Thus, the Cantor set (which is the complement of this union) is closed. The Cantor set has a convenient representation using decimal expansions in base 3. Last active Jan 6, 2017. I have a lot of problems with these ternary expansions so I was wondering if you guys could check my work quick, and let me know where I have gone wrong. The Cantor ternary set consists of all real numbers in the unit interval that doesn't contain the digit 1 in their ternary decimal representation. THE CANTOR SET 25 Hedda: I think I know what you’re driving at. Cantor's (ternary) set, Cantor-(Drittel)Menge. The next order number represents itself times 3. (a)45. b)37. c) 101 (d) 1/4 (e) 3/5 Theorem: The cardinality of Cantor’s set is the continuum. Autre nom : poussière de Cantor. Endpoints.
View A real number between 0 and 1 is in the Cantor set precisely when none of the digits in its base-3 expansion equals 1; the exception is that a fraction whose denominator is a power of 3 (whose base-3 expansion therefore terminates) is allowed to end in a 1. Pages 4 Ratings 100% (2) 2 out of 2 people found this document helpful; This preview shows page 3 - 4 out of 4 pages. Embed. Embed Embed this gist in your website. What would you like to do? Georg Cantor (1845-1918) : mathématicien allemand. Base 3 or ternary has three numbers: 0, 1, and 2. The problem is to find out for what values of p (for integers between 0 and 13), is p/13 in the cantor set.
The next order of magnitude would be 3 x 3 x 3, or itself times 27. I think you have that wrong.
In base 3 the fraction 1 3 = .1 and 2 3 = .2, and this eliminates the need for the awkward fractions. … Cantor Set. 3. We will show that in fact Cantor’s set has amuch larger cardinality (i.e. The digits in base 3 are 0,1 and 2. topologicallytony / index.html. 0 3: 0.01 3: 0.02 3: 0.1 3: 0.2 3: 0.21 3: 0.22 3: 1 3: Step 0: Step 1: Step 2: Step 3: Step 4 : Step 5 . There are only countably many such endpoints. Convert each of the following numbers to base 3 then, determine and justify which of the numbers are in Cantor Set. GitHub Gist: instantly share code, notes, and snippets.
The next order number represents itself times 3 x 3, or itself times 9. The endpoints are given in base 3 notation. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.A copy of the license is included in the section entitled GNU Free Documentation License. THE CANTOR SET 25 Hedda: I think I know what you’re driving at. 3.
One way to characterize points in the Cantor set is to look at the base 3. 3. All gists Back to GitHub. "number" of elements). A number is in the set formed after the kth iteration of the process of removing middle thirds used to create the cantor set iff the kth digit in its base 3 expansion is not 1 (more specifically (to handle endpoints like 1/3=0.1=0.0222...), if it can be written in such a form).
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That is, it contains all of its accumulation points. School University of Utah; Course Title MATH 3220; Type. Homework Help. In this research another idea has been developed which is done by deleting the middle 737 th . The Cantor set has a convenient representation using decimal expansions in base 3. Moreover, every point of the Cantor set is an accumulation point, since within any neighborhood of a number whose ternary expansion consists entirely of …
The digits in base 3 are 0,1 and 2. 3. Since this is true, maybe it will be easier to prove. Assuming that we are writing all our numbers in base 3, rather than decimal, notation, the middle third of [0,1] is (.1,.2).
CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The middle-third Cantor set C3 is a fractal consisting of all the points in [0,1] which have non-terminating base-3 representations involving only the digits 0 and 2. I know that they are only in the cantor set if they can be expressed in the ternary expansion to base 3 using only 2's and 0's. Assuming that we are writing all our numbers in base 3, rather than decimal, notation, the middle third of [0,1] is (.1,.2). Shown here are results for cantor_dust 2 and for cantor_dust 3 . 1j1 #"1 expands the lines to improve aspect ratio on character cell (console) display. The lowest order number represents itself times one. Sign in Sign up Instantly share code, notes, and snippets.