Ever growing integers?

[BWG]

The decimal system, in mathematics, is a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot (decimal point) to represent decimal fractions. In this scheme, the numerals used in denoting a number take different place values depending upon position.

Then within this scheme for every number x between 0 and 1 (0 > 1) there must be a number y = x*10^n and y >= 1 that consists of the same number of digits as x, no? (not taking into account the zeros that go before the 1st non zero digit of x).

Examples of what I try to describe:

0,1 becomes 1 (10^1)

0,01 becomes 1 (10^2)

0,11 becomes 11 (10^2)

0,999 become 999 (10^3)

0,22456 becomes 22456 (10^5)

What happens with the infinitely repeating numbers like e.g., 1/3 (0,333…)? Then we get a repeating number greater than 1, an infinitely growing integer or something like that? Is there an annotation for this?

 

[Deleted member 4993]

The decimal system, in mathematics, is a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot (decimal point) to represent decimal fractions. In this scheme, the numerals used in denoting a number take different place values depending upon position.

Then within this scheme for every number x between 0 and 1 (0 > 1) there must be a number y = x*10^n and y >= 1 that consists of the same number of digits as x, no? (not taking into account the zeros that go before the 1st non zero digit of x).

Examples of what I try to describe:

0,1 becomes 1 (10^1)

0,01 becomes 1 (10^2)

0,11 becomes 11 (10^2)

0,999 become 999 (10^3)

0,22456 becomes 22456 (10^5)

What happens with the infinitely repeating numbers like e.g., 1/3 (0,333…)? Then we get a repeating number greater than 1, an infinitely growing integer or something like that? Is there an annotation for this?

Yes - infinitely repeating decimal numbers can be expressed as fraction s with '9' s in the denominator, e.g.

0.33333 ... = 3/9

0.88888... = 8/9

etc.

 

[blamocur]

Then within this scheme for every number x between 0 and 1 (0 > 1) there must be a number y = x*10^n and y >= 1 that consists of the same number of digits as x,

If you require 'y' to be integer than this statement is not true. Moreover, "the same number of digits" sounds ambiguous in cases when the number if digits is not finite, e.g. [imath]\pi[/imath].

 

[JeffM]

The decimal system, in mathematics, is a positional numeral system employing 10 as the base and requiring 10 different numerals, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It also requires a dot (decimal point) to represent decimal fractions. In this scheme, the numerals used in denoting a number take different place values depending upon position.

Then within this scheme for every number x between 0 and 1 (0 > 1) there must be a number y = x*10^n and y >= 1 that consists of the same number of digits as x, no? (not taking into account the zeros that go before the 1st non zero digit of x).

Examples of what I try to describe:

0,1 becomes 1 (10^1)

0,01 becomes 1 (10^2)

0,11 becomes 11 (10^2)

0,999 become 999 (10^3)

0,22456 becomes 22456 (10^5)

What happens with the infinitely repeating numbers like e.g., 1/3 (0,333…)? Then we get a repeating number greater than 1, an infinitely growing integer or something like that? Is there an annotation for this?

You have stumbled on to some of the stranger aspects of mathematics.

You must be very careful about the word “number” because it covers many different but closely related concepts.

I am going to define a “natural number” as the whole numbers starting with zero. (Many mathematicians define the natural numbers as starting with one.) I am going to define the “grade school numbers” as the natural numbers plus any fraction with a numerator that is a natural number and a denominator that is a natural number other than zero. Already we have two different concepts that are called numbers.

Decimal notation does not represent a new kind of number. It is just a way to specify which number we are talking about. We should really say “decimal numerals” rather than “decimal numbers.” The system works perfectly for representing all the natural numbers, but it is not perfect for representing all the grade school numbers.

As you saw, your process always generates a natural number if the decimal representation involved terminates. That is because your process then involves multiplying by a finite power of ten, which is a natural number. When you ask about the decimal representation of 1/3, there is no decimal representation that terminates.

One way to think about this is just to say that decimal representation is an imperfect system for representing grade school numbers and that there is no decimal representation of 1/3. In that case, your question makes no sense because you are trying to apply your process to something that does not even exist. (This is the finitist position that talking about the infinite is nonsense.)

Alternatively, you can say that infinity is a new kind of number (called a “transfinite number”), and apply your process to the decimal 0.3……… forever. That requires multiplying some number greater than zero but less than one by 10 to the infinite power. What is the result of that operation?

Hint: look at arithmetic operations in the following citation

Projectively extended real line - Wikipedia

en.wikipedia.org

In other words, once you let infinity into your mathematical wotld, you are no longer dealing with the natural numbers and have no basis to expect answers that are natural numbers.

Personal note: I find transfinite numbers and one-point compactification to be intellectually interesting but without any practical utility of any sort.

 

[Deleted member 4993]

You have stumbled on to some of the stranger aspects of mathematics.

You must be very careful about the word “number” because it covers many different but closely related concepts.

I am going to define a “natural number” as the whole numbers starting with zero. (Many mathematicians define the natural numbers as starting with one.) I am going to define the “grade school numbers” as the natural numbers plus any fraction with a numerator that is a natural number and a denominator that is a natural number other than zero. Already we have two different concepts that are called numbers.

Decimal notation does not represent a new kind of number. It is just a way to specify which number we are talking about. We should really say “decimal numerals” rather than “decimal numbers.” The system works perfectly for representing all the natural numbers, but it is not perfect for representing all the grade school numbers.

As you saw, your process always generates a natural number if the decimal representation involved terminates. That is because your process then involves multiplying by a finite power of ten, which is a natural number. When you ask about the decimal representation of 1/3, there is no decimal representation that terminates.

One way to think about this is just to say that decimal representation is an imperfect system for representing grade school numbers and that there is no decimal representation of 1/3. In that case, your question makes no sense because you are trying to apply your process to something that does not even exist. (This is the finitist position that talking about the infinite is nonsense.)

Alternatively, you can say that infinity is a new kind of number (called a “transfinite number”), and apply your process to the decimal 0.3……… forever. That requires multiplying some number greater than zero but less than one by 10 to the infinite power. What is the result of that operation?

Hint: look at arithmetic operations in the following citation

Projectively extended real line - Wikipedia

en.wikipedia.org

In other words, once you let infinity into your mathematical wotld, you are no longer dealing with the natural numbers and have no basis to expect answers that are natural numbers.

Personal note: I find transfinite numbers and one-point compactification to be intellectually interesting but without any practical utility of any sort.

I totally misunderstood the "question"....

 

[JeffM]

I totally misunderstood the "question"....

Not surprising. The original poster did not have the vocabulary to express the question clearly, and the question is unlikely to occur to anyone with the vocabulary to express it clearly. It is a question that arises as one first tries to think about what words like “number” mean. Despite its lack of clarity, it is a good question.

 

[BWG]

@JeffM Thanks for the explanation! It's indeed my lack of knowledge of how to describe what I want to ask and also english is not my mother language. The question came to my mind reading an article in new scientist about infinity and the different sizes of infinity. I started to create sequences in a spreadsheet deviding integers by 10^n (1) (0,1) (0,01) (0,001) (etc.) and I thought, this is an infinte process but the formula would be something like r = z / 10^n But then ofcourse there are the infinite numbers. 1/3 wouldn't fit in there being r, writing it as 0,333 wouldn't give me and end point due to it's infinity, so then I wondered if there existed an anotation in math that could describe the same type of infinity "the other way around". Like 0,3333... the three points are sometimes used to indicate infinity becasue we read from left to right it easy to understand, but what happens with ...3333. We don't see that as an ever growing or infinite integer. Then again the process of creating this sequence is infinite as well so somewhere in the process you get the next decimal of and infinite number. Eg. with π the sequence would give first 3 when we "reach" 31 for example we would get 3,1 and then getting to 314 it would give 3,14 "etc".
Example of the spreadsheet method:

-5​

-4​

-3​

-2​

-1​

0​

1​

2​

3​

4​

5​

6​

1​

-0,5​

-0,4​

-0,3​

-0,2​

-0,1​

0​

0,1​

0,2​

0,3​

0,4​

0,5​

0,6​

2​

-0,05​

-0,04​

-0,03​

-0,02​

-0,01​

0​

0,01​

0,02​

0,03​

0,04​

0,05​

0,06​

3​

-0,005​

-0,004​

-0,003​

-0,002​

-0,001​

0​

0,001​

0,002​

0,003​

0,004​

0,005​

0,006​

4​

-0,0005​

-0,0004​

-0,0003​

-0,0002​

-0,0001​

0​

0,0001​

0,0002​

0,0003​

0,0004​

0,0005​

0,0006​

5​

-0,00005​

-0,00004​

-0,00003​

-0,00002​

-0,00001​

0​

0,00001​

0,00002​

0,00003​

0,00004​

0,00005​

0,00006​

6​

-0,000005​

-0,000004​

-0,000003​

-0,000002​

-0,000001​

0​

0,000001​

0,000002​

0,000003​

0,000004​

0,000005​

0,000006​

7​

-0,0000005​

-0,0000004​

-0,0000003​

-0,0000002​

-0,0000001​

0​

0,0000001​

0,0000002​

0,0000003​

0,0000004​

0,0000005​

0,0000006​

8​

-0,00000005​

-0,00000004​

-0,00000003​

-0,00000002​

-0,00000001​

0​

0,00000001​

0,00000002​

0,00000003​

0,00000004​

0,00000005​

0,00000006​

 

[JeffM]

I am at a partial loss on how to respond. One reason is that I am not sure exactly what your question is. Another is that the best response for you is likely to be in your native language. A third reason is that, ultimately, I am a finitist and do not believe that the real numbers, infinity, etc. correspond to anything observable in the physical world. They are challenging games we play with in our minds. So a person of faith can probably do a better job than I can.

But I suggest that you read some articles about Georg Cantor and his work. Here is one written in the style of Galileo’s ”Two New Sciences.”

https://www.mathpages.com/home/kmath371/kmath371.htm

 

[BWG]

Thank you very much again, this article clears most of my doubts.