# Binary Decimals - Dual Expansion

• clg211
In summary: To prove the bijection between (0,1) and the set of all binary sequences, the set of binary decimals on [0,1] is divided into two sets: one containing all the repeating tails of 1 and the other containing all the non-repeating tails of 1. To eliminate the dual expansions, the repeating tails of 1 are removed from the set of binary decimals. This may help in explicitly defining (0,1) in terms of the set of all sequences of 0's and 1's.
clg211
Hi,

I'm trying to prove that there's a bijection between the open interval (0,1) and the set of all sequences whose elements are 0 or 1 in order to show cardinality continuum.

So let C={a1, a2, a3,...|ai is either 0 or 1} which is the set of all sequences of 0's and 1's
and let D={0.b1b2b3...|bi is either 0 or 1} which is the set of all binary decimals on the closed interval [0,1]
I think it's pretty clear that there's a bijection between these two sets.

Then the open interval (0,1)=D\{0.000..., binary decimals with tails of repeating 1's} which is the part that gives me problems.

I'm trying to get rid of the dual expansions by getting rid of the tails of 1's. For example, I have both 0.1 and 0.0111... in D which are the same number, and I want to get rid of 0.0111...

Can someone please explain to me the rational numbers that will have this dual expansion in binary? A denominator of what form will cause this? I think this will help me in explicitly defining (0,1) in terms of D. Any other thoughts on what I've already stated would be appreciated as well.

Repeating tails of 1 in the binary expansions correspond to numbers with denominators that are powers of two.

## 1. What are binary decimals?

Binary decimals, also known as binary numbers, are a base-2 number system used in computers to represent numeric data. They consist of only two digits, 0 and 1, and are used to store and process information in a digital format.

## 2. How are binary decimals converted to decimal numbers?

To convert a binary decimal to a decimal number, each digit is multiplied by its corresponding power of 2 and then added together. For example, the binary decimal 1011 would be converted to decimal by multiplying 1 by 2^3, 0 by 2^2, 1 by 2^1, and 1 by 2^0, and then adding the results (8+0+2+1) to get the decimal number 11.

## 3. What is dual expansion in binary decimals?

Dual expansion is a method of representing fractions in binary decimals. It involves multiplying the fraction by 2 and then separating the whole number and fractional parts. The whole number part becomes the first binary decimal digit, and the process is repeated with the fractional part until the desired accuracy is achieved.

## 4. How are binary decimals used in computing?

Binary decimals are used in computing as the fundamental language of computers. They are used to represent and process data, perform calculations, and communicate instructions between hardware and software components.

## 5. What are some advantages of using binary decimals?

There are several advantages of using binary decimals in computing. They are simple and efficient to process, require less storage space compared to other number systems, and are easily transmitted and interpreted by digital devices. Additionally, binary decimals can accurately represent and process complex data and allow for precise control and manipulation of information in computer systems.

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