Is 1 = 0.999... ? Really?

[Ecmandu]

... in base 10 is .111... in base 3.

Yeah .. just a matter of recycling on the evens or recycling on the odds.

My point still stands... 3 is an even number in base 3

[Carleas]

You could then argue that the divisor of base and its multipliers are "even"

Sure, if you change the definition of 'even', then you could get any result you want. But that's not usually how we think of it (5 isn't even in base 10, even though it's also a prime factor of 10).

No matter the base, numbers that have 2 as a factor continue to have 2 as a factor. And changing the base doesn't affect the properties of odd and even numbers under arithmetic operations.

[Ecmandu]

You could then argue that the divisor of base and its multipliers are "even"

Sure, if you change the definition of 'even', then you could get any result you want. But that's not usually how we think of it (5 isn't even in base 10, even though it's also a prime factor of 10).

No matter the base, numbers that have 2 as a factor continue to have 2 as a factor. And changing the base doesn't affect the properties of odd and even numbers under arithmetic operations.

So, I'm going to ask you this question...

I pointed out earlier that addition and multiplication, even subtraction of wholes never yield regresses.

3*3 never equals 999999999....

So the fact that you wouldn't find this issue in factors and such is no surprise - division is inverse to multiplication, yet has very strange properties relative to the inverse of addition and subtraction.

So, my question is... why?

I was trying to point to a clue, since we brought up bases... if base is number, it sets up its lowest common denominator as the even. 2 is the lowest of ten, so no matter what you divide in base 10, that is a multiple of two, will terminate.

In base 3, 2 and 2 add to higher than base, 2 cannot be the lowest common denominator, it has to be 3 or multiples of 3

[Carleas]

In base 3, 2 and 2 add to higher than base, 2 cannot be the lowest common denominator, it has to be 3 or multiples of 3

I don't disagree, but that isn't what "even" means. 2 is still even in base 3.

division is inverse to multiplication, yet has very strange properties relative to the inverse of addition and subtraction.

So, my question is... why?

You're looking at "whole" numbers, by which I take you to mean either integers. And you're right that there's an asymmetry between multiplication and division there: multiply two integers and you always get an integer, divide one integer by another and you don't necessarily get another integer. But you can look at larger and smaller sets of numbers. If you look at just the natural numbers, you see a similar asymmetry between addition and subtraction: add two natural numbers and you always get a natural number, but 3-5 is not a natural number.

Conversely, the rational numbers are closed on division (except for zero), as are the real numbers.

I'm not sure if there are any sets that are closed under division but not multiplication, or subtraction but not addition.

[Ecmandu]

In base 3, 2 and 2 add to higher than base, 2 cannot be the lowest common denominator, it has to be 3 or multiples of 3

I don't disagree, but that isn't what "even" means. 2 is still even in base 3.

division is inverse to multiplication, yet has very strange properties relative to the inverse of addition and subtraction.

So, my question is... why?

You're looking at "whole" numbers, by which I take you to mean either integers. And you're right that there's an asymmetry between multiplication and division there: multiply two integers and you always get an integer, divide one integer by another and you don't necessarily get another integer. But you can look at larger and smaller sets of numbers. If you look at just the natural numbers, you see a similar asymmetry between addition and subtraction: add two natural numbers and you always get a natural number, but 3-5 is not a natural number.

Conversely, the rational numbers are closed on division (except for zero), as are the real numbers.

I'm not sure if there are any sets that are closed under division but not multiplication, or subtraction but not addition.

Smaller and larger sets of numbers??

There are as many naturals as negatives, you're just using an extra operator to separate the two.

There are also as many positive AND negative integers as positive or negative integers.., just depends how you explicate the operators.

But that's not the point of this discussion.

The big question, and I still maintain that when you switch base, an entirely different psychology is needed about what counts as even, and you divide, that infinite regress kicks in, for some instances of the odd numbers in that base.

This isn't, nor was it ever a discussion of whether rationals are ordered or not...

This is a discussion about why this method doesn't terminate if you keep running the method of dividing integers, such that one has to take a step back, convert the decimal to a fraction that it really isn't anymore, and then add the fractions together and magically call it an equality

[gib]

There's an even finer point to this mystery that is often overlooked. When you place a bar over the 3 to signify its rational repetition, the threes actually regress, technically, at no point are those 3's actually repeating!!

You seem to think that just because you have learned how to divide into decimals that you actually understand why some numbers terminate and others infinitely regress. I assure you, you have no clue.

You're not making any sense. In what way do the 3's at no point ever repeat? You mean on paper or in concept? Of course, they don't repeat on paper--at least not infinitely--but in concept, of course they do!

I have not just learned to divide, I have understood the reason why division churns out the (rational) number that it does. Take 1 and divide it by 2. You get .5. But this is not just a rule I have learned to blindly follow. I understand that dividing one thing into two equal halves gives you some something that we represent as "0.5". I know what a "half" is. Just as in long division, every digit you get in the decimal expansion, you can understand why you got that digit. With irrational numbers, I can understand your point (sort of)--you don't actually know what the full number is (what series of digits the decimal expansions is represented with), but with decimal expansions that terminate or with decimal expansions that repeat (rational numbers), you can understand how each digit is derived.

[James S Saint]

Well, if we are just going to play, why not look at it from base 0.3 ... or less fun; "base 1/3".

[Ecmandu]

Gib,

You're doing the same thing Magnus was doing.

You're saying that you understand it because there's a process that consistently executes it, and then you take a leap, you then say, without any support, that you actually understand WHY this process sometimes causes infinite regress... !!!

Actually, and I made this finer point before, BECAUSE it is a regress issue, technically, the number is irrational!! You're not actually repeating any of those 3's!!!

Think of it this way!!

What does 999999999... eventually equal??

But!! Somehow, it makes perfect sense to you that 0.999999... equals 1????


99999 makes sense. 99999... somehow makes sense to you as a rational number???

It's the same argument behind the decimal point as well.

[phyllo]

What does 999999999... eventually equal??

This doesn't converge.

But!! Somehow, it makes perfect sense to you that 0.999999... equals 1????

But this does converge.

[Ecmandu]

What does 999999999... eventually equal??

This doesn't converge.

But!! Somehow, it makes perfect sense to you that 0.999999... equals 1????

But this does converge.

I had a big long post about expansion and contraction, divisors and multipliers... but really I just want to ask you this.

Let me ask you this.

Does 9 ever divide equally into 10?

If so where does it terminate ?

If not, where does it converge?

Huh? At infinity??

Where exactly is infinity??

Show it to me.

It actually makes more sense for expansions to converge than contractions...

I'm curious why you think a contraction converges, and how... are you collapsing space every time you divide? Expansions contain their prior objects, divisions don't ...

There's a bit more to this, but I'm super-curious where you think this convergence occurs

[phyllo]

If so where does it terminate ?

If not, where does it converge?

Huh? At infinity??

Where exactly is infinity??

Show it to me.

There is no space and no time in mathematics. The concepts of 'when' and 'where' do not apply as you seem to think that they do.

It actually makes more sense for expansions to converge than contractions...

I'm curious why you think a contraction converges, and how...

Series which converge add ever smaller numbers to the total until ultimately they add nothing more. That's the basic concept. It's nothing mysterious or inaccessible. One can see it by preforming many calculations on a computer.

[James S Saint]

Series which converge add ever smaller numbers to the total until ultimately they add nothing more. That's the basic concept. It's nothing mysterious or inaccessible. One can see it by preforming many calculations on a computer.

That is the fallacy in your reasoning.

There is no point where "ultimately they add nothing more". That point, "at infinity", is erroneous imagining.

[Ecmandu]

Hmm... this is a very confused reply phyllo.

First, math is the abstraction of space and time.

One orange, half an orange...

Apparently, you'd have all us flatlanders believe that when you do math on the 5th dimension, infinite sequences terminate because there's no more space-time, but you need space time to make the argument that there's no space time (which means in the purest sense that the division never hits "no output")

Even if there was no output because space time is finite... what causes it to converge?

[phyllo]

Series which converge add ever smaller numbers to the total until ultimately they add nothing more. That's the basic concept. It's nothing mysterious or inaccessible. One can see it by preforming many calculations on a computer.

That is the fallacy in your reasoning.

There is no point where "ultimately they add nothing more". That point, "at infinity", is erroneous imagining.

The results are confirmed by other methods. The 0.999... =1 result can be achieved by series expansions or with straight divisions, multiplications, additions and subtractions - probably a dozen have been shown in this thread. In order to accept "your logic", we would have to throw out the work in many fields of mathematics.

You can start another thread where you claim that series don't converge. I'm sure that you can make it run for 50 pages or more.

[phyllo]

First, math is the abstraction of space and time.

It is not.

Mathematics can be used to model time and space.

[Ecmandu]

First, math is the abstraction of space and time.

It is not.

Mathematics can be used to model time and space.

Intetesting... what does mathematics model without time and space? Thoughts are discrete ... what does math model without thought?

[phyllo]

Intetesting... what does mathematics model without time and space? Thoughts are discrete ... what does math model without thought?

You're confusing space and time and thoughts about space and time - reality and representations of reality.

[Ecmandu]

Intetesting... what does mathematics model without time and space? Thoughts are discrete ... what does math model without thought?

You're confusing space and time and thoughts about space and time - reality and representations of reality.

So... you're saying that math is just a representation of reality and not reality proper??

I have a question for this pocket of exploring these concepts...

What do you think moves faster ... thought or light?

[phyllo]

Light moves in space. Does thought move in space? From where to where?

[Ecmandu]

Light moves in space. Does thought move in space? From where to where?

Technically it's a trick question... thought can make light move faster... thought can traverse all space and time much faster than light, practically speaking.... you can talk to embodied beings in their state that was trillions upon trillions of light years ago
Or even what they are now... thought moves faster than anything in existence

[phyllo]

Technically it's a trick question... thought can make light move faster... thought can traverse all space and time much faster than light, practically speaking.... you can talk to embodied beings in their state that was trillions upon trillions of light years ago
Or even what they are now... thought moves faster than anything in existence

Okey-dokey

[Ecmandu]

Technically it's a trick question... thought can make light move faster... thought can traverse all space and time much faster than light, practically speaking.... you can talk to embodied beings in their state that was trillions upon trillions of light years ago
Or even what they are now... thought moves faster than anything in existence

Okey-dokey

*shrug*. Analogies and metaphors have their limits... experience reigns supreme still. So tell me, this infinity that you hang out at where all these series converge for you... is it fun? Oh, that's right... it's just a metaphor! Like I said, experience reigns supreme...

[phyllo]

*shrug*. Analogies and metaphors have their limits... experience reigns supreme still. So tell me, this infinity that you hang out at where all these series converge for you... is it fun? Oh, that's right... it's just a metaphor! Like I said, experience reigns supreme...

Yeah, unfortunately I have to work within my limits. I have never talked to God or spirits or demons, been to heaven or hell, touched infinity, had out of body experiences, been reincarnated, been one with the universal consciousness, made a perpetual motion machine, circumvented the laws of thermodynamics ...

I'm making lemonade with the lemons that I have gathered over the years.

[Ecmandu]

*shrug*. Analogies and metaphors have their limits... experience reigns supreme still. So tell me, this infinity that you hang out at where all these series converge for you... is it fun? Oh, that's right... it's just a metaphor! Like I said, experience reigns supreme...

Yeah, unfortunately I have to work within my limits. I have never talked to God or spirits or demons, been to heaven or hell, touched infinity, had out of body experiences, been reincarnated, been one with the universal consciousness, made a perpetual motion machine, circumvented the laws of thermodynamics ...

I'm making lemonade with the lemons that I have gathered over the years.

Well... sounds to me like you're off to a good start.

When these things do occur, try not to go insane like I did!

I'll tell you something very important to keep with you...

The question of whether there is a grand creator, is actually yours to decide. There isn't a wrong answer.
That's how big this cosmos actually is

[Ecmandu]

*shrug*. Analogies and metaphors have their limits... experience reigns supreme still. So tell me, this infinity that you hang out at where all these series converge for you... is it fun? Oh, that's right... it's just a metaphor! Like I said, experience reigns supreme...

Yeah, unfortunately I have to work within my limits. I have never talked to God or spirits or demons, been to heaven or hell, touched infinity, had out of body experiences, been reincarnated, been one with the universal consciousness, made a perpetual motion machine, circumvented the laws of thermodynamics ...

I'm making lemonade with the lemons that I have gathered over the years.

Well... sounds to me like you're off to a good start.

When these things do occur, try not to go insane like I did!

I'll tell you something very important to keep with you...

The question of whether there is a grand creator, is actually yours to decide. There isn't a wrong answer.
That's how big this cosmos actually is

viewtopic.php?p=2668152#p2668152

Sorry James, I felt obligated to add this to my prior comment, perhaps we can get back to the question phyllo avoided about whether math being a representation of reality is in fact not reality itself, as we ponder ideas of presumed convergences