Rational numbers == infinitely repeating sequences of digits

[Jokodo]

I assure you I invented none of the definitions I use.

I am not the first to say that 3 is a number and a digit.

You're sure right about that. A lot of six year olds find it confusing that 1 + 3 is not in fact 13. By seven or eight, most have wrapped their head around the fact that "3" and 3 are different concepts, and usually guess which one is meant even when writers fail to distinguish them orthographically.

You do however deserve full credit for recognising "0.333..." as a digit, and by extension recognising that the digit is a recursive concept, that digits have digits as attributes. Don't be shy about that!

Unless of course, I'm mistaken and you can quote a published mathematician (or anyone else older than 8, for that matter) talking about the digit 0.333...

 

[untermensche]

I assure you I invented none of the definitions I use.

I am not the first to say that 3 is a number and a digit.

You're sure right about that. A lot of six year olds find it confusing that 1 + 3 is not in fact 13. By seven or eight, most have wrapped their head around the fact that "3" and 3 are different concepts, and usually guess which one is meant even when writers fail to distinguish them orthographically.

You do however deserve full credit for recognising "0.333..." as a digit, and by extension recognising that the digit is a recursive concept, that digits have digits as attributes. Don't be shy about that!

Unless of course, I'm mistaken and you can quote a published mathematician (or anyone else older than 8, for that matter) talking about the digit 0.333...

The repeating digit in 0.333....

Not the digit.

What a waste of time.

 

[Jokodo]

I assure you I invented none of the definitions I use.

I am not the first to say that 3 is a number and a digit.

You're sure right about that. A lot of six year olds find it confusing that 1 + 3 is not in fact 13. By seven or eight, most have wrapped their head around the fact that "3" and 3 are different concepts, and usually guess which one is meant even when writers fail to distinguish them orthographically.

You do however deserve full credit for recognising "0.333..." as a digit, and by extension recognising that the digit is a recursive concept, that digits have digits as attributes. Don't be shy about that!

Unless of course, I'm mistaken and you can quote a published mathematician (or anyone else older than 8, for that matter) talking about the digit 0.333...

The repeating digit in 0.333....

Not the digit.

What a waste of time.

To quote:

You can't say a digit repeats infinitely AND has a final digit.

In English, that sentence implies that a digit has digits, but I suppose you were writing untermensche

 

[untermensche]

The repeating digit in 0.333....

Not the digit.

What a waste of time.

To quote:

You can't say a digit repeats infinitely AND has a final digit.

In English, that sentence that a digit has digits, but I suppose you were writing untermensche

The digit repeating is 3.

You can't say 3 repeats infinitely and also say there is a final 3.

 

[Jokodo]

To quote:

In English, that sentence that a digit has digits, but I suppose you were writing untermensche

The digit repeating is 3.

You can't say 3 repeats infinitely and also say there is a final 3.

Once you're old enough to understand that digits/digit sequences and numbers are different concepts and 1 + 3 is not in fact 13, you'll understand why that's irrelevant.

 

[untermensche]

To quote:

In English, that sentence that a digit has digits, but I suppose you were writing untermensche

The digit repeating is 3.

You can't say 3 repeats infinitely and also say there is a final 3.

Once you're old enough to understand that digits/digit sequences and numbers are different concepts and 1 + 3 is not in fact 13, you'll understand why that's irrelevant.

You have nothing.

1 + 3 is defined.

Repeating infinitely is a definition.

Really an outcome from carrying out a defined operation of defined entities.

 

[Jokodo]

Once you're old enough to understand that digits/digit sequences and numbers are different concepts and 1 + 3 is not in fact 13, you'll understand why that's irrelevant.

You have nothing.

1 + 3 is defined.

Indeed it is. And so is "1" + "3". Once you've understood the two are not the same we can continue this conversation.

How old are you again? I want to know if we're talking months or years.

 

[Jokodo]

This is abject nonsense. A number doesn't have a changing value.

If you try to express the number you will get an ever changing number.

You will add 3's forever.

It will never finish.

If you don't express the number but merely imagine it has expressed you have violated the definition which says it can never be fully expressed. Infinite digits means there can never possibly be a last digit.

0.333... is not a value. It has no final value.

It is something else. It is approaching a value. It can by definition never get there.

I didn ask for more rambling nonsense from you. I challenged you to find any instance of any mathematician anywhere who has used the phrase "final value" in the way you are using it here - to refer to a number.

Can you meet that challenge, or is this entirely your own stupidity?

bears repeating.

untermensche, can you meet that challenge?

 

[untermensche]

I didn ask for more rambling nonsense from you. I challenged you to find any instance of any mathematician anywhere who has used the phrase "final value" in the way you are using it here - to refer to a number.

Can you meet that challenge, or is this entirely your own stupidity?

bears repeating.

untermensche, can you meet that challenge?

The only question is whether you can understand it.

Can a string of 3's that goes on infinitely have a final 3?

 

[untermensche]

Once you're old enough to understand that digits/digit sequences and numbers are different concepts and 1 + 3 is not in fact 13, you'll understand why that's irrelevant.

You have nothing.

1 + 3 is defined.

Indeed it is. And so is "1" + "3". Once you've understood the two are not the same we can continue this conversation.

How old are you again? I want to know if we're talking months or years.

Hand waving.

You have nothing.

The idea that 0.333... refers to other things is religion.

 

[Jokodo]

I didn ask for more rambling nonsense from you. I challenged you to find any instance of any mathematician anywhere who has used the phrase "final value" in the way you are using it here - to refer to a number.

Can you meet that challenge, or is this entirely your own stupidity?

bears repeating.

untermensche, can you meet that challenge?

The only question is whether you can understand it.

So the answer is no, you cannot find any instance of any mathematician anywhere who has used the phrase "final value" in reference to a number.

Just say so.

Can a string of 3's that goes on infinitely have a final 3?

No. Also, Gandalf has a beard while "Gandalf" starts with a "G". When talking numbers, your question makes about as much sense as asking whether a beard comes before or after a "G" in the alphabet. The number 0.333... doesn't intrinsically have any "3"s, not a final one, not 13th one, not a first one, only its decimal notation may or may not have any of these. Its duodecimal notation is "0.4", its binary notation is "0.0101..."

 

[untermensche]

The only question is whether you can understand it.

So the answer is no, you cannot find any instance of any mathematician anywhere who has used the phrase "final value" in reference to a number.

Just say so.

Can a string of 3's that goes on infinitely have a final 3?

No. Also, Gandalf has a beard while "Gandalf" starts with a "G".

That is meaningless.

There is nothing a string of 3's refers to but itself.

Just because you have a mind and a memory and know some equation that can create the endless string does not make the equation and the endless string the same thing.

 

[Jokodo]

So the answer is no, you cannot find any instance of any mathematician anywhere who has used the phrase "final value" in reference to a number.

Just say so.



No. Also, Gandalf has a beard while "Gandalf" starts with a "G".

That is meaningless.

There is nothing a string of 3's refers to but itself.

Just because you have a mind and a memory and know some equation that can create the endless string does not make the equation and the endless string the same thing.

If numbers and their decimal representations are the same thing, 1+3 Must be 13. please explain how what you say makes sense and yet 1+3!=13. Alternatively, have some 7-Year-old explain this stuff to you.

 

[untermensche]

If numbers and their decimal representations are the same thing, 1+3 Must be 13.

Doesn't follow in any way.

The decimal is many digits.

It can abstractly be thought of as a single number.

But it takes a mind capable of abstract thought to do it.

And a mind taught to do it.

It is in no way self evident.

 

[Kharakov]

Go to the original post and read the first sentence again: "A feature of rational numbers is that their decimal representations always have infinitely repeating sequences of digits" (emphasis added).

The property of having an infinite sequence of "3"s is a property of the decimal representation of the number that is expressed as "1/3" in fractional notation, as "0.1" in base 3, as "0.4" in base 12, or, as the previous poster hints at, as "10" in base 1/3. It is not a property of the number.

Saying about the number in question that is has infinitely repeating digits "3" is like saying that the continent Australia has nine characters. Continents don't have character counts and numbers don't have digits, infinitely repeating or otherwise, they have, respectively, e.g. areas and average altitudes and longest diameters, or prime factors and multiples and reciprocals etc. The labels "Australia" and "0.333..." have properties of their own, including a character counts, but those are uninformative about the properties of the continent/number.

- - - Updated - - -

ok, 1/3 IS 0.333...

what is wrong with you?

Are you just itching for a fight? Go to politics.

If you choose to discount definitions and have faith they are the same thing they are.

That is allowed. Nothing stops that kind of thinking.

I like 1/3 in base three. So easy. But we can make it easier. Base 1 third!

I think we should just make all numbers be specific bases. So 1/9th would be written in base 1/9th. And pi would be base pi. And 2/pi would be base 2/pi.

It makes things easier. All numbers are #1. They all win!

BTW, you owe me 1 dollar.

Actually, 1/3 in base 1/3 would be 10, not 1.

Say ten. It's the most ambiguous phrase in English.

1 is 10, therefore all numbers are 1, except 0. Which isn't, yet is.

 

[Jokodo]

If numbers and their decimal representations are the same thing, 1+3 Must be 13.

Doesn't follow in any way.

Of course it does. If the only thing a "string of 3's refers to" is the self-same string of 3s, that includes a string of 3s with length == 1 - and we have to assume that the same holds for strings of 1s.

It follows that the result of 1 + 3 is the concatenation of the strings "1" and "3".

Either that, or decimals do refer to something else than the strings used to represent them. Pick one.

 

[untermensche]

1 would be represented by the string 1.00000....

But of course here it doesn't matter since the entire string, no matter how long, is equal to zero and not having a last zero doesn't change anything.

But there is a difference between 0.33 and 0.333

 

[Jokodo]

1 would be represented by the string 1.00000....

You've been telling us for the better part of 14 pages of discussion that numbers do not exist and that numeric strings only represent themselves...

It seems you're finally coming around to the conclusion that that was a stupid thing to say.

But of course here it doesn't matter since the entire string, no matter how long, is equal to zero and not having a last zero doesn't change anything.

No, the string is not equal to zero. It may represent the number 0.

But there is a difference between 0.33 and 0.333

Sure there is. What exactly the difference is depends on whether you're talking about the numbers 0.33 and 0.333 (the difference being exactly 0.003) or about the strings "0.33" and "0.333" commonly used to reference them (the difference being exactly one character "3").

None of that is relevant to the fact that the number referred to by "0.333..." is exactly 1/3 or 0.4₍₁₂₎. The fact that there exists a system of notation in which you cannot fully actualize its label without the use of ellipsis no more changes its identity than the fact that you cannot print his name on an ASCII-only typewriter changes the identity of the Czech tennis player Jiří Hřebec. They're labels.

 

[untermensche]

You've been telling us for the better part of 14 pages of discussion that numbers do not exist and that numeric strings only represent themselves...

It seems you're finally coming around to the conclusion that that was a stupid thing to say.

?

That is bizarre and from left field.

My conclusions are strengthening.

There is nothing referred to by 1 except 1.

No, the string is not equal to zero.

0 + 0 = 0

0 + (N x 0) = 0

But there is a difference between 0.33 and 0.333

Sure there is. What exactly the difference is depends on whether you're talking about the numbers 0.33 and 0.333 (the difference being exactly 0.003) or about the strings "0.33" and "0.333" commonly used to reference them (the difference being exactly one character "3").

There is the character 3 that is an abstract representation of a value. And 0.003 is a representation of a different value.

So the value is not 3 just because the digit is 3.

The value is determined by where the 3 is.

None of that is relevant to the fact that the number referred to by "0.333..." is exactly 1/3

If your faith says so. You have not demonstrated it once. You have waved your arms a lot.

But 0.333... has no final value. It is not equal to anything.

1/3 just happens to be a random equation that produces an infinite string based on a recursive operation dealing with the same elements and locked into dealing with those elements.

But simple recursive operations like this that yield infinite products are not very interesting.

The recursive operation goes on without end.

That is what produces infinite digits. In other words a string of digits that never can have a last digit.

 

[Jokodo]

?

That is bizarre and from left field.

My conclusions are strengthening.

There is nothing referred to by 1 except 1.

And yet minutes ago you were finally coming around to the idea that strings represent numbers.

0 + 0 = 0

0 + (N x 0) = 0

That's numbers. Not digits and not strings.

But there is a difference between 0.33 and 0.333

Sure there is. What exactly the difference is depends on whether you're talking about the numbers 0.33 and 0.333 (the difference being exactly 0.003) or about the strings "0.33" and "0.333" commonly used to reference them (the difference being exactly one character "3").

There is the character 3 that is an abstract representation of a value. And 0.003 is a representation of a different value.

So the value is not 3 just because the digit is 3.

So you're saying the digit "3" refers to something other than the number 3?

The value is determined by where the 3 is.

None of that is relevant to the fact that the number referred to by "0.333..." is exactly 1/3

If your faith says so. You have not demonstrated it once. You have waved your arms a lot.

But 0.333... has no final value. It is not equal to anything.

1/3 just happens to be a random equation that produces an infinite string based on a recursive operation dealing with the same elements and locked into dealing with those elements.

But simple recursive operations like this that yield infinite products are not very interesting.

The recursive operation goes on without end.

That is what produces infinite digits. In other words a string of digits that never can have a last digit.

So what about that Czech Tennis player? Does he become undefined when you can't type his name?