Master's Dissertation
DOI
Document
Author
Full name
Daví Carlos Uehara Approbato
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2019
Supervisor
Committee
Galvez, Americo Lopez (President)
Bertoncello, Luciene Nogueira
Ebert, Marcelo Rempel
Pires, Benito Frazão
Title in Portuguese
Distância, na matemática e no cotidiano
Keywords in Portuguese
Códigos
Distância
Distância de Hamming
Distância genética
Espaço Métricos
Abstract in Portuguese
Title in English
Distance, in math and everyday life
Keywords in English
Code
Distance
Genetic distance
Hamming distance
Metric Space
Abstract in English
This work has as objective to discuss the formal concept of distance in mathematics, aiming to present examples of the distance concept in everyday situations. In general with this work we want the less familiar reader to understand the importance of the mathematical concept of distance. Distance is much more than the length of the segment between two points and this will be presented in each chapter. The subject was inspired by the book Encyclopedia of Distances Deza Michel Marie (2009), in which are presented, metric spaces, metrics in different areas and applications. In the second chapter, the definition of metric spaces will be presented. In the third chapter some examples of metrics will be presented. The first three metrics, the most common: usual, Euclidean, and maximum metrics in R and R2. Also the generalizations of each of them were presented in Rn. The next chapter, the fourth, is intended to show the study on normed spaces, because through these concepts we can analyze the distances between vectors and matrices. We will see that the relevance of these distances helps in the understanding of systems solutions approximation. In the chapter on distance of functions, a brief comment about Fourier series was presented, regarding the method of approximation through the decomposition of periodic functions. In order to analyze how the trigonometric functions are approaching, the concept of distance between functions is used, the measurements are made as the approximations increase, this distance "error" between them tends to zero. In codes theory, it is necessary to introduce the concept of distance between "words", this allows to verify if the code had some alteration, caused by an interference or noises during the trajectory. In some situations, the code can correct and understand the sent word even though it has undergone changes in the route. In these cases, there is Hammings metrics study. By the Hausdoorf metric, proposed by the mathematician of the same name, it is possible to calculate with more precision the distance between closed and limited sets. This metric can be used in face recognition studies, for example, because face images are transformed into clouds of dots. Then, through the Dijkstras algorithm will be presented the distance between the vertices of a convex graphic. There are several applications of distance between graphics and one of them is the issue of minimizing the cost of moving between a local carrier company and the place of delivery, for example. To finish the discussion about the importance of distance consensus, the distance between genes will be presented. Within this theme, the main scientist was Thomas Morgan, who through his studies managed to create the first genetic mapping. With this, he was able to relate the concept of distance between genes to the rate of gene recombination. Finally, an activity was elaborated with high school students with the objective of analyzing students knowledge about distance. This activity was also important so that the students could understand about a necessity to formalize this concept mathematically and, mainly, to motivate them through the presentation of applications on distance, in different scopes.