Final Revision in Math
Humood Theory
Important introduction
Do not rush to judgment after readings only some lines in this papers, read it to the end. I know many will see it as trying to dig in an empty well or argue with the unarguable , but during all history of Science, all big revolutions came only when we found something not very accurate in a field or point that no body could think it is able to be checked again.
Main Result
(0.999...) does not equal (1) They are two different concepts and qualifying them is like qualifying one orange and one apple.
(0.999...) represents the Unit for one Mathematical concept while (1) represent the Unit for another different mathematical concept.
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Let us look at these equations and reasoning
Lim xΰ ∞ ( 1- (1/x)) = 0.999 ( 1 minus an infinitely small fraction )
And Because 0.999 =1
Then Lim xΰ ∞ ( 1- (1/x)) = 1
By taking both sides to power x
[ Lim xΰ ∞ ( 1- (1/x))] ^ x = Lim xΰ ∞ ( 1 ^ x)
(1/e) = Lim xΰ ∞ ( 1 ^ x)
(1/e) = 1* 1 * 1* 1 * 1 * 1 * 1 * 1 * 1 * .
(1/e) = ( 1* 1 * 1)* (1 * 1 * 1 )* (1 * 1 * 1) * .
(1/e) = ( 1 * 1 * 1 ) * ..
(1/e) = ( 1 ) * ..
We can see that the right hand side of the equation well be always 1 But the left side of the equation is (1/e)
Which should mean there is something wrong in our first assumption 1=0.999 . However, later in this paper we will find out that the previous reasoning is a traditional ( proof by contradiction) which prove that the equation we began with
Lim xΰ ∞ ( 1- (1/x)) = 0.999 = 1 is a wrong equation.
The same contradiction appears also in this case:
Lim xΰ ∞ ( 1 - (1/x)) = Lim xΰ ∞ ( 1 + (1/x)) = 1
And by taking both sides to power x
[ Lim xΰ ∞ ( 1- (1/x))] ^ x = [ Lim xΰ ∞ ( 1+(1/x))] ^ x
This finally ends up with:
(1/e) = e
In fact The key to the whole issue can be generalized to be the detailed answer of this question: ( can a real number R minus or plus a fraction that is infinitely small equals any thing else than the real number R itself?)
Most mathematicians will answer NO, and when we say NO we end up with contradictions like (1/e) = 1 and e= (1/e) as we saw before.
So mathematicians escape from all that by saying that f(x)^ g(x) when f(x)=1 and g(x)=∞ is undermined form.
We saw that integer 1 to power ∞ gives always 1, the result become different only when we use 1 minus or plus an infinitely small fraction and raise that to power ∞, the result is always different from what we get when use integer 1
This should indicate that 1 does not equal to 1 minus or plus infinitely small fraction because if they were equal they should give the same result when raised to any power.
However, what mathematician were escaping from was in fact the other possible answer for the previous question which is :
Yes, a real number R minus or plus a fraction that is infinitely small does not equal R. such answer will end the contradictions we have.
But accepting such principle is a real nightmare for mathematicians because it simply means that 1 ≠ 0.999
Because 1 ( infinitely small fraction) = 0.999...
No one like to hear that 0.999 ≠ 1 because it will let them face other Proofs state that 0.999 = 1 and those proofs looks solid. The most famous proofs are the two below:
Proof (A) :
let
x = .999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
Proof B)
1/3 = 0.333
0.333...* 3 = 0.999
0.999 = 1
I will start by cracking Proof (A) and show it is in fact a Paradox not a proof.
Let x = 0.999...
10x = 9x + x = [ 9* (0.999...) ] + 0.999 I expressed 10x as (9x +x)
10x x = ( 9x + x ) x = [ 9* (0.999 ) ] + 0.999 - 0.999 Positive 0.999... goes with Negative 0.999...
9x = 9* (0.999 )
x = 9* ( 0.999 ) / 9 = 1 * 0.999
x =0.999
We did not end with x=1
In fact what made it looks like a proof was the mistake we did in this step:
10 x x = 9.999 - 0.999... = 9
This step assumes the value of infinite 9s after the decimal point in ( 10x ) equals the value of infinite 9s in (x)
The infinite decimals in x = 0.999... have different behavior than the infinite 9s in 10x =9.999... Because 9s in x= 0.999... Go to infinity faster than 9s in 10x=9.999...
(10 x ) is always one decimal behind (x)
When x = 0.99 then 10x=9.9
When x = 0.9999 then 10x=9.999
When x = 0.999999999 then 10x=9.99999999
Value of Infinite decimals in (x) Value of infinite decimals in (10x) =
Lim xΰ ∞ 9 / (10^x)
So the difference between the two values equals an infinitely small fraction which is Lim xΰ ∞ 9 / (10^x)
This means in Proof (A) when we say: 10 x x = 9.999 - 0.999... = 9
We are already Assuming that a number minus an infinitely small fraction is the number itself.
Which means we are already assuming that Lim xΰ ∞ 1 - ( 1/x ) = 0.999... = 1
So, in Proof (A) we already assumed 1= 0.999... Then no wonder we finally ended with our assumption, which make it not a Proof at all. You cannot prove any statement by assuming it to be true from the beginning.
But when we explain (10 x ) in a very fundamental method no body can argue with, we will say:
10x =0.999 + 0.999 + 0.999 + 0.999 +0.999 +0.999 +0.999 +0.999 +0.999 +0.999
And when we subtract x = 0.999 from it :
10x - x =0.999 + 0.999 + 0.999 + 0.999 +0.999 +0.999 +0.999 +0.999 +0.999 + 0.999 - 0.999 = 0.999... * 9
The infinite 9s in x after the decimal is gone with only one of those 0.999... Not all of them as we did in Proof (A).
For Proof (B) and any other proofs, after reading the next theory you will see such proofs is meaningless.
Humood Theory
When the man started to use numbers he used them to represent things in his daily life like representing how many sheep he has and how many time he can harvest his field. However as the mathematical theories and techniques developed during thousands of years, numbers kept representing one of two mathematical concepts depending on the conceptual ability for divisions into fractions:
1) Numbers which represent concepts that are conceptually able to be divided into fractions, like numbers which represents a distance, a period of time, quantity of flowers Etc.
2) Numbers which represent concepts that are conceptually unable to be divided into fractions. Like for example the number which represents how many solutions we have for an equation (you cannot have half a solution for the equation or quarter of a solution or any fraction of a solution for an equation).
Also another example, number which represent number of attempts for a player to kick a football (you cannot have a half or quarter or any fraction of an attempt, either you have an attempt or you do not have it at all),
Now, a question appears: Do numbers which represent different mathematical concepts in their ability for division into fractions have exactly the same forms?
There are two forms for a number :
1) Infinite decimals form (as sum of infinite series) like 0.999 ,3.999 ,158.999 ,Etc
2) Exact form (not as sum of infinite series ) like 1, 6, 100,158 , 14789 Etc.
But is having two forms for a number is just a luxury touch from the mathematical world? Or is there another meaning for that, as we always found during all the history of science?
In fact having two forms for a number is not for luxury purposes, every form has only one mathematical concept to represent!
When we get back to the entire structure of the infinite decimals form we find:
0.999 = 0.9 + 0.09 + 0.009 + 0.0009 +
By definition, a number in infinite decimals form is the result of an infinite addition for fractions. This mean you must have the fractions of the number first before you have the number itself.
This property is only valid for mathematical concepts that are able to be divided into fractions, for example if you have a distance that equals 1 meter then it is accepted to express it as this:
1 meter = 0.9 meter + 0.09 meter + 0.009 meter +
Also:
1 second = 0.9 second + 0.09 second + 0.009 second +
1 flower = 0.9 flower + 0.09 flower + 0.009 flower +
But for mathematical concepts that are conceptually unable to be divided into fractions it is a different case.
Mathematical concepts which conceptually unable to be divided into fractions cannot be represented in infinite decimals forms.
Those concepts cannot be defined as the sum of fractions.
For example when you have one solutions for an equation, you cannot define it as :
1 solution = 0.9 solutions + 0.09 solutions + 0.009 solutions +
Because fractions like 0.9 solutions, 0.09 solutions, 0.009solutions, etc Does not exist and assuming their existence is Out of Logic.
However, it is accepted to represent them in the Exact Forms, because Exact forms are not defined to be the result of adding any fractions.
Result 1.1: Concepts that are conceptually unable to be divided into fractions can be represented only in Exact Forms.
Result 1.2 Concepts that are conceptually able to be divided into fractions can be represented only in infinite decimals form and cannot be represented in Exact form, and here is why:
If you say a number X can be expressed as the sum of fractions this means the number consists of those fractions and the sum of those fractions should give back the number X
We all know the equation ( 1/3 ) = 0.333
This equation states that we can take an Exact form number (1) and through a division operation we get the other form (Infinite decimals form number) 0.333
When you assume the possibility to divide Exact form number into some infinite decimals form numbers, then it must be possible to add some infinite decimals numbers and get an exact number.
If X = A+B+C
Then A+B+C= X
If X/Y = Z
Then Z * Y = X
But the problem is: this operations is irreversible as it should, because we cannot take the infinite decimals number (0.333 ) and through a multiplication operation get the Exact number (1), whatever we do we will always get an Infinite Decimals Number.
In fact it is a general case: It is impossible to get an Exact Form number by the addition of any Infinite decimals form numbers. all what you get from adding numbers in infinite decimals forms is another numbers in infinite decimals forms)
If we say 1 = 0.333 + 0.333 + 0.333 .
We find 0.333 + 0.333 + 0.333 . = 0.999
The sum of infinite decimals fractions did not give the Exact form number (1) although we have assumed the Exact form Number (1) could be divided to give those infinite decimals numbers
So if A+B+C ≠ X
Then X ≠ A+B+C
And if Z *Y ≠ X
Then X/Y ≠ Z
This mean the equation we began with 1/3 = 0.333 is a wrong statement and led us to the wrong statement 1= 0.999
Which make it a traditional Proof by contradiction. That 1≠0.999
So Proof (B) in page which states:
1/3 = 0.333
0.333...* 3 = 0.999
0.999 =1
It is not as thought to be a direct proof that 1=0.999
In fact It is a proof by contradiction that 1≠0.999
This mean for the operation 1/3=0.333 the only correct form is
0.999 /3 = 0.333
** Note
It may be said that the Geometric series 0.9 + 0.09 + 0.009 +0.0009 +0.0009 + has the sum S = a / ( 1- r) = 0.9/0.9 = 1 , that is not true.
The problem with most of what so called proofs that 1=0.999 , they are somewhere in the reasoning assuming that 1=0.999 but do not notice that as we saw in Proof (A) in page , in fact it is not easy to notice such thing , such error hides very well, it is very tricky and very slight but at the end, there is no proof that 1=0.999 because somewhere before it reach that result, it make a step that assume 1= 0.999 and for sure that takes us to nowhere.
When we make revisions for the reasoning of sum of geometric series depending on the previous results we got here, we can see the sum = 0.999 not 1
Look at these other geometric series :
0.6 + 0.06 +0.006+ the sum of this series is 0.666..
0.8 + 0.08 +0.008+ the sum of this series is 0.888..
Also 0.9 + 0.09 +0.009+ the sum of this series is 0.999..
Result 1.3
When mathematicians found that numbers can be expressed as sum of infinite series in fact they discovered the right and only way to express numbers that represents things that are conceptually able to be divided into fractions.
The Unit for concepts that are conceptually able to be divided into fractions = 0.999
Result 1.4
The Unit for concepts that are conceptually unable to be divided into fractions =1
Hint from Physics
There is a strong support form physical world for this because things in physical world that can be divided into fractions cannot be Exact at all.
There is no Exact 1 meter where error is Zero. And that is not because our measurement equipment is not able to achieve that accuracy, but because it is a physical property for objects and matter in the real world. Objects in physical world do not have exact edges which mean there is no exact mathematical point where a physical objects start from or end to and that is what we discovered from modern Physics and quantum mechanics. When you say the end of a physical object (a table for example ) is where the last electron from the last atom of that object ends this is not helping you at all, because electron behave like a wave and there is no Exact location for that wave . So in real world assuming a dividable concept to be expressed in an Exact form number is a wrong statement because it cannot be exact.
Effects on Mathematics and Physics
The effects of the previous
theory and results are so powerful on Mathematics and Physics, It has a strong
potential to solve many old problems and dilemmas that cannot be solved
without a new mathematical tool like it. All that will be published soon in my
next paper with god willing.