The meaning of infinity

[Juma]

Or maybe better would be:

1/3 ~ 0.33333333333333333333

but:

1/3 = 2/6

Well, if so, then why not 1/3 = 0.333... and 1 = 0.999...
EB

Why ”why not”? They are. Its what 10 year olds are teached in school...

 

[Juma]

OK, that's a good point.

Personally, I take 0.999... to be equal to 1, and I take equality between numbers to be the same as them being exactly the same thing.

This is because I tend to take "0.999..." and "1" as two different formal expressions of the same number. Same things for "1/3" and "0.333...".

So, as I see it, the set of fractions is potentially the same as the set of all rational decimals. This needs to be proved because it's absolutely evident but the result is just that.

Still, I understand what you're saying. I could use the word "quantity" instead and reserve the word "number" for the formal expression of a quantity. I guess that's what you're doing and this may be more in line with how most mathematicians think of it.

So, if there are five children in the courtyard, there is no number in the courtyard, but there is a quantity, which is a quantity of children...

But I have already shown that each real has a decimal representation.

No, you haven't. You've merely expressed how you intuitively think of the problem. It's no proof.

That each decimal corresponds to a really the same; think if the decimals as ever smaller scale markings.

I understand your point. I just don't take it as a proof.

For finite number of decimals it is obvious that a decimal always maps to a real.

I think this isn't the issue. The question is whether decimals map all the Reals, i.e. whether there aren't any Reals somehow left out.

For infinite numbers of decimals its a bit trickier but it is not so hard to see that it must converge (it is always within the interval of the decimal your are on).

Sorry, you can't assume any "convergence" without having specified the series you're talking about.

If you're talking about a series of intervals of decimals only defined by the intervals being included in each other, the this series doesn't converge. It's obvious there is an infinity of possible ways it can go.
EB

I did specify the series: the serie of decimals. Since each decimal an is less than 10^(-n) we know that the serie sum(an) is convergent.

 

[Speakpigeon]

Or maybe better would be:

1/3 ~ 0.33333333333333333333

but:

1/3 = 2/6

Well, if so, then why not 1/3 = 0.333... and 1 = 0.999...
EB

Why ”why not”? They are. Its what 10 year olds are teached in school...

Because saying that 1/3 ~ 0.33333333333333333333 "may be better" suggests it may be also better than 1/3 = 0.333...

And, you forgot Steve...

0.9999 is a real number. 0.9999... is not, it is an infinite sequence.

If Steve thinks that 1 is a number but 0.999... is not a number then Steve must think that 1 isn't equal to 0.999... and therefore 1/3 isn't equal to 0.333...

And, why not? This is a free country.

No?
EB

 

[Speakpigeon]

I did specify the series: the serie of decimals. Since each decimal an is less than 10^(-n) we know that the serie sum(an) is convergent.

Sorry, I don't see what you mean. What series exactly?
EB

 

[Juma]

Why ”why not”? They are. Its what 10 year olds are teached in school...

Because saying that 1/3 ~ 0.33333333333333333333 "may be better" suggests it may be also better than 1/3 = 0.333...

And, you forgot Steve...

0.9999 is a real number. 0.9999... is not, it is an infinite sequence.

If Steve thinks that 1 is a number but 0.999... is not a number then Steve must think that 1 isn't equal to 0.999... and therefore 1/3 isn't equal to 0.333...

And, why not? This is a free country.

No?
EB

Skeptical probably meant ”Better example” not ”b
Etter representation of”...
Free country maybe.. but not free math...

 

[Speakpigeon]

I don't think that is true.

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Then does the = sign only work for reals? Or do we have to put =~ ?

Eh? The decimal number is a textual representation. 1 and 0.9999... are different textual representations.
Both representations represents the same real number.

Here you seem to be saying the opposite:

The sets are not identical. There are more decimal numbers then there are reals.
Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

I wouldn't know how to interpret this if not as you saying that 0.999999... and 1.0 are two different numbers.

A recurrent problem with your pronouncements.
EB

 

[Speakpigeon]

I did specify the series: the serie of decimals. Since each decimal an is less than 10^(-n) we know that the serie sum(an) is convergent.

Sorry, I don't see what you mean. What series exactly?
EB

Oh, this one? Sorry...

prove the decimal numbers map any continuum.

Since Real numbers are the continuum by definition we just have to prove that any real number can be written as a decimal number (not necesarily with finite number of decimals).

To prove that intuitively :

start with a line marked with integers.
Pick any point on this line. (Between, or on the integers)

Now pick the nearest smaller integer a0 and the nearest larger integer b0
Divide the interval between a0 and b0 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a1 and the nearest to the right b1
Divide the interval between a1 and b1 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a2 and the nearest to the right b2
Divide the interval between a2 and b2 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Etc

Working like this you get a serie of numbers a0 a1 a2 a4 etc that will be the decimal number.

Sorry, I'm confused now. It seems to me you could do the same thing using the Rationals instead of the Decimals and therefore prove similarly that the Rationals is the same as the Reals and yet the Rationals are missing the part of the Reals that's called the Irrationals.

Can you explain where I'm wrong here?
EB

 

[steve_bank]

Tje results of numerical calculations in a general sense are all approximations. In practical work what matters is having enough decimal places that results do not become inaccurate from rounding. You can see interesting results with a hand calculator from rounding and finite number size.


0.9999 is a real number. 0.9999... is not, it is an infinite sequence.

Sigh...
0.9999 is a decimal representation of a Real number.
0.9999... is another decimal representation of (another) Real number.

I have done a fair amount if numerical analysis.

You are arguing semantics.

O.9999 can be found on the real number number, I word say 0.999 is a finite instantiation of the set of real numbers. 0.999... is not a number, unless you can provide an example where it can be applied. The thread is math not metaphysics..

Another example is continuous probability distributions. The probability of a continuous distribution is the limit as the interval of integration goes to zero. In a continuous distribution the probability of a single number is zero, there are an infinite number of possibilities, at least mathematically. In real applications the infinite nature of reals appears. It is not just abstract.

Elaborate on the source of your discomfort.

PI is 3.14159... When I set up calculations I use the number of digits needed to maintain a minimum accuracy along with physical constants and data resolution.

 

[Speakpigeon]

(...) 0.999... is not a number, unless you can provide an example where it can be applied. The thread is math not metaphysics..

0.999... = 1
0.333... x 3 = 0.999... = 1
0.111... x 9 = 0.999... = 1
0.999... + 1 = 1.999... = 2
0.61290322580645161290322580645161... x 31 = 18.999... = 19
...

I think.

I think, therefore you are therefore you are...

Show

wrong.

Dots are real handy.

Just three dots gives you infinity!
EB

 

[Speakpigeon]

0.999... is not a number

PI is 3.14159... When I set up calculations I use the number of digits needed to maintain a minimum accuracy along with physical constants and data resolution.

So, 0.999... isn't a number but 3.14159... is?!

Beats me!

The guy isn't even consistent...
...
EB...
...

 

[Juma]

Oh, this one? Sorry...

prove the decimal numbers map any continuum.

Since Real numbers are the continuum by definition we just have to prove that any real number can be written as a decimal number (not necesarily with finite number of decimals).

To prove that intuitively :

start with a line marked with integers.
Pick any point on this line. (Between, or on the integers)

Now pick the nearest smaller integer a0 and the nearest larger integer b0
Divide the interval between a0 and b0 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a1 and the nearest to the right b1
Divide the interval between a1 and b1 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Of these smaller intervals pick the nearest to the left a2 and the nearest to the right b2
Divide the interval between a2 and b2 in 10 intervals and denote with 0,1,2,3,4,5,6,7,8 and 9

Etc

Working like this you get a serie of numbers a0 a1 a2 a4 etc that will be the decimal number.

Sorry, I'm confused now. It seems to me you could do the same thing using the Rationals instead of the Decimals and therefore prove similarly that the Rationals is the same as the Reals and yet the Rationals are missing the part of the Reals that's called the Irrationals.

Can you explain where I'm wrong here?
EB

Sorry if was unclear...
The sequence of narrowing down intervals that I descibed was the build up of an infinite long decimal number as
In 6.7375674785365377...
as a contrast a rational number, as for example 567/397 is the completed number. The corresponding example with rational numbers would be a series 1/2+ 1/4+7/230+...

 

[skepticalbip]

Why ”why not”? They are. Its what 10 year olds are teached in school...

Because saying that 1/3 ~ 0.33333333333333333333 "may be better" suggests it may be also better than 1/3 = 0.333...

No. It suggests no such thing.
0.3333333333333333 is a truncated decimal series that is approximately 1/3.
0.333..... indicates an infinite decimal series that equals 1/3.

 

[Juma]

Here you seem to be saying the opposite:

The sets are not identical. There are more decimal numbers then there are reals.
Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

I wouldn't know how to interpret this if not as you saying that 0.999999... and 1.0 are two different numbers.

A recurrent problem with your pronouncements.
EB

Read the posts again. Everything is there. You just have to actuslly read what is there, not what you expect to read.
A hint: ”decimal numbers are a textual representation” is central.

 

[ryan]

I don't think that is true.

Okay, sure. But suppose I think it is. Where are we now?

You made the claim first. Reference it.

 

[ryan]

Then does the = sign only work for reals? Or do we have to put =~ ?

How do you feel about 1l2 = 2/4.

How is this the same thing as my other issue? 2/1 = 2, so... ?

 

[Phil Scott]

Then does the = sign only work for reals? Or do we have to put =~ ?

How do you feel about 1l2 = 2/4.

How is this the same thing as my other issue? 2/1 = 2, so... ?

I couldn't say. To be honest, I'm not entirely sure what your issue is. There appears to be a lot of confusion in this thread generally, which is the usual for threads like this, which are often led by people who's experience with real numbers doesn't extend behind playing about with a calculator in high school. No offence meant there: these foundational issues over the real numbers didn't get settled until the mid-19th century, which is incredibly late in the history of maths. It takes a bit of sophistication to finally settle them.

Maybe this helps, maybe not: one of the best bits of ranting I ever read of Quine was in his Mathematical Logic, where he spent a good deal of time talking about the distinction between use and mention. The idea is that one uses signs to mention what's signified, and that we should never confuse the two. To avoid this confusion with the pedantry required of a mathematical logician, we have to be exact in our use of quotation, quotes being devices which we can use to mention signs.

For example, a dog is a four legged carnivore. "Dog", however, is a word which I can use to mention dogs. I can even iterate this, and say that "'dog'" (I just used double quotation) is a quotation I use to mention the word "dog", which I use in turn to mention dogs.

Now we can use this to get all pedantic and correct something Juma said earlier:

Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

What one should say is that "0.999..." and "1.0" are two different decimals. "0.999..." is not equal to "1.0". They are different decimals. But the reals which these decimals mention are 0.999... and 1.0, which are, in fact, the same real, a matter we can express with the equation "0.999... = 1". Indeed, 0.999... = 1.

There's no "=~" involved in this discussion (does not mean "approximately equal to"?). 0.999... and 1.0 are identical reals, just as 1/2 and 2/4 are identical reals, and just as 1 + 1 and 10 - 8 are identical reals.

 

[ryan]

I don't think that is true.

- - - Updated - - -



Then does the = sign only work for reals? Or do we have to put =~ ?

Eh? The decimal number is a textual representation. 1 and 0.9999... are different textual representations.
Both representations represents the same real number.

Yes I know, but why does the equals symbol work if 1 and 0.999... are not the same thing? Is the equals symbol only good for reals (or standard math) is what I am asking.

 

[ryan]

How is this the same thing as my other issue? 2/1 = 2, so... ?

I couldn't say. To be honest, I'm not entirely sure what your issue is. There appears to be a lot of confusion in this thread generally, which is the usual for threads like this, which are often led by people who's experience with real numbers doesn't extend behind playing about with a calculator in high school. No offence meant there: these foundational issues over the real numbers didn't get settled until the mid-19th century, which is incredibly late in the history of maths. It takes a bit of sophistication to finally settle them.

Maybe this helps, maybe not: one of the best bits of ranting I ever read of Quine was in his Mathematical Logic, where he spent a good deal of time talking about the distinction between use and mention. The idea is that one uses signs to mention what's signified, and that we should never confuse the two. To avoid this confusion with the pedantry required of a mathematical logician, we have to be exact in our use of quotation, quotes being devices which we can use to mention signs.

For example, a dog is a four legged carnivore. "Dog", however, is a word which I can use to mention dogs. I can even iterate this, and say that "'dog'" (I just used double quotation) is a quotation I use to mention the word "dog", which I use in turn to mention dogs.

Now we can use this to get all pedantic and correct something Juma said earlier:

Example: 0.999999... and 1.0 are two different decimals that maps to the same real.

What one should say is that "0.999..." and "1.0" are two different decimals. "0.999..." is not equal to "1.0". They are different decimals. But the reals which these decimals mention are 0.999... and 1.0, which are, in fact, the same real, a matter we can express with the equation "0.999... = 1". Indeed, 0.999... = 1.

There's no "=~" involved in this discussion (does not mean "approximately equal to"?). 0.999... and 1.0 are identical reals, just as 1/2 and 2/4 are identical reals, and just as 1 + 1 and 10 - 8 are identical reals.

This was originally about whether or not the reals form a continuum. I was taught that they do in a very rigorous calculus course.

So I need some reference that your claim is true.

As for the equals symbol. Like I asked Juma, is it only an equals symbol for reals? If so, then I have no issue with 1 = 0.99.... If not, then the previous claims about 1 not being the same as 0.99... using hyperreals naturally concerns me when I see 1 = 0.99....

 

[Speakpigeon]

Why ”why not”? They are. Its what 10 year olds are teached in school...

Because saying that 1/3 ~ 0.33333333333333333333 "may be better" suggests it may be also better than 1/3 = 0.333...

No. It suggests no such thing.

Sorry but as far as I'm concerned it did.
EB

 

[Speakpigeon]

Oh, this one? Sorry...


Sorry, I'm confused now. It seems to me you could do the same thing using the Rationals instead of the Decimals and therefore prove similarly that the Rationals is the same as the Reals and yet the Rationals are missing the part of the Reals that's called the Irrationals.

Can you explain where I'm wrong here?
EB

Sorry if was unclear...
The sequence of narrowing down intervals that I descibed was the build up of an infinite long decimal number as
In 6.7375674785365377...
as a contrast a rational number, as for example 567/397 is the completed number. The corresponding example with rational numbers would be a series 1/2+ 1/4+7/230+...

Sorry, but that doesn't address my point.
EB