On 8/11/22 11:01 AM, wij wrote:
> On Thursday, 11 August 2022 at 17:29:00 UTC+8, Skep Dick wrote:
>> On Tuesday, 9 August 2022 at 02:30:19 UTC+2, richar...@gmail.com wrote:
>>> On 8/8/22 8:27 AM, wij wrote:
>>>> If 0.999... is rational, then:
>>>> 0.999....= p/q (p,q∈ℕ)
>>>> <=> 0.999...*q=p
>>>> If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
>>>> since 0.999... is defined as infinite repeating, and q is finite,
>>>> but 0.999...*q is never finite.
>>>> Conclusion: Repeating decimals are irrational.
>>> Wrong, 9/9 can generate the pattern 0.9999...
>>>
>>> This can be proven by 9 = 3+3+3 and the assocative property thus 9/9 =
>>> (3+3+3)/9 = 3/9 + 3/9 + 3/9 = 0.3333... + 0.3333... + 0.3333... =
>>> 0.9999....
>> Ooooh! The associative property! I know how to abuse that too.
>>
>> S = 1 + 2 + 3 + 4 + 5 ...
>> S = 1 + (2+3+4) + (5+6+7) + (8+9+10)...
>> S-1 = 9 + 18 + 28 ...
>> S-1 = 9(1 + 2 + 3 ...)
>> S-1 = 9S
>> -8S = 1
>> S = -1/8
>> 1+2+3+4+5... = -1/8
>
> I think you should provide a solution to really solve the issue.
>
> My idea of infinity is simple. '∞' denotes a unique number like the i (unit of
> imaginary number).
>
> Definition of ∞:
> 1. ∀n∈ℕ, n<∞
> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>
> The meaning of ∞ in 'thinking' is merely (a process/procedure) 'never end'.
>
> I think I solved the basic 'paradoxes' of infinite series. The basics is that
> the addend of an infinite series cannot be rearranged.
> Everybody seems to agree this point, but really as did?, or I formalized the idea.
> Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

Yes, you can start to build a number system like this, and in fact there
are a number of them based on slightly different details.

The key is that none of them are "The Real Number System", but an
expansion of it, and when you do this to express the concept of
"infinity" you tend to also lose some of the properties of the Real
Number system. (Which properties you lose depends on details of how you
flesh out the details of this new number).

Note, it takes some work to fully define how your "infinity" works, and
one of the problems is that some (many) definitions turn out to actually
end up with inconsistent systems (which is one of the reasons you lose
some of the normal basic properties, to eliminate those inconsistencies).

>
> +-----------------+
> | Infinite Series |
> +-----------------+
> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
> a(n) is called the general term, addend, summand. n is referred as the
> index. Series S is the sum from the first term a(0) to the last term a(k).
> The sum of those first terms (n<k) is called the partial sum.
> "a(0)+...+a(k)" is called expanded form.
>
> Infinite Series::= If the series S refers to infinite terms (n=∞), S is called
> an infinite series. Note that there are infinite addend. The sum cannot be
> completed by enumeration (∞ means unreachable, by definition).
>
> In the concept that number-is-an-expression-of-computation, infinite series is
> a number, no such concern of converge/diverge (statement when number converges
> is a number, diverges is not, is self-controdictory). The computaion rule of
> infinite series is based on the expanded form and concepts mentioned above.
> Noteworthy difference is that the interpretation of "..." in the expanded form
> is a "fixed/unique" number of terms, i.e. "∞+1≠∞" (not the notion of Cantor's
> infinite correspondence).
>
> Arithmetic of expanded form:
> Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
> S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
> <=> S= 1+a*S-a^(∞+1)
> <=> S(1-a)=1-a^(∞+1)
> <=> S= (1-a^(∞+1))/(1-a)
>
> Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
> S= 1+2+3+...+n // (1)
> S= n+...+3+2+1 // (2)
> 2S= n*(n+1) // (1)+(2)
> <=> S= n*(n+1)/2
>
> ∴ Basically, formula for 'finite' series is applicable to infinite series.
> (note that mathematical inducion cannot prove such formulas because by
> definition, ∞ is unreachable by counting.)
>
> Rule: Handling of the expanded form of infinite series must list the last
> addend. Otherwise, the expanded form is ill-formed (obscure semantics and
> information being lost cannot conduct valid deduction).
>
> Ex.1 (the last addend is omitted):
> A=1+2+3+4+5+...
> =(1+2)+(3+4)+5+...
> =3+7+5+... // ill-formed, obscure semantics.
>
> Last addend listed:
> A=1+2+3+4+5+...+∞ // well-formed, the exanded form of Σ(n=1,∞) {n}
>
> Ex.2:
> S=1+2+4+8+... // ill-formed
> <=> S=1+2(1+2+4+8+...)
> <=> S=1+2S
> <=> S=-1
>
> Last addend listed:
> S=1+2+4+8+...+2^∞
> <=> S=1+2(1+2+4+...+2^(∞-1))
> <=> S=1+2S-2^(∞+1)
> <=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
> // S=-1 can be found in youtube. (the term containing the
> // last addend ∞ is ignored)
>
> Ex.3:
> "f(n)= Σ(k=0,n) 1/k! => f(∞)=e(The base of natural logarithm)"?
> We know for sure ∀n∈ℕ, f(n)∈ℚ. To get the result f(n)=e (f(n)∉ℚ), the only
> current option is n=∞. But the issue whether or not f(∞)=e (exact equal by
> definition) can only be decided via definition, e.g. e≡f(∞). Otherwise, we
> can only say f(∞)≈e. (In considering the definition of the equal sign '=',
> other forms of e are likely not mutually replaceable with f(∞))
>
> Ex.4: x= Σ(n=1,∞) 1/n²
> A common expression is x= Σ(n=1,∞) 1/n²= π²/6, therefore, π=√(6*x)
> The issue here is: Lots of π can be derived from various infinite serieses.
> But, according to the definition of '=', the result of mutual substitution
> may become inconsistent.
> For now, the uncontroversial definition of π is the ratio of the
> circumference of a circle to its diameter (no computable definition), it is
> more correct to use '≈'.
> Therefore, Σ(n=1,∞) 1/n² ≈ π²/6 is what it is.
>
> [Note1] "..." in expression is normally indeterminant, of vague semantic.
> "0.999..." is also indeterminant before the "..." is eliminated, the
> number "0.999..." represents is uncertain, must be removed to ensure
> what the number is.
> Ex1: Let x=0.999...
> 10*x= 9+x // This is the result of x after interpreted, not necessarily
> // the result followed from "x=0.999..."
> // This equation must be given to define x (eliminate the
> // ambiguous "...")
> Ex2: Let x=√(2+√(2+√(2+...))). Then, possible interpretation of x are:
> x=√(2+x)
> x=√(2+√(2+x))
> x=√(2+√(2+√(2+x)))
> ...
>
> Ex3: "0.999..." usual 'repeating decimal' cannot denote a unique number.
> Let A= Σ(n=1,∞) 1/2^n = 0.999...
> B= Σ(n=1,∞) 9/10^n = 0.999...
>
> Let A=B
> <=> 1-1/2^∞= 1-1/10^∞ // converted from the formula of geometric series
> <=> 1/2^∞= 1/10^∞
> <=> 10^∞= 2^∞
> <=> 5^∞=1
> <=> false
>
> [Note2] Expanded form is prone to magic tricks, perhaps owing to conceptional
> generalization of visual illusion too easy to form. It is an error
> because the regrouping of the expanded form hides the fact that the
> original way of computation is reformulated.
> Ex: S can be the sum of any sequence of natural numbers.
> S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
> = Σ(n=1,∞) n+1 // S is modified
>
> Axiom: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
> = a(∞)+ Σ(n=0,∞-1) a(n)
> Theorem1: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
> Theorem2: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
> Proof: Omitted (Can be derived from the expanded form)
>
> Ex1: Σ 2*n =Σ (n+n) =Σ n + Σ n
> If Σ 2*n is said the sum of all even numbers, Σ n the sum of all natural
> numbers, the notion that the whole is greater than the part is conflicted
> by this rule (many paradoxical and current text book arithmetic have the
> same issue using Theorem2 like in Ex3).
> But, how do we express "the sum of even numbers"? Or Σ(n=0,∞/2) 2*n ?
> An idea that using C-language's for loop expression might solve this
> problem (or, at least, better than the traditional Σ notation):
> for(n=0;;++n) n; or f(n=0;;n+=2) n;
> Benefit of such a notation is 1.the symbol '∞' can be omitted 2. the
> meaning is more concrete, reducing mathematical imagination of 'Σ'.
>
> Temporary Conclusion: The essence of an infinite series may be a number whose
> computation never terminates because of infinite number of non-zero
> addends), or could be imagined as a 'running' number (density property
> requires the existence of such an 'irrational' number).
> ------------------------- End of Quote