Anonymous
6/2/2025, 9:39:29 AM
No.16684085
>>16684067
I will respond as plainly as possible. I was never intending to have an argument, I was simply stating how the rationals and irrationals are defined and pointing out how it doesn't have to do with the choice of a base. I apologize if I came off as flippant, I was assuming you knew more about the topic, but the question you asked afterward makes it clear that you also have your own questions about numbers. I will answer some.
>You can't "express" something if your number system doesn't allow it.
It is possible to produce extensions of a number system. In the real numbers, this is done by something called a completeness property. The overall idea is pretty abstract, but one way to explain it is to create a sequence of numbers which can be thought of as an approximation, of a number that does not yet exist, then defining algebraic operations on these approximations. See this on Wikipedia: https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Cauchy_completeness
>Show me the formula or proof that 3.747234235 is rational or rational because it can or can't be expressed by a ratio of two integers.
That is a finite sequence of digits. It is rational. In particular, it is equal to the ratio 3747234235/1000000000
>Also, this is a genuine question, but what is the point of defining something as exactly two integers instead of a chain of ratios?
If you can express the chain of ratios in a way which provides for consistent and unambiguous algebraic rules (this is called a well-defined representation) then from a mathematical point of view, there is no problem. I am not sure if you are aware of this, but what you are talking about sounds like something called continued fractions: https://en.wikipedia.org/wiki/Continued_fraction
I will respond as plainly as possible. I was never intending to have an argument, I was simply stating how the rationals and irrationals are defined and pointing out how it doesn't have to do with the choice of a base. I apologize if I came off as flippant, I was assuming you knew more about the topic, but the question you asked afterward makes it clear that you also have your own questions about numbers. I will answer some.
>You can't "express" something if your number system doesn't allow it.
It is possible to produce extensions of a number system. In the real numbers, this is done by something called a completeness property. The overall idea is pretty abstract, but one way to explain it is to create a sequence of numbers which can be thought of as an approximation, of a number that does not yet exist, then defining algebraic operations on these approximations. See this on Wikipedia: https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Cauchy_completeness
>Show me the formula or proof that 3.747234235 is rational or rational because it can or can't be expressed by a ratio of two integers.
That is a finite sequence of digits. It is rational. In particular, it is equal to the ratio 3747234235/1000000000
>Also, this is a genuine question, but what is the point of defining something as exactly two integers instead of a chain of ratios?
If you can express the chain of ratios in a way which provides for consistent and unambiguous algebraic rules (this is called a well-defined representation) then from a mathematical point of view, there is no problem. I am not sure if you are aware of this, but what you are talking about sounds like something called continued fractions: https://en.wikipedia.org/wiki/Continued_fraction