At the first stage of the construction, you remove the open interval $(1/3,2/3)$, or in ternary notation $(0.1,0.2)$. That removes all the numbers that have a 1 as the first digit in their ternary expansion. The left-hand endpoint of the interval, the point 0.1, is not removed. At first sight, it looks as though this fails to remove a point with a 1 as the first digit in its ternary expansion. However, the point 0.1 can also be represented as 0.022222... (recurring). So this point can in fact be represented without any 1s in its ternary expansion, and we have only removed those points in the unit interval which must have a 1 as the first digit in their ternary expansion.dwsmith said:Explain why the Cantor set consists precisely of all the numbers between 0 and 1 (including 0 and 1) which can be represented by a ternary expansion in which the digit 1 does not appear anywhere in the expansion.
I believe this has to do with always taking a 3rd away.
A Cantor Set Number is a type of number that has a unique ternary expansion, meaning it can be represented using only the digits 0 and 2. It is named after the mathematician Georg Cantor, who first studied these numbers in the late 19th century.
Unlike regular numbers, which use a base-10 system, Cantor Set Numbers use a base-3 system. This means that instead of having 10 digits (0-9), they have only 2 digits (0 and 2). Additionally, Cantor Set Numbers have a unique ternary expansion, while regular numbers can have multiple decimal expansions (e.g. 0.999... and 1.000... both represent the number 1 in base-10).
Cantor Set Numbers are calculated using a specific algorithm called the Cantor Set Construction. This involves dividing an interval into 3 equal parts, removing the middle third, and repeating the process infinitely. The resulting number is a Cantor Set Number.
Cantor Set Numbers have some interesting properties, such as being self-similar, meaning that they contain smaller copies of themselves. They are also uncountable, meaning that there are infinitely many of them. Additionally, they have a fractal nature, meaning that they exhibit similar patterns at different scales.
Cantor Set Numbers have applications in various areas of mathematics, including number theory, topology, and dynamical systems. They are also used in computer science, specifically in the design of fractal-based algorithms. Additionally, they have connections to other mathematical concepts such as the Cantor function and the Cantor set.