0.(9) = 1 is based on infinite sum theory, which Euler flawed. He thought he was using perfect mathematics, but he was not, which let do a contradictory conclusion which was “amazing!” and minds were blown, that 2 different decimal numbers, which should all be unique for the “amount” or “value” they represent, had the same “value”. This is contradictory of course, as much as 1 = 2 is.
The infinite series formula was based on finite terms.
Then with a leap of faith, the finite equation was used with infinity.
However, limit concept prevents this equality in the first place (as a sum).
The leap of faith is the same one that mathematicians try and use to prevent the induction technique which disproves 1 = 0.(9):
0.9 < 1
0.99 < 1
0.9(n times) < 0.9(n+1 times) < 1
Here is a summary of Euler’s proof for infinite sums:
S = a + ar + ar² + … + arⁿ + …
rS = ar + ar² + … + arⁿ + …
Now what usually happens here is the magical INCORRECT subtracting of infinite series terms. You see, here is the correct way of subtracting infinite converging series:
Σ { a_i - b_i } = Σ a_i - Σ b_i
But you see in the proof, they subtract like this: a_i+1 - b_i:
S - rS = (a - 0) + (ar - ar) + (ar² - ar²) + …
This forces the results to be what is desired along with things such as 2 different decimal numbers being the same decimal number, ie, 0.(9) = 1
However, when done correctly, you get consistent results:
S = Σ a_i
rS = Σ r x a_i
S - rS = Σ {a_i - ra_i}
Apply to 9.(9) where a = 9, r = 1/10:
S = 9 + 9/10 + 9/100 + …
rS = (1/10) x S = 9/10 + 9/100 + …
S - rS = (9 - 9/10) + (9/10 - 9/100) + … = 8.1 + 0.81 + 0.081 + … = Σ 81/10ⁿ
(then we have coercions that there is no ‘1’ as the decimal form of the above sum, and that is is equal to 8.999… to which I reject and say then, that there is no decimal number to accurately represent this series, since there is a ‘81’ in every iteration added to the end)
You are led to believe that S - rS = a, but **clearly ** above:
S - rS = (a - ar) + (ar - ar²) + …
If a = 9, and r = 1/10, then S - rS = 9
But if we do it correctly, S - rS = Σ 81/10ⁿ
As we can see, the proof is flawed using incorrect infinite series subtractions to achieve the results that were desired. The proof was not a proof at all since it is invalid… unless you *assume *Σ 81/10ⁿ = 8.999… and also 9 = 8.999… which is completely circular, that is, assuming S - rS = a which is the basis of the entire “proof” of 0.(9) = 1.
Secondly, you can show using induction that 0.(9) - 0.(9) is never equal to 0, just as induction can be used to show that 1 = 0.(9) is never zero:
1 - 0.9 > 0
1 - 0.99 > 0
1 - 0.9(n times) > 1 - 0.9(n+1 times) > 0
Now I am sure the “density” theory has been discussed, which is also circular since it assumes 0.(9) is finite.
Rules exist for operations with finite numbers.
Rules exist for operations with infinite series.
I ask, where are the rules for subtracting infinite series and a finite number ??
Unless you assume the limit, none exist, and then of course your argument is circular since you already used the limit instead of the actual series.
Construction of a “Real Number”
Same argument here. Restrictions on the definition force it to only use finite “n” and “epsilon” which only prove that any finite number is less than infinity.
The “made up” rules are such that infinity automatically invokes a limit on itself.
This is strictly made up in order for them to get their own way.
In no way does 0.9(infinite times) automatically invoke any limit. Write a computer program to create endless 9’s. Unless you specifically put code in there to identify an endless loop (infinity) and break the loop, it will go on creating 9’s for infinity. If you wrote code to break the loop and use the limit, then you DECIDED to do this. It is a clear DECISION and it must be accepted by students and professors.
It is no different than saying 1/10^∞ = lim (n→∞) (1/10^∞)
So back to the density theory; the question is:
What is the number between 1 and 0.(9) ? This question makes no sense since you are comparing a finite number and an infinite series. You are, however, pointing out that infinity is flawed and is exploitable in the number system.
I say again, infinity is exploitable, and exploitation has no place in mathematics.
Question: What is the average of 1 and 0.(9) ?
Answer:
(1 + 0.(9) ) / 2 = (1 + Σ 9/10ⁿ ) / 2 = 1/2 + 1/2 x Σ 9/10ⁿ = 1/2 + Σ (1/2 x 9/10ⁿ)
= 1/2 + Σ 4.5/10ⁿ
So what is Σ 4.5/10ⁿ ?
s1 = 0.45
s2 = 0.495
s3 = 0.4995
Again, what is the decimal number that represents this series? Unless you make some assumptions or circular reasoning, you have no argument.
Σ (n=1 to N) 4.5/10ⁿ = 0.49(N-1 times)5
Now, if you set N = infinity (which is usually a NO-NO to set any variable equal to infinity), you end up with 0.49(∞-1 times)5
and ∞-1 is an illegal math statement.
These are just some of the flaws with these concepts and are usually just glossed over, or some hand-waving and authority is used to coerce you into “believing”.
