> q` ZrbjbjqPqP .::i%/
```t###8P#l#DtIA0$(4$"V$V$V$i1i1i1@@@@@@@$yChE@`3)1@i133@V$V$A::::::3rV$`V$@::3@::::V?@b
`d@V$$ \Z#8(@@DA0IA4@0F8<Fd@F`d@ i1Z1@::2472ui1i1i1@@9^i1i1i1IA3333ttt#ttt#ttt
Students perceptions of the Completeness Property of the Set of Real Numbers
Analia Berg
Department of Mathematics
College of Rimouski
Quebec, Canada
1. Introduction
The set of real numbers R is the natural domain used in Calculus and Analysis courses at the university level. However, we can hypothesize that first year students do not have a clear understanding of what this domain represents nor what the properties of R, on which the work in Calculus and Analysis is founded, are.
In most universities, approaching the set of real numbers is done in a progressive manner throughout several courses. In the first Calculus courses the set R is not explicitly defined; instead, what is used is the naive idea of considering R as all possible numbers, influenced by the image of the number line. This idea is compatible with the kind of work usually carried out in these courses, and it allows instructors to advance through the curriculum fairly quickly. Further on, when studies in Analysis start and the set R becomes the natural domain of functions, other properties become relevant. R is then usually introduced by means of axioms of complete ordered field. In advanced courses R is presented as the set of rational cuts or rational Cauchy sequences. What do students understand and misunderstand about R and completeness along that path? The purpose of this paper is to offer an account of undergraduate students' mathematics ideas on R and completeness. The set of real numbers and its defined property of completeness is a topic that can be studied from a didactic point of view on a long-term basis, since it is taught, in various depths, in several undergraduate courses. Above all, it is a topic involved in problems that are at the core of Analysis, and reflect some general principles of this domain, like determining an element by means of a nested sequence of sets, or defining one element as the limit of others under a certain condition, or achieving an extremum value, among others.
2. Background
The essential difference between R and the other numerical fields is given by the property of completeness. This property can be expressed by several equivalent characterizations, the most typical being the existence of the least upper bound of an upper-bounded and non-empty set.
The notion of completeness was developed at the end of the 19th century in order to allow a reliance on a numerical system in which all finite or infinite decimal representations represented a well-defined number; and also in order to rely on a numerical system that had the attributes that had been naturally assigned to the points of the line: order, density, and continuity. This way it is possible to prove some obvious Calculus theorems like the convergence of monotonic bounded sequences, the existence of intersection of nested intervals whose lengths tend to zero, the convergence of Cauchy sequences, the Intermediate Value Theorem, etc. These theorems could not be proven before on an arithmetical basis only (Berg and Sessa, 2003). The quotation marks around obvious require an explanation. Since lines and curves are graphically represented by a continuous movement, the existence of maximum, minimum and limits is not put in question, as they can be seen in a drawing. Completeness becomes necessary when we need to prove the existence of some numbers, like those previously mentioned, leaving the idea of natural continuity.
To the best of my knowledge there are few works in research in Mathematics Education that explicitly focus on the concepts of completeness of R and continuity of the line. C. Castela claims that the correspondence numbers-points is used in teaching, hoping to transfer to the set of numbers the properties intuitively assigned to geometrical objects. Nevertheless, not only does this correspondence implicitly entail that the line is provided with density and continuity, but it also assumes that the correspondence numbers-points is not problematic for the students: two conditions that are not necessarily fulfilled, as was showed in Castela (1997). R. Nuez et al. (1999, 2000) refer in their works to two conceptions of line. One spontaneous and intuitive, global not atomic and naturally continuous, where the points lie on. The other conception of line considers it as the union of [uncountable] points. Both lines are cognitively different and consequently for students, they provide different inferential structures.
There are several other works in Mathematics Education concerning real numbers which are mostly focused on teachers and students conceptions about rational and irrational numbers, such as the works of Robinet (1986), Barbosa and Da Silva (2001), Fischbein et al (1995), Sirotic and Zazkis (2006), Bronner (1997) to mention a few. The mathematical aspects that are considered in these works are the decimal expansions, the representation of numbers on the line, their density, cardinality and the necessity to introduce numbers other than rational ones. Beyond the diversity of their theoretical and methodological frames, there are some points of consensus among them. Firstly, students at different scholastic levels hold inconsistent conceptualization regarding rationality, irrationality, discrete order, dense order and continuity. Secondly, representing numbers on the line does not necessarily help to reduce those difficulties, as students do not conceptualize the line all in the same way, and consequently referring to its perceived attributes does not bring a secure support. These works, most of them centered at secondary level or at the end of that cycle, do not focus on completeness but they constituted for me a valuable departure point for analyzing students ideas on R when its properties become essential to offering proof and theoretical justifications.
3. A survey concerning completeness. Goals and methodology
My goal was to explore what students think R is, what representations they have about it and to what extent they overcome the idea of natural continuity through different correlative courses in Analysis. A central idea I considered, drawing on notions developed in the Theory of Conceptual Fields (Vergnaud, 1990) is that a concept cannot be reduced to its definition, but it acquires its meaning through problems and situations. Accordingly, my initial questions were of the sort: Is the statement a real bounded above non decreasing sequence has a limit taken by the students as a manifestation of completeness or as evidence that it is a powerful tool to show convergence? When students learn that a continuous function in a bounded closed interval reaches the maximum, or when they learn the intermediate value theorem, do they relate these results with the completeness of the domain or do they use them as evident theorems assuming completeness as something pre-existent or rather preconstructed (in the sense of Chevallard, 1985)? I assume that an analysis of personal conceptions cannot be done in isolation of what students have learned, consequently I observed the opportunities students had to work with completeness through four courses of undergraduate studies in mathematics at University of Buenos Aires (in what follows I will name them by Courses I, II, III and IV) analyzing the tasks students had to do, the techniques they used and their technical and theoretical justifications (Berg, 2008). Regarding these tasks, in Course I students are demanded to find maxima, minima, least upper bounds (sup) and greatest lower bounds (inf) of some subsets of R, which is done based on the graphic representations of the subsets in question. In Course II students have to find maxima, minima, sup and inf of more complex sets and to justify their answers by means of the definitions. I interpret his requirement of justification as a change in the didactic contract, where graphic representations are no longer accepted as a support for justifying an answer. Students do not understand the reasons of this change as no new tasks are proposed in terms of these notions. In Course III sup and inf appear in a more instrumental role (in defining distances for instance), and also as objects that admit more than one definition, which equivalence students are asked to prove. In Course IV, students have to prove the equivalence of five different ways of defining completeness.
I carried out two inquiries. The first one consisted of a written questionnaire that was answered by the majority of the students in Courses II, III and IV (124 out of 192 students in Course II, 11 out of 24 in Course III, 10 out of 16 of Course IV).
The second one consisted of interviews of volunteer students from the three courses. In this paper I present a part of the written questionnaire.
4. The written questionnaire
One of the written questions was: If you wished to explain to a younger student that a real non-decreasing bounded-above sequence has a limit, how would you do it?
I used the answers to this question to see to what extent completeness is problematized for the students. Do they question themselves about the existence of a limit? Do they assume its existence as something evident? If they translate, for instance, the statement to a graphic frame, the existence of the limit is no longer problematic. When students see in a drawing the sequence accumulating, the existence of a limit is for them, evident. It is important to analyze if the drawing is regarded as being the complete answer or if it illustrates an explanation. Another thing to observe in the answers is to what extent completeness of R appears as a necessary condition, as the same statement wouldnt be true if the sequence wasnt a real one.
For students in courses III and IV, who studied the property of completeness as a part of their syllabus and manipulated it on several occasions, I included the question: What does R is a complete set mean for you? I included this question to observe if students give their answer in terms of properties and conditions or mostly in terms of images of natural continuity. In the first case completeness can be used in exercises and problems, we may say it is operational.
For each course, after reading all copies, I grouped the answers in types having as a guide the degree of problematization of completeness for the first question and to what extent the idea of completeness the students have is operational for the second question. Ill present the answers of the first question for the three courses and then the answers to the second question for courses III and IV.
4.1 Answers to the question: If you wished to explain to a younger student that a real non-decreasing bounded-above sequence has a limit, how would you do it? , By students in Course II
I sorted them in five types.
Type 1: The existence of the limit is not put in question. Students explain the meaning of each and all terms of the statement. They explain what the notions of sequence, non-decreasing sequence, bounded-above sequence, and convergent sequence mean. At a certain moment they write "though, it necessarily tends to the limit" or "intuitively, it must converge" or " then, it has no option but to converge". There were 98 answers of this type, out of 124. Almost half of the answers included a drawing.
Examples of answers type 1
A sequence (an ) n( N is non-decreasing (( n00 ( n0 / ( n ( n0 : |an-L|<(. [Drawing 1] The terms of the sequence are near and nearer M (an upper bound of (an) n( N ).
I would firstly explain what a sequence is and what it means that it is non-decreasing and bounded above. From that, it is easy to see what the limit represents, it is a value to which the sequence approaches as I want, but I never touch it [Drawing 2].
[Drawing 2] As we can see in the drawing, the sequence is non-decreasing as it takes values bigger and bigger each time. At the same time, it is bounded above, that is, all the terms are lower than a number, M in this case. The sequence never takes the value M, then I can say that from a certain value, all the points of my sequence are nearer and nearer M. I can say that my sequence tends to M.
Firstly, if a sequence is non-decreasing, then from n=1 the values it takes will be bigger each time a10 ( (= ((() such that c -(/2 ( a( ( c, (
c -(/2 ( a( ( an ( c( n>( ( -(/2 ( an c( 0 , ( n>(, ( 0 ( c- an ( (/2, (
|c- an|=| an- c|<(.
I should explain that the sequence is non-decreasing and bounded above, which means that it is always lower than a real number. Therefore, as it grows and it is lower than a number the terms of the sequence will approach a number l (l may be different from the upper bound, not all bounds are limits, the limit is the unique which is the best upper bound, the supremum.
In this answer the distinction between upper bound and limit is correctly done, the limit is described as the best bound, and its existence is justified by the supremum.
Type 4: For two students (always out of 124) the sequence necessarily stabilizes, that is, it is constant after a certain sub index.
Example of answers type 4
A sequence can be thought as a function from N to R. If it moves only in a way, it is non-decreasing. It arrives to a point that is a top, it doesnt matter to what value the function is applied, it image will always be that point.
Here, students think the rank of the sequence as a discrete set.
Type 5: Eight students gave unclear answers, and one student did not answer.
Fifty-three students included a drawing in their answers:
Drawing 1:
SHAPE \* MERGEFORMAT
Drawing 2 :
SHAPE \* MERGEFORMAT
In my view, for almost all students in Course II, the existence of the limit is taken as being evident or transparent. For the majority of them (98 out of 124) it is so transparent that each term of the statement is explained, the exception being the existence of the limit, which seems to require no explanation. Other students (10 out of 124) consider an extra-mathematical context, where the word limit takes its daily use and it exists without any questioning. Through an explanation or a drawing students stress without making it explicit- the fact that the sequence is a Cauchy sequence, and the limit is there. Fifty-two out of 53 drawings go together with a type 1 answer. I interpret this major tendency as evidence of a preconstructed and natural vision of completeness.
4.2 Answers to the question: If you wished to explain to a younger student that a real non-decreasing bounded-above sequence has a limit, how would you do it? , by students in Courses III and IV
I found three types of answers for these courses:
Type 1: The existence of the limit is taken for granted, and completeness is used implicitly (4 answers out of 11 in Course III, 8 out of 10 in Course IV).
Examples of answers type 1
I would use a drawing. [Drawing 2] As the sequence is non-decreasing, it goes up, as it is bounded above, never can surpass a top. If we have a bounded sequence, it cannot diverge to (. Then the unique form that it does not converge is that it oscillates (like sinus, cosinus). If we add the hypothesis of non-decreasing, it cannot oscillate, therefore, it has a limit.
A sequence has three possibilities: it converges, it diverges, it oscillates. If it converges, thats it. If it diverges: ( n ( N ( an ( n, that is, the sequence is not bounded. If the sequence oscillates, it cannot be, as it is non-decreasing.
The argument of these answers is that a sequence converges, diverges or oscillates (something that is true only if the sequence is defined in R). Completeness passes implicitly, as a property something naturally verified.
Type 2: The statement is taken as if the situation were discrete, or in extra-mathematical contexts (2 out of 11 in Course III, 0 in Course IV).
Type 3: The existence of the limit is justified via the existence of supremum (5 answers out of 11 in Course III and 2 out of 10 in Course IV)
Example of answers type 3
If the student hasnt done Course II, I would explain that non-decreasing and bounded above sequences always have a limit. And that is true only if the sequences are defined in R, otherwise it is not true. For instance, an= 1-1/n is non-decreasing and bounded above, and therefore it must have a limit, the limit is 1. But also we can have a sequence like Fn , such that Fn+2 = Fn+1 + Fn , F2 = F1 = 1. It diverges, but an= Fn / Fn+1 verifies that an ( Q, an ( 1, an+1 (an (n(N. However, this sequence only converges in the set of real numbers, as its limit is ((5-1)/2, which does not belong to rational numbers. It does not converge to a value in Q
In this answer the student considers completeness as a necessary condition for the existence of the limit.
The majority of the answers from the two courses are similar to those of type 1 in Course II. They are somehow out of phase with the exercises and problems that the students are supposed to have done in the course. It is surprising the use of images and non-mathematical terms (a man walking up to a wall, a barrier) that were never used in textbooks or courses. Those terms constitute a support for some students, and belong to their private ideas. It is interesting to think what kind of problems would get them to put aside these images and offer them to use more operational definitions.
4.3 Answers to the question: What does R is a complete set mean for you? by students in Course III
- Some answers evoke the everyday meaning of the word complete as replete, there are no places, there are no gaps (3 out of 11). For instance: R is complete, as it contains all the numbers R has no gaps
- Some answers refer to the line (4 out of 11): Complete is a line without gaps, an infinity density of points R is complete, you can see it if you represent R as a continuous line, every point is a real number. If we take Q, for instance, that does not happen, because between two rational numbers there is a real number. If I cut a line with another line, there is a real number there. That is, it has no gaps. I think of complete as replete, without empty spaces. An example could be the line, the line of Euclid, a sequence of infinite points, one glued to another...
The last answer shows that the representation on the line does not necessarily help to better understand what completeness is, as the line is viewed as a sequence of points [numerable].
- Some answers are nearer to being operational, describing R as a set such that the cuts of numbers, in the sense of Dedekind, are neither jumps nor lacunas (3 out of 11) R is complete is the same as R is continuous. That is, it has no jumps, nor lacunas. Between two points of Q there is one irrational, and between two irrational there is a point of Q. Then, theres no gap on R.
- One answer gives an operational definition. R is complete, that means it has no jumps, nor lacunas. It has the property of supremum, and it is continuous.
Seven out of 11 students seem to have an intuitive idea of what completeness is, but this idea is not sufficient to be used in proof or in exercises. Still a big proportion of students recall the notion of line, but it does not necessarily help them to better conceptualize the notion of completeness.
4.4 Answers to the question: What does R is a complete set mean for you? by students in Course IV
- Seven out of 10 students give operational answers (there are as many real numbers as points on the line, upper bounded sets have supremum, every Cauchy sequence is convergent to an element of the set, it has no gaps, every bounded-above set has a supremum and every Cauchy sequence is convergent, completeness guarantees the existence of several elements, via the axiom of supremum or other equivalent.
- Three answers are non-operational, referring to the line or comparing to Q (real numbers cover the line, to each point corresponds a real number real number complete the line, between two real numbers there is another, one cannot take the former number
Synthesis and conclusions
The answers to the questionnaire show that only a few students of Course IV see completeness as a tool to define new elements. A possible explanation to this can be related to the fact that students can use strong theorems, so that they are not obliged to directly face completeness. Most of the students do not know which problems completeness solve.
I think that the aspect object is surprisingly weak for students in Course III, if we compare with the type of tasks they solve in this course. The majority of the students expressed completeness in a non-operational way: by means of the daily use of the word complete or by means of images. In Course IV this changes: 7 students out of 10 give an operational characterization.
The expression complete means that it has no gaps in referring to completeness can be thought as a degenerate version of the statement every cut of the set has an unique element of separation. In a similar way, the expression getting closer is a way of talking about approximation, which is posing the distance between two objects less than some positive number. Both are examples of weak and non-operational images of mathematical definitions. It might be interesting to design learning situations to help students to turn these images into mathematical statements.
To understand completeness as a property or an axiom that answers to a veritable problem demands to situate oneself in a certain perspective that is not the most natural. It requires also a reflection that neither seems to appear spontaneously as a result of solving the exercises, nor belongs to the natural role of students. For most of students, doing the typical exercises of supremum does not have as a consequence understanding that R is the set that contains all the suprema of its bounded above subsets. Few students can perceive that Cauchy sequences come from the necessity of characterizing the kind of sequences that must converge to develop analysis, and that completeness has to do with the fact that the limit belongs to the set. All these aspects, that in general remain in the sphere of private work of the student, are elements that I think are important to take in account when preparing syllabi, and designing exercises or a course.
References
Barbosa S. y Da Silva B. (2001). Concepes dos alunos sobre os nmeros reais, en Educao Matemtica, a prtica educativa sob o olhar de professores de Clculo, 39-67. Ed FUMARC.
Berg, A and Sessa, C. (2003) Continuidad y completitud revisadas a travs de 23 siglos. Aportes a una investigacin didctica. Revista RELIME (Revista Latinoamericana de Investigacin en Matemtica Educativa, Centro de Investigacin y de Estudios Avanzados del IPN, Mxico), 6(3) 163-197.
Berg, A. (2008) The Completeness Property of the Set of Real Numbers in the Transition from Calculus to Analysis. Educational Studies in Mathematics, vol. 67(3) 217-235
Bronner, A. (1997) tude didactique des nombres rels: idcimalit et racine carr. Thse de doctorat, Universit de Grenoble I.
Castela, C. (1997). La droite des rel en seconde: point dappui disponible ou enjeux clandestin? IREM de Rouen, Universit de Rouen
Chevallard, Y. (1985). La transposition didactique. La Pense Sauvage, Grenoble.
Fischbein, E.; Jehiam, R. and Cohen, D. (1995) The concept of irrational number in High-School Students and Prospective Teachers. Educational Studies in Mathematics, vol. 29 (1)/29-44.
Nuez, R.; Lakoff, E. & Matos J. (1999), Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics, vol. 39/1-3
Nuez, R.; Lakoff, E. (2000). Where mathematics comes from. How the emboidiedmind bring mathematics into being. Basic Books. New York.
Robinet, J. (1986). Les rels: Quels modles en ont les lves. Educational Studies in Mathematics 17 pp 359-386
Sirotic, N and Zazkis, A (2007) Irrational Numbers: The Gap between Formal and Intuitive Knowledge. Educational Studies in Mathematics, vol. 65 (1) 49-76
Vergnaud, G. (1990). La thorie des champs conceptuels Recherches en Didactique des Mathmatiques, Vol 10/2.3, 133-170
For clarifying ideas: all cuts in the set of integer numbers are lacunas, all cuts in the set of rational numbers are jumps.
PAGE
PAGE 1
(
(
(
(
(
(
(
(
(
(
Q
CE
6"!N!!("())n***0+..//11(2D2a2b2c2i2X4߸߭߭߭ѥ߭߭߭߭߭߄h/5CJmH sH h/6 h/<h/<B*CJphh/CJPJh/6CJmH sH h/CJH*mH sH h/B*CJmH phsH h/CJOJQJmH sH h/CJmH sH
h/CJ
h/5CJh/ h/54QR_z ~d $d a$xd $d a$ $d xa$d xd xqr*rYr"()n*o**2+./1D2c2X4t45689::`d
x`d x
xx`xd x`X4t4444444444444444444444444444444444444444444c5d5h5i5j5˽֖˽{ jh/CJH*mH sH j$h/CJmH sH jh/CJmH sH j"h/CJmH sH jh/CJmH sH jh/CJH*mH sH h/CJH*mH sH h/6CJH*mH sH h/6CJmH sH h/CJmH sH h/mH sH 0j5k5m5n5q5r5s5t5x5y5{5|555555555555555555555668888888µϛϐϵσϐϨϐugh/B*CJmH phsH jh/CJH*mH sH jh/CJmH sH h/CJH*mH sH j$h/CJmH sH jeh/CJmH sH j"h/CJmH sH jh/CJmH sH h/CJmH sH h/6CJH*mH sH h/6CJmH sH jh/CJH*mH sH (8888798999:9c9d9g9h9u9v9x9y9z9{9|9~999999999999999999999:ݶݩݞݑݞu jh/CJH*mH sH jh/CJH*mH sH j\h/CJmH sH h/CJH*mH sH jh/CJmH sH j"h/CJmH sH jh/CJmH sH j$h/CJmH sH h/CJmH sH h/6CJH*mH sH h/6CJmH sH (:::;<==>>>>>H>K>L>h@i@@@AAAAAAAAAAAɼܓ܌܁|qqcqV j$h/CJmH sH jh/CJH*mH sH h/CJH*mH sH h/5*h/CJmH sH
h/6CJh/6CJmH sH jh/CJ(aJ(mH sH jh/CJ aJ mH sH jh/CJmH sH jh/CJaJmH sH h/h/CJmH sH h/5CJmH sH h/B*CJmH phsH :;<<,==>>>?i@@@AABB(C?CD^E_EEEF,G-G`d
x`AAAAAAAABBBBBBBBBBBBBBBBBBBBBB B#B$B%B(BdBeBqBrBsBtBuBvByBzB{B|B}B~BBBBBBBBBܴܴܜܴ jh/CJmH sH jh/CJmH sH h/CJH*mH sH j"h/CJmH sH jh/CJmH sH h/6CJH*]mH sH h/CJmH sH jh/CJmH sH h/6CJ]mH sH 8BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBΧΚΚδδreδr jh/CJmH sH jnh/CJH*mH sH jh/CJmH sH h/6CJ]mH sH jnh/CJmH sH j$h/CJmH sH jeh/CJmH sH j"h/CJmH sH h/CJmH sH h/CJH*mH sH jh/CJH*mH sH h/6CJH*]mH sH 'BBBBBBBBBBBBBBBBBBBBBBBBCCCC
CCC
CCCCCCCCCCCCC C!C"C&C'C)C*C+C.C/C3C4Ch/6CJH*]mH sH jeh/CJmH sH jh/CJmH sH jnh/CJmH sH j"h/CJmH sH h/CJH*mH sH h/6CJ]mH sH h/CJmH sH jh/CJmH sH 74C5C;Cr?r@rArBrCrDrErFrGrHrIrJrKrLrMrNrOrPrQrRrSrTrUrVrWrYrZrh/CJmH sH jh/CJ,
h/CJ,ht]0JmHnHu
h/0Jjh/0JUh/jh/U15r6r7r8r:r;r=r>r@rArCrDrFrGrIrJrLrMrOrPrRrSrTrUrWrXrYrZr
&Fx,1h/ =!"#$%Dd
RD
3@"?Dd
D
3@"?@@@NormalCJ_HaJmH sH tH
J@J Heading 1$@&5CJ\mH sH uJ@J Heading 2$@&5CJ\mH sH uDA@DDefault Paragraph FontVi@VTable Normal :V44
la(k@(No ListlOlICMI TextStyle$d P`a$OJPJQJaJmH sH uXQ@XBody Text 3$a$CJOJQJaJmH
sH
tH u8B@8 Body Text$a$CJB@"B
Footnote TextCJaJtH @&@1@Footnote ReferenceH*RPBRBody Text 2 B*OJQJaJmH ph3fsH u4 @R4Footer
!.)@a.Page Number`Or`
ICMI Heading2 $P"5CJOJPJQJaJmH sH uLCLBody Text Indentx^aJ44Header
!XR@XBody Text Indent 2$d `a$CJSZj
#&)+.Zj
YZ[\]^_`abcd
#&)+.1
ZjQR_z ~
" !n"o""2#&')D*c*X,t,-.0122344,556667i88899::(;?;<^=_===>,?-?|??????@@CC CCDDDDDF;GHH;IUIKPLNNOOORRRVTWTTT'V(VVV*X,X/Y0Y1Y2YLYZ+\k^&b'b2b3bb
dd8eeffyghsh
iijjjj
jj
jjjjjjj'j(j)j*j+j,j-j.j/j0j1j2j3j4j5j6j7j8j:j;j=j>j@jAjCjDjFjGjIjJjLjMjOjPjRjSjTjUjWjXj[jI0I0I0I0I0I0I0I0I0I0I0I0I0I0I0I0I0I0I000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0@000@0I00@0I00@0I00@0I00I00I00I00@0@0I00I00I0I0I0I0I0I0I0I0I0I0I0I00I00I00I00I00I0
90I090I090I090I090I090I090I0I0 $$$'X4j58:ABB4CKR1arZr:>?@ACDEFHIKN:-GS2ar5rZr;=BGJLMOYr<????@ @Zj__ '!!48rs@r((
Tb &(/
r#"f
T
s*X99?`'(/`B
U
c$D$ &$ Q/Z p&!D-
qp&!D-`B
W
c$DpD-!D-ZB
X
SDp+(X +(`
Y
C
T&(
`
Z
C0t*
,
`
[
C
)+
`
\
C)x'+
`
]
CX(hs*
`
^
C'X)
`
_
CY'Ht)
`
`
C'8))
`
a
C&)
`
b
Cm
+%-
`
c
Cd&(
`
d
C&(
pX
e3 S"`
f
c$X99?pX
ZB
g
SD$ L L TB
h
CD TB
i
CD TB
j
CD TB
k
CDGH TB
l
CD TB
m
CDde TB
n
CD TB
o
CD ZB
p
SD1 B
S ???Zjetr!
t ǜM^
ǜL^ǜ_ǜH^
ǜT_ǜ_zz[j[jB*urn:schemas-microsoft-com:office:smarttagscountry-region8*urn:schemas-microsoft-com:office:smarttagsCity=*urn:schemas-microsoft-com:office:smarttags PlaceName=*urn:schemas-microsoft-com:office:smarttags PlaceType9*urn:schemas-microsoft-com:office:smarttagsplace.5LS\chjyLT:=bbbbbbbbccc c!c*c-c3c:c@cBcIcLcOcPc]c^cgcicpcyccccccccccccccccdddddddddddddee eeeeee(e8e?eOeUeZe^ebeiereyezeeeeeeeeeeeeeeeffff%fffffg"gyg~ggggghhhh!h&h'h.h2h5h:h@hshzhhh
ii%i,i8iCiDiNiRi\iainipisiijjjjjjj
jj
jjjjjj&j)j*jXj[jYijjjjjj
jj
jjjjj&j[j33'+*-*--44W>e>zBBDDRRCVDVWWw[[]]^^'__iiiiijjjjjjjj
jj
jjjjj&j[jjjjjjj
jj
jjjj)j*j[j| ..J4?^BoL^`B*OJQJo(h^`OJQJo(h^`OJQJo(ohpp^p`OJQJo(h@@^@`OJQJo(h^`OJQJo(oh^`OJQJo(h^`OJQJo(h^`OJQJo(ohPP^P`OJQJo(h^`OJQJo(h^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@@^@`OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hH|J?^/t]ijj
j
jj)j[j@(TZj@Unknown Gz Times New Roman5Symbol3&z ArialI&??Arial Unicode MS9 Webdings3z TimesESouvenir Lt BT;Wingdings?5 z Courier New"1͊F͊FuRFY5Y5!4dOiOi!2QHX?t]2PStudents perceptions of the Property of Completeness of the Set of Real NumbersAnalia BergeJohann EngelbrechtOh+'0 ,8L ht
TStudents perceptions of the Property of Completeness of the Set of Real NumbersAnalia BergeNormal.dotJohann Engelbrecht2Microsoft Office Word@F#@@VZ@VZY՜.+,0Dhp
% Analia Berge5OiQStudents perceptions of the Property of Completeness of the Set of Real NumbersTitle
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPRSTUVWXZ[\]^_`abcdefghijklmnopqrstuvwxyz{|~Root Entry FP\ZData
Q1TableYFWordDocument.SummaryInformation(}DocumentSummaryInformation8CompObjq
FMicrosoft Office Word Document
MSWordDocWord.Document.89q