I’m comfortable with this, although we have to admit it is irrational. We can’t let our imagination get away with us. That makes it too complex. As long as zero can have a direction, all is right with me.
The number that corresponds to this Cauchy sequence does:
s(0) = 1
s(n+1) = ( s(n) + 2/s(n) ) / 2
Each value is a rational number. Here are the first few past 0:
s(1) = 3/2
s(2) = 17/12
s(3) = 577/408
s(4) = 665857/470832
s(5) = 886731088897/627013566048
s(6) = 1572584048032918633353217/1111984844349868137938112
s(7) = 4946041176255201878775086487573351061418968498177/
3497379255757941172020851852070562919437964212608
s(8) = 48926646634423881954586808839856694558492182258668537145547700898547222910968507268117381704646657/
34596363615919099765318545389014861517389860071988342648187104766246565694525469768325292176831232
Tell me why that’s not a well-behaved real number, because Dr. Wildberger sure didn’t. MF94 mentions Cauchy, then MF95-MF107 talk about other stuff, like 1/0 being a useful extension to the rational numbers. MF108 and higher don’t exist.
(And wow, that converges a WHOLE lot faster than I thought it would. s(8) is within 3×10[sup]-196[/sup] of √2.)
However this goes beyond just irrationals.
By this logic it depends even on which integer base you use what numbers exist - and that different bases exclude different rational numbers.
So in base 10, we can have 1/4 (0.25), 1/5 (0.2), but not 1/3, or 1/7.
However in base 3 we can have 1/3, (0.1) but not 1/4, 1/5 and 1/7.
In general the deal is that you can only represent in base 10 rationals that have denominators built from 2 and 5, and no other numbers. Or further generalising, no base can represent any rational whose denominator is not perfectly factored by only the factors of the base.
This clearly stems from the idea that any infinite extension to the right of the decimal point is not valid, no matter what the base.
Now there is a clear difference between the rationals and the irrationals here. We can create a perfectly well understood deterministic notation that expresses the representation of the missing rationals in any base. This is because they always contain repeating patterns.
In base 10 1/7 is the pattern 142857, and 1/7 is 0.(1428157) and is perfectly and consistently represented as such.
The objection seems to be, as usual, a dislike of the idea of an infinite number of repetitions of the pattern.
So where 1/3 is fine as 0.1[sub]3[/sub] and 1/10 is fine as 0.1[sub]10[/sub] the numbers “don’t exist” if you swap bases.
Most people would find this a very extreme interpretation of the notation. When a perfectly good well understood unambiguous notation exists, it seems churlish to not use it. However it does require understanding what the notation means.
In this discussion we mean that the following are equivalent (in base 10 - similar for other bases):
0.abcde… = 0.(abcde) = a10[sup]-1[/sup] + b10[sup]-2[/sup] +c10[sup]-3[/sup] +d10[sup]-4[/sup] +e10[sup]-5[/sup] + a10[sup]-6[/sup] + b10[sup]-7[/sup] +c10[sup]-8[/sup] +d10[sup]-9[/sup] +e10[sup]-10[/sup] + …
and all are the same as
limit as n: 0 -> ∞ ∑(a10[sup]-(n5 +1)[/sup] + b10[sup]-(n5 +2)[/sup] +c10[sup]-(n5 +3)[/sup] +d10[sup]-(n5 + 4)[/sup] +e10[sup]-(n5 + 5)[/sup]
With no special dealing going on here, but only as a way of making simple notation.
You are talking that the sum of a convergent series should be accepted as true. And the convergent series is 0.99999…
I have already said that this is correct
9x10[sup]-1[/sup] + 9x10[sup]-2[/sup] + 9x10[sup]-3[/sup] + 9x10[sup]-4[/sup] + 9x10[sup]-5[/sup] + … = 0.99999…
here you can see the limit of this convergent series, it is 0.9999…
but this is not correct
9x10[sup]-1[/sup] + 9x10[sup]-2[/sup] + 9x10[sup]-3[/sup] + 9x10[sup]-4[/sup] + 9x10[sup]-5[/sup] + … = 1
the limit of this convergent series is not equal to 1
I know, there is a formula S = a/(1-q)to calculate the sum of a geometric series
a + aq[sup]-1[/sup] + aq[sup]-2[/sup] + aq[sup]-3[/sup] + aq[sup]-4[/sup] + aq[sup]-5[/sup] + …
and it gives in our case the result
9x10[sup]-1[/sup] + 9x10[sup]-2[/sup] + 9x10[sup]-3[/sup] + 9x10[sup]-4[/sup] + 9x10[sup]-5[/sup] + … = 0.9/(1 - 1/10) = 1
What is wrong?
The correct way of writing the formula is
lim[sub]n -> ∞[/sub] S = a/(1 - q)
The geometric series formula does not give an exact value as a result. It gives only
the limit.
I write the series now as
9x10[sup]-1[/sup] + 9x10[sup]-2[/sup] + 9x10[sup]-3[/sup] + 9x10[sup]-4[/sup] + 9x10[sup]-5[/sup] + … 9x10[sup]-n[/sup] +…
It is clear that the formula ignores the infinitesimal because it involves infinity so that the n’th term
9x10[sup]-n[/sup] becomes 9x10[sup]-∞[/sup]
It is clear that the n’th term now
9x10[sup]-∞[/sup] = 0
but it is an error to treat infinity like a number which can be inserted in a geometric
series formula in this way. But it is happening in your way of understanding how the sum of a convergent series should be calculated.
You just seem to arrive at an exact value 1 of the convergent series 0.99999…
but you make errors:
First you think that the limit is the exact value
Then you ignore the infinitesimal and think that it is equal to zero
You provide an illusion of the truth. You should seek for truth not an illusion of it.
I am intrigued at the direction this has taken. I didn’t think it was going to go there.
At first I thought it was the usual misconceptions related to limits or how we treat infinities.
then I thought it was misconceptions related to non-standard use of notations.
Now we are deep in stuff loaded with layers of bizarre and unintuitive weirdness.
[ul]
[li]Irrationals are not actually numbers. Fictitious I believe was the word used[/li][li]Any number where we cannot pin down the last decimal digit is not truly a number[/li][li]Numbers change whether they are rational or irrational depending on the base being used[/li][li]Recurring decimals are not precisely defined and do not equate to the fractions that they are normally interpreted to represent[/ul][/li]By this logic all recurring decimals are not truly numbers since they can not be precisely defined. and this would extend to the number 1.0000… too.
I wonder at the sense in eroding a perfectly functional, useful, well defined, well-behaved, rigorous number system with a long history and which is both intuitive and models our world well and replacing it with an ill-defined system with inconsistent use of notation where all numbers fail to have a proper representation and nothing can be perfectly defined.
The bind moggles.
So where does n appear on the right hand side in order to run it towards infinity? Currently n is unbound, and the value of the RHS can take on a value irrespective of n’s value.
And around we go again with no progress. You assert these without proof or reasonable justification. Again you seem to be stuck in the rationals, and unable to accept either the nature of infinity or the existence of the irrationals. Sure, it isn’t a trivial issue to make the leap. But this is becoming some sort of article of faith. We can show that in the Real numbers there are no infinitesimals other than zero. You don’t seem to want to address the Reals, and are stuck in the Rationals. It isn’t just the limit of 0.999… here, but almost all numbers. Almost all numbers are irrational, in a precise way of defining almost all. As has been noted earlier, the Pythagoreans knew about √2 and that it was both a constructable number, and was not rational. It gave them just about as much trouble as we are seeing here. They hid the proof, and didn’t discuss it, as it went against everything they believed. Legend has it that they drowned Hippasus of Metapontum at sea for revealing it. We seem to be back there again.
RHS value is dependent of the n’s value, if n is finite the result of the sum of the finite series is exact value:
S[sub]n[/sub]= a(1 - q[sup]n[/sup])/(1 - q)
whereas if there are infinite terms in the series, the value is the limit
lim[sub]n -> ∞[/sub] S = a/(1 - q)
The limit is not anymore the same as exact value.
You cannot ignore the LHS of the equation and say that the RHS is independent of
the LHS. The RHS represents the limit of the LHS when n is unbound. You can try to deny a clear fact, but that is a mistake. If you take away “lim” from the LHS , the
equation is not correct anymore.
Yes, it is true that we go around with no progress. The reason is that you are
constantly ignoring the fact that there are infinitesimals other than zero.
You refuse even to consider this possibility. I can consider the fact that
the infinitesimal can be equal to 0, but at the same time I have no problem
considering that the infinitesimal can also be non-zero. Why this issue is such
a leap?
I don’t know why you want to insist that I am unable to accept either the nature of infinity or the existence of the irrationals. It is rather you who don’t know
infinity. I have no problem accepting irrationals, I don’t know where do you
get your conclusions, I did not even talk about irrationals.
There is no symbol n on the RHS. I really can’t be more clear than that. None. The value of the expression has no dependency on n. Its value remain the same whether n is one or -10000. Adding a limit notation has no effect on the evaluation of the expression. You may as well add "when z equals Thursday as well, it doesn’t change the evaluation.
You don’t seem to understand the difference between the Rationals and the Reals.
Do you accept that infinity is the cardinality of the Natural numbers?
The value of the expression of the sum of a geometric series has a dependency on n.
If n is finite, the value is the exact value S[sub]n[/sub]= a(1 - q[sup]n[/sup])/(1 - q)
If n is infinite, the value is the limit lim[sub]n -> ∞[/sub] S = a/(1 - q)
You are denying a clear fact. You write that there is no symbol n on the RHS. There is symbol n on the LHS in any case and you cannot separate the RHS and LHS of each others, because they together form the expression. You cannot remove
the “lim” from the LHS as I already said. You are trying to treat the RHS as
independent of the LHS, independent of the fact that it represents the limit, but it is wrong.
OK, I see what you are writing. We are at cross purposes about the meaning.
So, how about the others?
Do you accept that infinity is the cardinality of the natural numbers?
Can you define the Reals and the Rationals? In particular can you elucidate why they are different?
But as said previously, 0.99… is defined as the limit of that series! The formula is both accurate and adequate.
This is simply not accurate. The equation does nothing of the sort. What it does is establish a cardinality that is shared by both parts of the equation (i.e. you can perfectly match each term in the upper line to a term in the lower line, thus ensuring that they cancel out - this works due to the nature of infinity) then subtract each kth term on the top with each k+1th term on the bottom, leaving only the first few terms. At no point does any term become 9*10^-inf. The terms simply cancel each other out in a way which is entirely valid given our understanding of infinity. The infinitesimal, even if it exists and is valid in the set of real numbers (it doesn’t and it isn’t), doesn’t play into it at all!
No. Again, you simply fail to understand the formula. Look at it again. For each term in
S = x + xr^-1 + xr^-2 + xr^-3…
You can match up with perfect cardinality that term to a term in the other row
Sr = xr^-1 + xr^-2 + xr^-3…
Which is equal! So when you subtract one from the other, you end up with S-Sr = x! That is how the formula works! It has nothing to do with making the nth term 0, or anything to do with infinitesimals. It simply doesn’t matter what the xr^-n is, because unless it’s n=0, it’s already been canceled out of the equation!
The claim that 0.99… is equal to the limit is not an error, it is a definition. 0.99… is defined as the limit of the infinite sum (0.9, 0.09, 0.009, …). There is no other definition that is coherent.
At no point is the infinitesimal even a part of this. In fact, if anything, this equation shows in no uncertain terms how useless the concept is.
You’re soooooo deep… :rolleyes:
Meanwhile, what you’re saying goes against literally centuries of mathematical teachings. I really do wonder - at what point does someone like you look at the opinion of various professors and teachers and mathematicians and <insert essentially anyone with a damned clue about mathematics here>, see that they are distinctly opposed to one’s own, and wonder, “Hmm… I wonder if there isn’t something here I’ve missed?” Or does Dunning-Krüger just ensure that no matter how much every authority on the subject disagrees with you, you just write off their expertise?
Is this type of behavior accepted on these forums ?? I don’t attack people, but I will certainly retaliate if that is the norm around here. I don’t like to, since it will likely derail the conversation in a hurry.
I have nothing to be embarrassed about. I am satisfied with my current education and my plans to further it.
The answer is √2
√2 and 1.414213562373095… are each just notations, just ways to represent a number. Do you think the latter has some “purity” the former lacks?
In base-10, 1/2 can be represented 0.5 while 1/3 has only the non-terminating 0.3333333… Yet in base-3 the situation reverses; now 1/3 = .1 while 1/2 = .11111111… How does that affect your argument?
[Quote=7777777]
…99999 = -1
1=0.99999…
…99999 +1 = 0
…99999.99999…= 0
1/3 ≈ 0.333333333…
1/3 = 0.333333333…
-1 + 1 = 0
-1 + 0.99999…= 0
…99999.99999… = ∞
…99999.99999… ≈ ∞
…99999.99999… = 0
…99999.99999… ≈ 0
1 - 1/∞ = 1
1 - 1/∞ = 0.99999…
0.88888…+ 0.11111… = 1
0.88888…+ 0.11111… = 0.99999…
[/QUOTE]
You just think that I think that you should take each pair as having one correct and one incorrect statement. That was originally my purpose, that each pairs were mutually
exclusive, in the same way as the pair 2=1, 2≠1, for example.
But this time there seems to be at least one pair where both are correct or both
are incorrect depending on how they are to be understood.
What is this pair?
Not sure why you concluded that at all.
Is this a false statement? : “At any finite (n) of the definition of 0.999… you are infinitely far from the limit”
"
I am saying you tell me the “number”. It is a downfall of certain base-number systems. Just like an infinite loop of computer programming… you get stuck in the loop and can never get out of it. So you make up a rule and say “we’ll just use the limit instead”.
The limit is finite. The original object is infinite (endless), (as in “infinite decimal”).
They are certainly NOT the same thing.
infinite decimal NOT EQUAL finite decimal ! 
For what (n) is s(n) exactly equal to √(2) ?
This is exactly correct.
given:
S(n) = (a - arⁿ) / (1 - r)
Then by the usual rules of algebra, we take the limit of *both *sides:
limit (n→∞) S(n) = limit (n→∞) (a - arⁿ) / (1 - r) = a / (1 - r) = L
They are defined. Just improperly so in my view.
We both know that this is strawman. Zeros have no value and are most definitely not analagous to non-zero numbers.
…0001.000… is what the number 1 actually is. 1 is a shorthand way to write it. That is was a “terminating decimal” is: One that has “infinite zeros” after it.
This is why it is important to understand an infinite series as defined for each index.
[/QUOTE]
He skipped to the conclusion. It wasn’t a proof, but rather an equality, and a very simple one of which I would expect understanding. Just look up this post by me, where I quoted him. I actually put in the few missing steps.
We don’t define limits. We analyze limits and figure out what L is. Defining it before analysing it is circular as it assumes the conclusion before you do anything.
Sqrt is a funtion performed on the variable, say: f(x) = sqrt(x)
What is the result of this operation ?? You gave me a function. I want to know the base-10 number please. If you are going to write […] that implies you left out some digits for which for the sake of argument, I don’t know.
If a child asked you what 25 times 25 equals, would you say 6… ?? where they have to guess the rest ??
Here is one of the problems with infinity:
1, 2, 3, … , n , … , ??? , … ∞
I always here the argument “You can’t use induction because infinity it involved” I disagree completely because if by infinity we mean the cardinality of natural numbers, then induction CAN be used to show that for any n, we can show that any n+1 is still less than 1:
0.9(n times) < 0.9(n+1 times) < 1 where n = 1, 2, 3, …
Notice the […] there ??
No amount, including infinite, of 9s after the decimal will equal 1. it will be infinite 9’s only…
But then: You perform a voluntary operation, like the limit, and claim equality because “it is a definition!”
The statement you’re replying to doesn’t mention defining a limit, it explains that what mathematicians mean when they write 0.999… is a limit, and it is a limit by definition. And in the world of math, when we analyze the limit of the sum of all terms 9*10[sup]-n[/sup], from n=1 to infinity, we get 1.
I disagree because the limit is:
limit (1 - 1/10ⁿ) = 1
where f(n) = 1 - 1/10ⁿ is the partial sum function for the sequence of partial sums.
Again, we don’t define limits. That is contradictory to the entire concept.
Unless you mean that 0.999… really means: Limit 0.999…, where 0.999… = Σ (n=1 to inf) 9/10ⁿ , but why would someone ever leave out the word “limit” for a limit ?
As others have explained already, you’re stuck in a world where sqrt is a function. In the world of math its a way to describe a number. The square root of 2, for instance. One exact way to write that is “square root of 2”, this tells us a lot about the number and allows us to manipulate it in a number of convenient ways. The length of the hypotenuse in a right angled triangle with two sides equal to one, is exactly sqrt 2, it’s not approximately “however many digits I calculate of the non-existent number sqrt 2”.