Base changing for transcendental numbers

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Discussion Overview

The discussion revolves around the possibility of converting transcendental numbers into terminating decimals through base changing, specifically within integer-based number systems. Participants explore the implications of base conversion on the nature of transcendental numbers and the characteristics of their representations in different bases.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether a transcendental number can be expressed as a terminating decimal in any integer-based number system.
  • Another participant clarifies that changing a transcendental number from one base to another does not alter its transcendental nature, and provides an example of expressing a number in its own base.
  • A later reply emphasizes that while the base change does not change the nature of the number, the representation of rational numbers can vary between bases, leading to terminating or repeating decimals.
  • Some participants express confusion about terminology, particularly regarding the distinction between transcendental and transfinite numbers.
  • One participant asserts that all numbers are finite and that any repeating pattern of digits indicates a rational number, suggesting a misunderstanding of the original question.

Areas of Agreement / Disagreement

Participants generally agree that changing the base of a transcendental number does not change its classification. However, there is disagreement regarding the possibility of representing transcendental numbers as terminating decimals in integer bases, with some participants expressing uncertainty about the implications of base conversion.

Contextual Notes

Participants have not reached a consensus on the original question regarding the conversion of transcendental numbers into finite representations in integer bases. There are also varying interpretations of the definitions and properties of transcendental and rational numbers.

Stress2Death
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Hi All,
This might be a silly question but can anyone tell me with certainty if it is possible to convert a transcendental number into a terminating decimal by base changing?

If that is possible that is insanely awesome.

[edit] Sorry that was completely not what I was wondering. I meant this:

Does an integer based number system exist, wherein some transcendental number when converted into this number system is a finite number.

I think I worded it ok that time.
 
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First, decimal means base 10. You can't take a transcendental number written in base 10, and change to base 10, and expect the expression to be different. (This is not correct for all real numbers. But you only asked for transcendental.)

If you are asking "given a transcendental number x, does there exist a base, so that x can be written finitely in that base?" Then the answer is yes: base x. For example the golden ratio is "10" in base phinary.

Edit: I just realized that phi is not transcendental. But the idea is the same.
 
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In other words, given any transcendental number, x, we can write x in the number system having base x, as "10".
 
Hi Folks & thanks for responding. I think I should have been more explicit in my question though. I'm going to go back and change it. I meant for the base to be an integer.

As far as me calling it a decimal... yeah that was pretty stupid.
 
like how 2^{\aleph_0}=10^{\aleph_0} which equals \aleph_1
is that what you mean.
edit: I confused transcedental with transfinite. but I still think you could change the base.
 
Stress2Death said:
Hi Folks & thanks for responding. I think I should have been more explicit in my question though. I'm going to go back and change it. I meant for the base to be an integer.

As far as me calling it a decimal... yeah that was pretty stupid.

Change of base from one integer to another integer will not change the nature of a real number. Transcendental will stay transcendental, algebraic will stay algebraic, and rational will stay rational.
The only thing noticeable is that the decimal expression of rational numbers will terminate in some bases, but not in others. Example: 1/3 = .3333... in decimal, = .1 in base 3.
 
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Stress2Death said:
Hi All,
This might be a silly question but can anyone tell me with certainty if it is possible to convert a transcendental number into a terminating decimal by base changing?

If that is possible that is insanely awesome.

[edit] Sorry that was completely not what I was wondering. I meant this:

Does an integer based number system exist, wherein some transcendental number when converted into this number system is a finite number.

I think I worded it ok that time.

All numbers are finite. Any repeating pattern of digits no matter what the base will give you a rational number.
 

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