.999 = 1?

[Budget_Player_Cadet]

@7777777

I said that Z cannot be equal to infinity because adding 1 to infinity does not increase
its value, whereas adding 1 to Z increases its value.

…But then Z + 1 = infinity cannot possibly be true. Is Z an element of the set of Real numbers? Because 1 certainly is, and R is closed under addition. So either infinity is an element of the real numbers (in which case your definition of “infinity” is woefully different than ours), or Z is not. You cannot define Z with the qualities “element of R” and “Z+X = Y, where X element of R and Y not element of R”. That is self-contradictory.

You must see what you yourself say: “add+1 to get 2,3,4,5,etc. The thing about this, though, is that at no point does this end”.
You say that in this way you will get to infinity,

No. I do not. In fact, I will say with absolute clarity that you could use the fastest supercomputer conceivable to count until the heat death of the universe and you will never reach infinity. I mean, that wouldn’t even make sense - infinity is not an element of R, but 1 and every other one of those numbers is. You do not understand what infinity means. At no point does a function or term “reach” infinity. For example, in practical terms it’s useful to say “at 0, 1/x = ±infinity”; in reality, the term is simply not defined. The function never reaches “infinity”; it simply diverges further and further the closer you get to the undefined location. This is a different understanding of “infinity” than the one that comes into play when you say there are an infinite number of 9s to the right of the decimal point in 0.999…

and you get to infinity with numbers which are real numbers. Number 1 is a real number, and add infinite amount of 1s equals infinity: 1+1+1+1+…=∞.

Fundamentally, this is the basis of your misunderstanding. 1+1+1+1+… = infinity is, strictly speaking, false. You’re misusing infinity. You can only approach it, and while using it as such colloquially is acceptable, no valid proof will ever speak of infinity like that. Rather, they’d consider it as such:

lim(x->∞)[sum(0 to x)[1]].

That is the limes of the sum. Because that sum will never equal infinity. It simply diverges forever, and as x approaches infinity, so does the sum. It will never, however, reach infinity.

(Also, mathematically speaking, sum(0 to ∞)[…] is an invalid term - you’re simply not allowed to apply ∞ that way.)

Why should Z not be a real number as well
if we can get to infinity by adding 1 to it?

Because we cannot. There’s your problem.

You write also that " There is not a single real number Z for which Z+1 does not equal another real number". Apparently this rule does not apply if you are adding an infinite amount of real numbers because their sum is infinity and not a real number.

Again, you misapply infinite sums. They simply don’t work that way.

It seems that there are exceptions to the rules when dealing with infinity.

No, “infinity” works off a fundamentally different set of rules than most numbers because it is NOT a number! Not even conceptually.

Can we be sure if any rules work with infinity?

Yes - it has its own rules.

The main problem with Z is, as I said already, it does not seem to have a value.
If you write down a largest number, for example 100000, then you can add 1 can
arrive at 100001 and so on. You did not tell a solution to this problem.
I said that the wrong answer is that there is no Z. You can never write down all the decimals of π, it does not mean there is no π.

Yes, but you can define pi in a way that is internally consistent and meaningful. You cannot define Z in any consistent way. Either Z is not a real number, or infinity is.

Z = ∞ is a wrong answer as I have said many times. The main property of Z is that
its value increases if you add 1 to it, whereas adding 1 to ∞ does not increase its
value because ∞ = ∞ + 1. Of course , if you assume that Z = ∞ then x = 0
as you said, but it is false. The infinitesimal is not equal to 0.

You claim that this is false but have yet to give any convincing argumentation for why that is the case. Can you prove that it is false?

[iljitsch]

In the real world there is no such thing as an “equals” operation on real numbers, because there’s always measurement errors and storage limitations that impose rounding.

So sometimes 0.99999 = 1 and sometimes 0.99999 <> 1. Deal with it.

[Great_Antibob]

@iljitsch

In the real world there is no such thing as an “equals” operation on real numbers

In the real world, there’s no such thing as “real numbers”.

So if we’re already playing in our little mathematical sandbox where “real numbers” do exist, true equality can just as easily exist.

[The_Great_Unwashed]

@Half_Man_Half_Wit

OK, so you accept that the value of the geometric series s = Σar[sup]n[/sup] is a/(1 - r). Now then, let’s see what this means for the sum 0.9 + 0.09 + 0.009 + 0.0009 + …

Plainly, in this case, a = 0.9 and r = 0.1. So, can you tell me what the value of s = a/(1 - r) then is?

It does seem utterly bizarre that 7s would offer exactly that which he denies (implicitly that lim n -> infinity r[sup]n[/sup] = 0 for some values of r.)

Unfortunately, if r = 1 the formula yields 0/0 and not 1/0.

[Indistinguishable]

No, when r = 1, the formula yields a/0, which is infinite for nonzero a, just as you might have hoped for.

[The_Great_Unwashed]

@Indistinguishable

No, when r = 1, the formula yields a/0, which is infinite for nonzero a, just as you might have hoped for.

I can’t be utterly numb in two threads:

S[sub]n[/sub] = a(1-r[sup]n[/sup])/(1-r) = a 0/0 fpr r = 1

[Indistinguishable]

Oh, I thought you meant the formula for the infinite summation: a + ar + ar^2 + … [ad infinitum] = a/(1 - r). Which does still work out: you get a/0 when r = 1, which is infinite for nonzero a. (For a = 0, this is the indeterminate form 0/0, which is compatible with the correct answer 0 for 0 + 0 + 0 + …)

The finitary summation formula a + ar + ar^2 + … + ar^(n - 1) [with only n many terms] = a (1 - r^n)/(1 - r) also works out for r = 1, insofar as it yields the indeterminate form 0/0 which is compatible with the correct answer. But, yes, it’s not that useful of a formula compared to the more precise S[sub]n[/sub] = a * n in that case.

[j_sum1]

@Wendell_Wagner

j_sum1, The Psychology of Learning Mathematics was first published in 1971. We’ve learned a lot since then. I believe that the notion that there are centers of the brain for particular skills like mathematics or language is no longer considered to be clearly true.

Well, I did say it was a while back.

Not surprised that our understanding has changed since then. However I was under the impression that the role of the temporal lobe as the centre for processing language and communication was well established. It certainly would not surprise me if some processing skills that fall under the broad heading of Mathematics were indeed innate.

[Wendell_Wagner]

My impression is that the temporal lobe is involved with many of the highest mental functions:

Temporal lobe

It could be that both language and mathematics are just particular examples of the application of the capacity of the brain to use the proper tools for a situation. This is the viewpoint of Daniel Everett, for instance. He thinks that mankind has the mental capacity to use tools, and both language and mathematics could be just the application of that capacity:

Repeat After Me (Published 2012)

[j_sum1]

I am going to defer to hour greater knowledge here, WW. In any case, we have deviated somewhat from the general thrust of the conversation at the moment.

Besides which, I don’t actually have a temporal lobe (at least not on my left side.) So said my brain surgeon. And no – I don’t know how that works either.

[Exapno_Mapcase]

@j_sum1

Besides which, I don’t actually have a temporal lobe (at least not on my left side.) So said my brain surgeon. And no – I don’t know how that works either.

“The doctors x-rayed my head and found nothing”. - Dizzy Dean

[7777777]

@Budget_Player_Cadet

Fundamentally, this is the basis of your misunderstanding. 1+1+1+1+… = infinity is, strictly speaking, false. You’re misusing infinity. You can only approach it, and while using it as such colloquially is acceptable, no valid proof will ever speak of infinity like that. Rather, they’d consider it as such:

lim(x->∞)[sum(0 to x)[1]].

That is the limes of the sum. Because that sum will never equal infinity. It simply diverges forever, and as x approaches infinity, so does the sum. It will never, however, reach infinity.

(Also, mathematically speaking, sum(0 to ∞)[…] is an invalid term - you’re simply not allowed to apply ∞ that way.)

I agree that you can normally only approach infinity, but now it is not enough,
we need to arrive at infinity.
You need to distinguish the limit of infinity and infinity from each others, how do you
do it?
How do you do it without the infinitesimal?
There is an infinitesimal difference between infinity and the limit of infinity,
the infinitesimal is x, and according to you x=0, which sets the limit of infinity
equal to infinity.
To illustrate further the concept of limit: exactly the same problem exists with
1=0.99999…which means that the limit of 0.99999…is 1,or maybe
we should talk about whether 1 has a limit. What is the limit of 1, what is the
limit of infinity? Or what is the limit of 10, is it 9.999999…? There is infinitesimal
difference x always in these cases, otherwise there is no way to distinguish between
a number and its limit.

You say that I misuse infinity, although I just showed how infinity is usually thought
to be accessed: 1+1+1+1+…∞. That is not my method, rather it is the
common method in use, I have my own approach: Z + 1 = ∞

You write that I don’t understand infinity. If you do understand it better than me,
can you write down the value of the limit of infinity:
lim(x->∞)[sum(0 to x)[1]] = ?
according to what has been written above, the limit of infinity is not equal to ∞, so that ∞ is a wrong answer.

Z = ∞ is a wrong answer as I have said many times. The main property of Z is that
its value increases if you add 1 to it, whereas adding 1 to ∞ does not increase its
value because ∞ = ∞ + 1. Of course , if you assume that Z = ∞ then x = 0
as you said, but it is false. The infinitesimal is not equal to 0.

@Budget_Player_Cadet

You claim that this is false but have yet to give any convincing argumentation for why that is the case. Can you prove that it is false?

Yes, I can prove it is false. First, I take it as a definition of Z that it has a property
that its value increases if the number 1 is added to it, so it fulfills the definition of what a number is. Infinity is not a number according to the same definition.

Z = ∞ is a misuse of infinity. Infinity should not be written
on a point occupied by a real number, it is not part of line of points describing
real numbers. We are all agreeing that infinity is not a real number, not element of R.
The elements of R can be thought to represent points on a line extending or approaching infinity , the co-ordinate axes extending infinitely to the right and left
from the origin. If you want to write infinity on this same line, there should be an
empty space, an unoccupied point for ∞, so where is its location?
This is the definition of the co-ordinate axes. Usually these are taken as granted,
so that there is a framework where one can plot the graphs of functions.
But what if the co-ordinate axes themselves are functions, then one has
to deal with infinity, axes extending into infinity,to the right and to the left from the origin. Where is the empty space, unoccupied point, reserved for ∞?
The place for the largest number is occupied by Z, so you cannot write that Z
is an empty space for ∞ or that Z=∞. That is the misuse of ∞. You need to find
a place for ∞, where is it?

[Francis_Vaughan]

If you want to write infinity on this same line,

But only you want to do this. By making a modification to the line to accept ∞ you have modified the definition of your model of R. Such a modified model can exist, and can even be useful - but it isn’t R. It fails some of the critical axioms of R. This whole argument then becomes circular.
You keep wanting to modify R in a way that allows you to reason about ∞ within it, but won’t accept that adding ∞, in any way, modifies R in a manner that precludes the rest of your arguments. Then you say that the results of your arguments require these modifications in order to make sense. You have to be consistent. You need to stick with one system.

[7777777]

If you want to write infinity on this same line,

@Francis_Vaughan

But only you want to do this.

No, it is not me.

It is you.

It is you who say that Z = ∞

It is you who say there is no largest number, or that if Z + 1 = ∞ then Z it is equal to infinity. It is you who mean that infinity is a number, although you say that infinity
is not a number, not part of R. It is you who have to be consistent. It is you who need to stick with one system. It is you who need to understand infinity better.

I say that there is no place where we can locate infinity on the real number line.
I asked where it is because you imply that it can be written on the same place as Z.
Although that place is already occupied. There should be an empty space on the
real number line for infinity, so where it is? If there is no empty place, infinity is
not part of R, that is what I say. I don’t know why you refuse to understand what
I say, or misinterpret what I say.

[Francis_Vaughan]

I say that there is no place where we can locate infinity on the real number line.

Good.

OK, the next step is why you say Z exists? We keep trying to show the problem. You claim Z exists. Yet you claim that Z is a number on the line. But you also say that adding one to Z - ie moving along the line one single unit, takes you to ∞. We say that isn’t consistent. Adding one to any real number takes you to another real number. That is by definition. If you claim there exists a number on the line for which adding one to it takes you off the line - well that violates the definition of the line. Therefore Z cannot exist on the line.

[7777777]

@Francis_Vaughan

Good.

OK, the next step is why you say Z exists? We keep trying to show the problem. You claim Z exists. Yet you claim that Z is a number on the line. But you also say that adding one to Z - ie moving along the line one single unit, takes you to ∞. We say that isn’t consistent. Adding one to any real number takes you to another real number. That is by definition. If you claim there exists a number on the line for which adding one to it takes you off the line - well that violates the definition of the line. Therefore Z cannot exist on the line.

There can be a singularity on the real number line. And this singularity is reached by Z + 1 = ∞. Adding 1 to Z takes us to the singularity.

The nature of this singularity is what I am wondering.

It might be compared to a point where a function is discontinuous. When a graph of a discontinuous function is plotted there is an empty space at the point of
discontinuity. Could also an empty space exist on a real number line?
How do we know that the real number line is continuous? Can we imagine a line infinitely long if continuity requires it?
If so where is infinity itself located on such a line? Or might the line be broken at
a singularity?

[Francis_Vaughan]

Not really. There is no singularity on the real number line. What there are, are locations where some functions of the value at that location have a singularity. But that are an infinite number of such functions, and as such every value on the line can denote such a singularity. A trivial case is

f(x) = 1/(x-y), where for any y you choose, there will be a singularity at y.

But there are an infinite number of ways you can generate such functions, and they can have an infinite number of singularities. Analysis of such things is a pretty important thing by itself. The counterpoint are those points where the function is zero. Between them, poles (where the function has a singularity) and zeros (where the function has a zero value) are the bread and butter of many professionals. However you are moving away from the properties of the reals.

[Senegoid]

If space and time are quantized at Planck scales, maybe the Real Number System Itself must also be quantized at Planck scales. Everything we know about Math is wrong!

[j_sum1]

I don’t know if the world of physics is quantised like you suggest. But I know that the real number system is not. It is everywhere continuous. Which is why this talk of a discontinuity between Z and infinity is such nonsense.

[watchwolf49]

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