Relevant bibliographies by topics / Visualisation of rational numbers

Academic literature on the topic 'Visualisation of rational numbers'

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Published: 4 June 2021
Last updated: 28 January 2023

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Journal articles on the topic "Visualisation of rational numbers"

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., Jyoti. "Rational Numbers." Journal of Advances and Scholarly Researches in Allied Education 15, no. 5 (July 1, 2018): 220–22. http://dx.doi.org/10.29070/15/57856.

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Scott Malcom, P. "Understanding Rational Numbers." Mathematics Teacher 80, no. 7 (October 1987): 518–21. http://dx.doi.org/10.5951/mt.80.7.0518.

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Understanding is sometimes an elusive goal in mathematics. Although we may believe we have a complete understanding of a concept, another approach to this same concept may bring us additional insight.
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Lennerstad, Håkan, and Lars Lundberg. "Decomposing rational numbers." Acta Arithmetica 145, no. 3 (2010): 213–20. http://dx.doi.org/10.4064/aa145-3-1.

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PEYTON JONES, SIMON. "12 Rational Numbers." Journal of Functional Programming 13, no. 1 (January 2003): 149–52. http://dx.doi.org/10.1017/s0956796803001412.

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Gong, Linming, Bo Yang, Tao Xue, Jinguang Chen, and Wei Wang. "Secure rational numbers equivalence test based on threshold cryptosystem with rational numbers." Information Sciences 466 (October 2018): 44–54. http://dx.doi.org/10.1016/j.ins.2018.07.046.

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Marcos, J. E. "Topological completions of the field of rational numbers which consist of Liouville numbers and rational numbers." Journal of Pure and Applied Algebra 138, no. 3 (May 1999): 251–77. http://dx.doi.org/10.1016/s0022-4049(98)00053-x.

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Hill, Rosemary Lucy. "The political potential of numbers: data visualisation in the abortion debate." Kvinder, Køn & Forskning, no. 1 (September 5, 2017): 83–96. http://dx.doi.org/10.7146/kkf.v26i1.109789.

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Data visualisation has been argued to have the power to ‘change the world’, implicitly for the better, but when it comes to abortion, both sides make moral claims to ‘good’. Visualisation conventions of clean lines and shapes simplify data, lending them a rhetoric of neutrality, as if the data is the whole story. It is imperative, therefore, to examine how data visualisations are used to shape women’s lives. This article draws on the findings of the Persuasive Data project . Google Image Scraper was used to locate abortion-related visualisations circulating online. The images, their web locations, and data use were social semiotically analysed to understand their visual rhetoric and political use. Anti-abortion groups are more likely to use data visualisation than pro-choice groups, thereby simplifying the issue and mobilising the rhetoric of neutrality. I argue that data visualisations are being used as a hindrance to women’s access to abortion, and that the critique of such visualisations needs to come from feminists. This article extends discussions of how data is often reified as objective, by showing how the rhetoric of objectivity within data visualisation conventions is harnessed to do work in the world that is potentially very damaging to women’s rights.
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Sándor, József. "On certain rational perfect numbers." Notes on Number Theory and Discrete Mathematics 28, no. 2 (May 12, 2022): 281–85. http://dx.doi.org/10.7546/nntdm.2022.28.2.281-285.

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Atanasiu, Dragu. "Laplace Integral on Rational Numbers." MATHEMATICA SCANDINAVICA 76 (December 1, 1995): 152. http://dx.doi.org/10.7146/math.scand.a-12531.

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Detorie, Rick. "Let's Be Rational about Numbers." Mathematics Teaching in the Middle School 20, no. 7 (March 2015): 394–97. http://dx.doi.org/10.5951/mathteacmiddscho.20.7.0394.

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Dissertations / Theses on the topic "Visualisation of rational numbers"

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Tolmie, Julie, and julie tolmie@techbc ca. "Visualisation, navigation and mathematical perception: a visual notation for rational numbers mod1." The Australian National University. School of Mathematical Sciences, 2000. http://thesis.anu.edu.au./public/adt-ANU20020313.101505.

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There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. ¶ The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. ¶ The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.
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Tolmie, Julie. "Visualisation, navigation and mathematical perception : a visual notation for rational numbers mod 1." View thesis entry in Australian Digital Theses Program, 2000. http://thesis.anu.edu.au/public/adt-ANU20020313.101505/index.html.

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Coward, Daniel R. "Sums of two rational cubes." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.320587.

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Ketkar, Pallavi S. (Pallavi Subhash). "Primitive Substitutive Numbers are Closed under Rational Multiplication." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278637/.

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Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
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Brown, Bruce John Lindsay. "The initial grounding of rational numbers : an investigation." Thesis, Rhodes University, 2007. http://hdl.handle.net/10962/d1006351.

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This small scale exploratory research project investigated the grounding of rational number concepts in informal, everyday life situations. A qualitative approach was taken to allow for the identification and then in depth investigation, of issues of importance for such a grounding of rational number understanding. The methodology followed could be seen as a combination of grounded theory and developmental research. And the data was generated through in-depth and clinical interviews structured around a number of grounded tasks related to rational numbers. The research comprised three cycles of interviews that were transcribed and then analysed in detail, interspersed with periods of reading and reflection. The pilot cycle involved a single grade three teacher, the second cycle involved 2 grade three teachers and the third cycle involved 2 grade three children. The research identified a number of different perspectives that were all important for the development of a fundamental intuitive understanding that could be considered personally meaningful to the individual concerned and relevant to the development of rational number concepts. Firstly in order to motivate and engage the child on a personal level the grounding situation needed to be seen as personally significant by the child. Secondly, coordinating operations provided a means of developing a fundamental intuitive understanding, through coordination with affording structures of the situation that are relevant to rational numbers. Finally, goal directed actions that are deliberately structured to achieve explicit goals in a situation are important for the development of more explicit concepts and skills fundamental for rational number understanding. Different explicit structures give rise to different interpretations of rational numbers in grounding situations. In addition to these perspectives, it became evident that building and learning representations was important for developing a more particularly mathematical understanding, based on the fundamental understanding derived from the child's grounded experience. The conclusion drawn in this research as a result of this complexity, is that to achieve a comprehensive and meaningful grounding, children's learning of rational numbers will not follow a simple linear trajectory. Rather this process forms a web of learning, threading coordinating operations for intuitive development, interpretations for explicit grounding and representations to develop more formal mathematical conceptions.
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LORIO, MARCELO NASCIMENTO. "APPROXIMATIONS OF REAL NUMBERS BY RATIONAL NUMBERS: WHY THE CONTINUED FRACTIONS CONVERGING PROVIDE THE BEST APPROXIMATIONS?" PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2014. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=23981@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Frações Contínuas são representações de números reais que independem da base de numeração escolhida. Quando se trata de aproximar números reais por frações, a escolha da base dez oculta, frequentemente, aproximações mais eficientes do que as exibe. Integrar conceitos de aproximações de números reais por frações contínuas com aspectos geométricos traz ao assunto uma abordagem diferenciada e bastante esclarecedora. O algoritmo de Euclides, por exemplo, ao ganhar significado geométrico, se torna um poderoso argumento para a visualização dessas aproximações. Os teoremas de Dirichlet, de Hurwitz-Markov e de Lagrange comprovam, definitivamente, que as melhores aproximações de números reais veem das frações contínuas, estimando seus erros com elegância técnica matemática incontestável.
Continued fractions are representations of real numbers that are independent of the choice of the numerical basis. The choice of basis ten frequently hides more than shows efficient approximations of real numbers by rational ones. Integrating approximations of real numbers by continued fractions with geometrical interpretations clarify the subject. The study of geometrical aspects of Euclids algorithm, for example, is a powerful method for the visualization of continued fractions approximations. Theorems of Dirichlet, Hurwitz-Markov and Lagrange show that, definitely, the best approximations of real numbers come from continued fractions, and the errors are estimated with elegant mathematical technique.
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Carbone, Rose Elaine. "Elementary Teacher Candidates’ Understanding of Rational Numbers: An International Perspective." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79565.

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This paper combines data from two different international research studies that used problem posing in analyzing elementary teacher candidates’ understanding of rational numbers. In 2007, a mathematics educator from the United States and a mathematician from Northern Ireland collaborated to investigate their respective elementary teacher candidates’ understanding of addition and division of fractions. A year later, the same US mathematics educator collaborated with a mathematics educator from South Africa on a similar research project that focused solely on the addition of fractions. The results of both studies show that elementary teacher candidates from the three different continents share similar misconceptions regarding the addition of fractions. The misconceptions that emerged were analyzed and used in designing teaching strategies intended to improve elementary teacher candidates’ understanding of rational numbers. The research also suggests that problem posing may improve their understanding of addition of fractions.
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Clark, David Alan. "The Euclidean algorithm for Galois extensions of the rational numbers." Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=39408.

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Let K be a totally real, quartic, Galois extension of $ doubq$ whose ring of integers R is a principal ideal domain. If there is a prime ideal p of R such that the unit group maps onto $(R/{ bf p} sp2$)*, then R is a Euclidean domain. This criterion is generalized to arbitrary Galois extensions.
Let E be an elliptic curve over a number field F. Suppose ($F: doubq rbrack le 4$ and $F(E lbrack q rbrack ) not subseteq F$ for all primes q such that F contains a primitive $q sp{ rm th}$ root of unity, then the reduced elliptic curve $ tilde{E}(F sb{ bf p})$ is cyclic infinitely often. In general, if $ Gamma$ a subgroup of $E(F)$ with the range of $ Gamma$ sufficiently large, there are infinitely many prime ideals p of F such that the reduced curve $ tilde{E}(F sb{ bf p}) = Gamma sb{ bf p}$, where $ Gamma sb{ bf p}$ is the reduction modulo p of $ Gamma$.
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Bruyns, P. "Aspects of the group of homeomorphisms of the rational numbers." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375224.

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Millsaps, Gayle M. "Interrelationships between teachers' content knowledge of rational number, their instructional practice, and students' emergent conceptual knowledge of rational number." Connect to resource, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1124225634.

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Thesis (Ph. D.)--Ohio State University, 2005.
Title from first page of PDF file. Document formatted into pages; contains xviii, 339 p.; also includes graphics (some col.). Includes bibliographical references (p. 296-306). Available online via OhioLINK's ETD Center
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Books on the topic "Visualisation of rational numbers"

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Rational numbers: Poems. [Kirksville, Mo.]: Truman State University Press, 2000.

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S, Bezuk Nadine, ed. Understanding rational numbers and proportions. Reston, Va: National Council of Teachers of Mathematics, 1994.

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Mary, Stroh, and Sopris West Inc, eds. TransMath: Making sense of rational numbers. Longmont, Colo: Cambium Learning/Sopris West, 2010.

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P, Carpenter Thomas, Fennema Elizabeth, and Romberg Thomas A, eds. Rational numbers: An integration of research. Hillsdale, N.J: Lawrence Erlbaum Associates, 1992.

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Rational number theory in the 20th century: From PNT to FLT. London: Springer, 2012.

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Lappan, Glenda. Bits and pieces I: Understanding rational numbers. Palo Alto, CA: Dale Seymour Publications, 1998.

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Lappan, Glenda. Bits and pieces I: Understanding rational numbers. Palo Alto, CA: Dale Seymour Publications, 1998.

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1948-, Silver Edward A., and Stein Mary Kay, eds. Improving instruction in rational numbers and proportionality. New York: Teachers College Press, 2005.

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H, Salzmann, ed. The classical fields: Structural features of the real and rational numbers. Cambridge: Cambridge University Press, 2007.

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Bellos, Alex. Here's looking at Euclid: A surprising excursion through the astonishing world of math. New York: Free Press, 2010.

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Book chapters on the topic "Visualisation of rational numbers"

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Eriksson, Kenneth, Donald Estep, and Claes Johnson. "Rational Numbers." In Applied Mathematics: Body and Soul, 71–86. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-05796-4_7.

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Bhattacharjee, Meenaxi, Rögnvaldur G. Möller, Dugald Macpherson, and Peter M. Neumann. "Rational Numbers." In Notes on Infinite Permutation Groups, 77–86. Gurgaon: Hindustan Book Agency, 1997. http://dx.doi.org/10.1007/978-93-80250-91-5_9.

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Bhattacharjee, Meenaxi, Dugald Macpherson, Rögnvaldur G. Möller, and Peter M. Neumann. "Rational numbers." In Lecture Notes in Mathematics, 77–86. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0092559.

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Shah, Nita H., and Vishnuprasad D. Thakkar. "Rational Numbers." In Journey from Natural Numbers to Complex Numbers, 47–60. Boca Raton : CRC Press, 2021. | Series: Advances in mathematics and engineering: CRC Press, 2020. http://dx.doi.org/10.1201/9781003105244-3.

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Noël, Marie-Pascale, and Giannis Karagiannakis. "Rational numbers." In Effective Teaching Strategies for Dyscalculia and Learning Difficulties in Mathematics, 236–94. London: Routledge, 2022. http://dx.doi.org/10.4324/b22795-6.

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Ovchinnikov, Sergei. "Rational Numbers." In Real Analysis: Foundations, 1–30. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64701-8_1.

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Stillwell, John. "Rational Points." In Numbers and Geometry, 111–42. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0687-3_4.

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Kramer, Jürg, and Anna-Maria von Pippich. "The Rational Numbers." In Springer Undergraduate Mathematics Series, 93–139. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-69429-0_3.

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Stillwell, John. "The Rational Numbers." In Elements of Algebra, 18–37. New York, NY: Springer New York, 1994. http://dx.doi.org/10.1007/978-1-4757-3976-3_2.

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Kay, Anthony. "Rational Numbers, ℚ." In Number Systems, 107–48. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9780429059353-6.

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Conference papers on the topic "Visualisation of rational numbers"

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Dejdumrong, Natasha. "Efficient Algorithms for Non-Rational and Rational Bézier Curves." In 2008 5th International Conference on Computer Graphics, Imaging and Visualisation (CGIV). IEEE, 2008. http://dx.doi.org/10.1109/cgiv.2008.62.

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Sarfraz, M., M. Z. Hussain, T. S. Shaikh, and R. Iqbal. "Data Visualization Using Shape Preserving C2 Rational Spline." In 2011 15th International Conference Information Visualisation (IV). IEEE, 2011. http://dx.doi.org/10.1109/iv.2011.91.

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Sarfraz, Muhammad, Farsia Hussain, and Malik Zawwar Hussain. "Shape Preserving Positive Rational Trigonometric Spline Surfaces." In 2015 19th International Conference on Information Visualisation (iV). IEEE, 2015. http://dx.doi.org/10.1109/iv.2015.77.

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Sarfraz, Muhammad, Malik Zawwar Hussain, and Tahira Sumbal Shaikh. "Visualization of Positive Data by Rational Cubic Spline Interpolant." In 2010 14th International Conference Information Visualisation (IV). IEEE, 2010. http://dx.doi.org/10.1109/iv.2010.82.

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Hussain, Malik Zawwar, Muhammad Sarfraz, Munaza Ishaq, and Sadaf Sarfraz. "Two Methods of Object Designing by Rational Splines." In 2014 18th International Conference on Information Visualisation (IV). IEEE, 2014. http://dx.doi.org/10.1109/iv.2014.76.

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Pion, Sylvain, and Chee K. Yap. "Constructive root bound for k-ary rational input numbers." In the nineteenth conference. New York, New York, USA: ACM Press, 2003. http://dx.doi.org/10.1145/777792.777831.

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Cheng, Howard, and Eugene Zima. "On accelerated methods to evaluate sums of products of rational numbers." In the 2000 international symposium. New York, New York, USA: ACM Press, 2000. http://dx.doi.org/10.1145/345542.345581.

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Aphirukmatakun, Chanon, and Natasha Dejdumrong. "An Approach to the Feature-Based Comparisons for the Rational Curves." In 2008 5th International Conference on Computer Graphics, Imaging and Visualisation (CGIV). IEEE, 2008. http://dx.doi.org/10.1109/cgiv.2008.60.

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May, John P., B. David Saunders, and David Harlan Wood. "Numerical techniques for computing the inertia of products of matrices of rational numbers." In ISSAC07: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2007. http://dx.doi.org/10.1145/1277500.1277520.

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Ge, Q. J., and Donglai Kang. "Rational Bézier and B-Spline Ruled Surface Patches." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/dac-1495.

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Abstract This paper presents a geometric method for constructing bounded rational Bézier and B-spline ruled surfaces directly from line-segments. Oriented line-segments in a Euclidean three-space are represented by vectors with four homogeneous components over the ring of dual numbers. Projective algorithms for rational Bézier and B-spline curves are dualized to yield algorithms for rational Bézier and B-spline ruled surfaces.
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Reports on the topic "Visualisation of rational numbers"

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Lu, Chao. A Computational Library Using P-adic Arithmetic for Exact Computation With Rational Numbers in Quantum Computing. Fort Belvoir, VA: Defense Technical Information Center, November 2005. http://dx.doi.org/10.21236/ada456488.

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Lutz, Carsten. Adding Numbers to the SHIQ Description Logic - First Results. Aachen University of Technology, 2001. http://dx.doi.org/10.25368/2022.117.

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Recently, the Description Logic (DL) SHIQ has found a large number of applications. This success is due to the fact that SHIQ combines a rich expressivity with efficient reasoning, as is demonstrated by its implementation in DL systems such as FaCT and RACER. One weakness of SHIQ, however, limits its usability in several application areas: numerical knowledge such as knowledge about the age, weight, or temperature of real-world entities cannot be adequately represented. In this paper, we propose an extension of SHIQ that aims at closing this gap. The new Description Logic Q-SHIQ, which augments SHIQ by additional, 'concrete domain' style concept constructors, allows to refer to rational numbers in concept descriptions, and also to define concepts based on the comparison of numbers via predicates such as < or =. We argue that this kind of expressivity is needed in many application areas such as reasoning about the semantic web. We prove reasoning with Q-SHIQ to be EXPTIME-complete (thus not harder than reasoning with SHIQ) by devising an automata-based decision procedure.
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