Real Numbers: Show Base b Analogy Properties

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In summary, the real numbers have b-nary expansions with analogous properties, where b is any integer greater than 1. This can be shown by using a bijective function between the real numbers base ten and any other base, and making slight changes to the range of values and denominators.
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Diffy
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Question:
Decimal (10-nary) expansions of real numbers were defined by special reference to the number 10. Show that the real numbers have b-nary expansions with analogous properties, where b is any integer greater than 1.

Attempt at solution:
I think if I show that there is a bijective function between the real numbers base ten, and any other base that will show they have analogous properties.

so let a0.a1a2... be any real number, where a0is any integer and ai i >0 and i /in {0,1,2,...,9}.

Then it has been shown (in the book) that this can be represented as
a0 + a1/10 + a2/102 + ...

I think now I need to show that this number can be changed into base b which I am not quite sure how to do. And even once I have done that, I am not sure that I am any closer to solving the problem.

Any help is appreciated.

Thanks

-Dif
 
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  • #2
The above proof for 10 carries over to base b with the following changes.
1) range of a's is (0,b-1).
2) denominators are powers of b rather than powers of 10.
 

1. What are real numbers?

Real numbers are numbers that can be found on the number line. They include both positive and negative numbers, as well as zero. Examples of real numbers include 3, -5, 0, 1/2, and the square root of 2.

2. What is the base b analogy for real numbers?

The base b analogy for real numbers is similar to the decimal system we use in everyday life. Just like how the decimal system is based on powers of 10, the base b analogy is based on powers of the base b. For example, in base 2, the numbers are represented using only 0 and 1, and each place value is a power of 2 (1, 2, 4, 8, etc.).

3. What are the properties of real numbers?

The properties of real numbers include the commutative property, associative property, distributive property, identity property, and inverse property. These properties dictate how real numbers can be manipulated and combined in mathematical operations.

4. How do you show the properties of real numbers using the base b analogy?

To show the properties of real numbers using the base b analogy, we can use examples and numerical proofs. For instance, to show the commutative property, we can demonstrate how the order of numbers does not affect the result when adding or multiplying in different bases.

5. Why is it important to understand the properties of real numbers?

Understanding the properties of real numbers is essential in solving mathematical problems and equations. These properties allow us to manipulate numbers and equations to simplify them and find solutions. They also help us in understanding the patterns and relationships between numbers, which can be applied in various real-world situations.

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