For example whatever the unit of length be, if we take a right angled triangle with base and height of unit length, then the length of its hypotenuse does not correspond to any ra-tional number. Thus, rationals when put on the line do not cover up the line, even though they are 'dense', i.e., between any two points representing rationals we can find a point in between them which also represents some rational. Let us call the points on the line which do not represent rationals as irrational points. They represent 'gaps 7 on the number line after the rationals have been put on the line. We can define a 'gap' in the line more precisely as follows. 148 Ch. 4: Rational Numbers 4.5.1 Definition. A pair (A, B) of nonempty subsets of
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