Deeper than primes

[doronshadmi]

Agree with whom, yourself? That you claim it "is at AND not at the given distinct 0()" shows that you can't even agree with yourself.



Again, stop simply trying to posit aspects of your own failed reasoning onto others.



Again, stop simply trying to posit aspects of your own failed reasoning onto others.



What contradictory claim of mine are you referring to?



Nope, as explained to you many times before it means specifically that the boundary points are not included in the set of points resulting from that interval.




Again, stop simply trying to posit aspects of your own failed reasoning onto others.

“≠” is still not a location on a line.

Again, please indentify any location on a line that is not and can not be covered by points.

The Man, enjoy your 0-only reasoning.

 

[doronshadmi]

Why do you feel the need for changing the traditional description? 1/3 is the "exact form" and 0.333... is called the "approximate form." Believe it or not, the distinction have had its own description.

1/3=0.333...[base 10] by the traditional description, and it is false.

 

[doronshadmi]

I never mentioned n^-n. Where do you see it?

I mentioned a formula particular to base 2, where I meant by "universal" the involvement of the limit, that means n → ∞. It wasn't exactly the way I should have explained it, so that's why you tried to say that 1 - 2^-n where n → ∞ doesn't apply to other number bases apart from base 2. So I need to make amends.

The universal formula that involves the limit n → ∞ and applies to all number bases is

(aⁿ - 1)/(aⁿ⁺¹ - aⁿ) = 0.1111... [base a]

where n → ∞ and a is the number base. You can substitute finite k for n and run a few examples.

You can play with the notations as much as you like, but it does not change the fact that, for example: 1 - 0.111...[base 2]=0.000...1[base 2], or 1-0.999...[base 10]=0.000...1[base 10], where 0.000...1[base 10] < 0.000...1[base 2].

You are also missing http://www.internationalskeptics.com/forums/showpost.php?p=6451665&postcount=12034.

 

[sympathic]

You can play with the notations as much as you like, but it does not change the fact that, for example: 1 - 0.111...[base 2]=0.000...1[base 2], or 1-0.999...[base 10]=0.000...1[base 10], where 0.000...1[base 10] < 0.000...1[base 2].

You are also missing http://www.internationalskeptics.com/forums/showpost.php?p=6451665&postcount=12034.

He still does not get that "...." means infinite times, and "0.0......1" is just something you can type but does not exist in reality.

 

[epix]

1/3=0.333...[base 10] by the traditional description, and it is false.

The equality between the exact and the approximate form 1/3 = 0.333... doesn't disprove the existence of a number where the decimal digit 3 repeats itself the way that the number of repetitions approaches infinity. The approximate form is used to display the final result, but it never enters any algebraic manipulation. You can see expressions, such as (1/3)² but never (0.333...)².

It's the same case with Pi = 3.1415... with the difference that you won't be able to trace the "complementary digit" of Pi, coz it is an irrational number. You can use your notation that says

0.999... + 0.000...1 = 1

but your notation cannot solve

3.1415... + 0.000...x = Pi

to "prove" anything. That's why the approximate form is never used in algebraic manipulations, coz 0.000...x is vastly inconsistent and cumbersome expression. You can use your notation as an argument that can be followed only with rational numbers, but it doesn't work with the irrational numbers. That's why jpfisher always says that your stuff is inconsistent and can't cover all the classes.

 

[epix]

He still does not get that "...." means infinite times, and "0.0......1" is just something you can type but does not exist in reality.

He is using the recurrence

0.9 + 0.1 = 1
0.99 + 0.01 = 1
0.999 + 0.001 = 1
.
.
.
0.999... + 0.000...1 = 1

to demonstrate that 0.999... doesn't equal 1. There has been a great deal of confusion regarding the usage and the interpretation of the term "0.999..." as it can be seen here:
http://www.mathforum.org/dr.math/faq/faq.0.9999.html

 

[epix]

You can play with the notations as much as you like, but it does not change the fact that, for example: 1 - 0.111...[base 2]=0.000...1[base 2], or 1-0.999...[base 10]=0.000...1[base 10], where 0.000...1[base 10] < 0.000...1[base 2].

You are also missing http://www.internationalskeptics.com/forums/showpost.php?p=6451665&postcount=12034.

You are trying to prove something using a notation the meaning of which you don't understand well and that you call a "non-strict number," which means an "approximate form" in the traditional language. You are not familiar with the usage and interpretation of the approximate form.

Have you ever seen an expression, such as e⁴ F?

You didn't, coz a thermometer scale cannot accommodate all possible exact results of various computations, such as f(x) = e⁴. Instead, an analogue thermometer scale is divided into equidistant segments and the points are marked with decimal numbers. That means a meteorologist needs to convert the exact form e⁴ into its approximate form, which is 54.598... to see if the temperature obtained by some formula is above or bellow the point of freezing.

The approximate form is necessary for scientific applications of math, and, as such, it's not a subject to any inquiries that concerns math itself.

The approximate form is used in computational devices, coz scientists use computers and calculators. The devices do the internal math in binary numbers and can't possibly handle any exact computations. So there is an agreement that when the result shows 0.111111111111, for example, it is equivalent to the exact form 1/9 The same goes for informal numerical definition in some math texts, such as x = 0.111..., where the expression implies that x is approaching the limit 1/9; it doesn't indicate number 10^-1 + 10^-2 + 10^-3...

You are trying to find a pointless line segment using . . . well, pointless arguments.

 

[epix]

The Man, enjoy your 0-only reasoning.

"Well, your fingers weave quick minarets
Speak in secret alphabets..."

Well, your fingers weave quick minarets, speak in secret alphabets...

 

[epix]

You are also missing http://www.internationalskeptics.com/forums/showpost.php?p=6451665&postcount=12034.

The reason why I might miss something is that you are missing stuff in the first place:

0.000...1 is an example of non-strict (infinitely smaller AND > 0) number.

I expect that the subordinating conjunction "than" shows up after "infinitely smaller," but big, fat "AND" sits there instead. "Infinitely smaller AND?" It should read Infinitely smaller than [something] AND greater than zero, coz you using AND as a Boolean operator. Or did you attempt to write Infinitely small AND greater than zero? Just make up your mind what you want to say and then make sure that stuff isn't missing from your sentences, otherwise the chances that someone would understand what you mean are infinitely small BUT > 0.

 

[doronshadmi]

He still does not get that "...." means infinite times, and "0.0......1" is just something you can type but does not exist in reality.

You still do not get that 0.000...1 is a result of the irreducibility of 1-dim space to 0-dim space.

 

[doronshadmi]

3.1415... + 0.000...x = Pi

to "prove" anything. That's why the approximate form is never used in algebraic manipulations, coz 0.000...x is vastly inconsistent and cumbersome expression.

Again epix, you simply can't comprehend that, for example, 1-dim space is irreducible to 0-dim space, which leaves a room for non-strict numbers, which are the results of 1-0.111...[base 2], 1-0.999...[base 10], or PI-3.14...[BASE 10], etc. ... ad infinitum.

The rest of your posts, are based on this miss-comprehension.

 

[doronshadmi]

The reason why I might miss something is that you are missing stuff in the first place:

I expect that the subordinating conjunction "than" shows up after "infinitely smaller," but big, fat "AND" sits there instead. "Infinitely smaller AND?" It should read Infinitely smaller than [something] AND greater than zero, coz you using AND as a Boolean operator. Or did you attempt to write Infinitely small AND greater than zero? Just make up your mind what you want to say and then make sure that stuff isn't missing from your sentences, otherwise the chances that someone would understand what you mean are infinitely small BUT > 0.

"Infinitely smaller AND > 0" means that given any arbitrary (0,1] number, the considered number
is always smaller than the given arbitrary (0,1] number AND > 0.

 

[The Man]

"Infinitely smaller AND > 0" means that given any arbitrary (0,1] number, the considered number
is always smaller than the given arbitrary (0,1] number AND > 0.

Not for the interval you indicated, a 1 is included in that interval. So while 1 > 0 it is not smaller than, well, 1. The interval should have been (0,1) thus any arbitrary number included in that interval is > 0 and < 1.

 

[The Man]

The Man, enjoy your 0-only reasoning.

Again, stop simply trying to posit aspects of your own failed reasoning onto others.

For example we can represent the resulting set from the interval (2,6] as a union of resulting sets from the half closed intervals (2,3], (3,4], (4,5] and (5,6]. The “0-only reasoning” you refer to is still only yours, so stop simply trying to attribute it to others.

By the way, have you identified that location or locations on a line that you claim are not covered by points?

 

[doronshadmi]

Let us take for example PI.

PI > than any non-strict number that is defined by the place value method.

For example, 11.0010010000111111011010101000100010000101101000110000100011010011...[base 2] < PI , where 11.0010...[base 2] < PI first 6 scale levels are clearly seen by the following diagram:

The bold horizontal black 1-dim element is PI.

The orange rectangle is unit 1, which is common to any given [base n>1] number system.

The purple objects are the [base 2] number system.

The vertical green line is the exact location of PI on 1-dim space, and it is clearly seen that 11.0010...[base 2] < PI and this fact is not changed upon infinitely many scale levels.

(The black numbers and the thin lines are [base 10] number system, which is different than [base 2] number system, but it is also < PI).

 

[zooterkin]

Let us take for example PI.

It's 'pi' or 'Pi', not 'PI'.

PI > than any non-strict number that is defined by the place value method.

Really? Did you mean to say that?

 

[doronshadmi]

Not for the interval you indicated, a 1 is included in that interval. So while 1 > 0 it is not smaller than, well, 1. The interval should have been (0,1) thus any arbitrary number included in that interval is > 0 and < 1.

Some reading problems?

The considered number is smaller than any arbitrary number of (0,1] AND it is also > 0.

In other words, this considered number is not 1 AND it is not 0.

 

[doronshadmi]

It's 'pi' or 'Pi', not 'PI'.

pi or Pi already used bi the wrong notion that the place value is Pi or pi.

So I use PI as the exact location on 1-dim space, which is > any place value system.

Really? Did you mean to say that?

Yes I do mean to say that!

 

[readme.txt]

http://en.wikipedia.org/wiki/0.999...#Skepticism_in_education

Skepticism in education

Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[35]
Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[36]
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.[37]

Just sayin'

 

[epix]

You still do not get that 0.000...1 is a result of the irreducibility of 1-dim space to 0-dim space.

Doron, you are building a humongous press to crack a walnut. A math reduction is accomplished by two common arithmetic operations: subtraction and division. The applicable difference is:

Subtraction: a - x = 0
Division: a/x = 0

Solve both equations and you find out that your concept of irreducibility crumbles under the hard friction that an eraser can cause onto your 1-dim space, but in the case of division, a trace of that 1-dim space always survives the vicious attack. That's because when you start to erase the word DIVIDE from the end and reduce it up to DIVI, a miracle takes place and DIVI becomes a full word again with the same cardinality reading "DIVIne." Some of us enjoy a divine protection and so does 1-dim space. That's why the solution to

a/x = 0

doesn't exist for a > 0, and you can't therefore reduce 1-dim space to 0-dim space by using division.

The divide/divine protection was used in the attempt to reduce a NUMBER (of people). The reduction came to a halt when NUMBER was reduced to NU and became divided into N:U. Since N=14 and U=21 in the alphabet, N:U = 14:21, so you can use 14:21 to find the practical example of the divine/divide protection:

Then Moses stretched out his hand over the sea, and all that night the LORD drove the sea back with a strong east wind and turned it into dry land. The waters were divided.
Exodus 14:21

See? If it wasn't for His Uttermost Omniscience God the Lord, Ph.D., you would be now sitting in Cairo, Egypt making $0.000...1 an hour.