# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Try explaining it to someguy1. He is still struggling with the solution.

Actually, you are lying, I explained it in great detail to someguy1, for some reason, it is not sinking in for him. Do you know how to do the partition using just a compass and a ruler? I think I asked you this several times but you evaded each time.
You are going to have to provide a link that supports your claim...because the last I heard you reckoned 1/infinity = 0. Also you have been trolling and denigrating this proposed solution since I first introduced it about a fortnight ago.
Any way Rpenner has already indicated why it is invalid...according to current axiomatic usage.

I don't answer to Tach.... When a poster demonstrates a genuine desire for discourse I may, but as yet you Tach, have not demonstrated a genuine desire for discussion...so uhm, how's the weather over there?

You are going to have to provide a link that supports your claim...because the last I heard you reckoned 1/infinity = 0.

Yes, this is basic calculus. It has nothing to do with either my criticism of someguy1's ability to solve the basic geometry problem of trisecting the circle nor with your unfulfilled challenge to solve a simple problem of geometry.

Also you have been trolling and denigrating this proposed solution since I first introduced it about a fortnight ago.

Stop deluding yourself, you have not produced any solution. How do you trisect the circle, Quack. Show us.

Any way Rpenner has already indicated why it is invalid...according to current axiomatic usage.

What is "invalid", Quack?

....because you can't.

A circle dissected equally into three congruent segments all sharing the same central point.

I would hazard a bet that you can't do it using your infamous claim that 1/infinity = 0

Remember we are not after an approximation here and I don't think beginners calculus can cut it Tach..
You gotta dig deeper and look at the meaning of what you offer and not just quote "rote learned methodology".
One of the key differences between a supposed Masters and PHD, I would guess is understanding what exactly you are trying to do.

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Clearly by using this "infinitesimal approach" to zero the central point CAN be shared equally amongst all 3 segments.

I don't think you can divide a single point(considering the center of a circle as a single point) into three equal parts. Because as per definition in Math a point is dimensionless, meaning it does not have any area.

The single center point can be 'either included or excluded' to all the three equal parts of the circle. Area-wise i don't think it will make any difference.

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As I posted sometime last month, you can divide a disc into congruent sections of equal measure and you can partition the points of disc into pointwise disjoint sets of equal cardinality, but it has not been shown how to do both at once.

I don't think you can divide a single point(considering the center of a circle as a single point) into three equal parts. Because as per definition in Math a point is dimensionless, meaning it does not have any area.

Little refinement here. The rational numbers are a set of measure zero, meaning they don't have any area. But you can divide the rational numbers into two nonempty subsets, for example the negative rationals and the nonnegative rationals.

You can't subdivide a mathematical point, but not because it has no area. It's because it's a point. It has no sub-parts. It has no nonempty proper subsets. But in math there are huge sets, infinite sets in fact, that nevertheless have no area.

I don't think you can divide a single point(considering the center of a circle as a single point) into three equal parts. Because as per definition in Math a point is dimensionless, meaning it does not have any area.

The single center point can be 'either included or excluded' to all the three equal parts of the circle. Area-wise i don't think it will make any difference.
posted by someguy1
You can't subdivide a mathematical point, but not because it has no area. It's because it's a point. It has no sub-parts. It has no nonempty proper subsets. But in math there are huge sets, infinite sets in fact, that nevertheless have no area.
...well do the thought experiment yourself!
Reduce a ball so that it is the smallest it can be with out being zero.
and then ask yourself some questions about it's 3 dimensional vs zero dimensional reality

Then ask yourself : "Does currently held math use of a zero dimensional point still hold as totally valid?"

...well do the thought experiment yourself!
Axioms are needed before a thought experiment is more than mental masturbation.
Reduce a ball so that it is the smallest it can be with out being zero.
Not possible in axiom systems where length is parametrized by the rational numbers, the real numbers, the hyperreal numbers or the surreal numbers for if x is claimed to be the smallest positive number, x/2 must be a legal number, positive and smaller yet.
and then ask yourself some questions about it's 3 dimensional vs zero dimensional reality
If the radius is positive, three degrees of freedom are requires to unambiguously address a point on its surface in all the above number systems.
Then ask yourself : "Does currently held math use of a zero dimensional point still hold totally valid?"
That isn't the question such an exploration answers. Your exploration answers some the ways a sphere of finite radius differs from a point. You have not begun to address the topic of a number system other than the integers that has a smallest positive element. You haven't shown that such a number system exists and is sensibly compatible with Euclidean geometry which says if the length is not zero then half the length is smaller than the whole.

Axioms are needed before a thought experiment is more than mental masturbation.
Not possible in axiom systems where length is parametrized by the rational numbers, the real numbers, the hyperreal numbers or the surreal numbers for if x is claimed to be the smallest positive number, x/2 must be a legal number, positive and smaller yet.
If the radius is positive, three degrees of freedom are requires to unambiguously address a point on its surface in all the above number systems.
That isn't the question such an exploration answers. Your exploration answers some the ways a sphere of finite radius differs from a point. You have not begun to address the topic of a number system other than the integers that has a smallest positive element. You haven't shown that such a number system exists and is sensibly compatible with Euclidean geometry which says if the length is not zero then half the length is smaller than the whole.
it is not for me to provide anything!
the thought experiment stands on it's own merit. take it or leave it.
to believe in something with out rigorously testing it is also a form of "mental masturbation" .
I don't have to relate the thought experiment to potentially failed systems, however I am confident that it would be in your own interests to.
If all you can do is say that the thought experiment is irrelevant and a act of mental masturbation due to axiomatic calls to authority then that is what you have to live with as a mathematician. Not me.
Reduce a ball so that it is the smallest it can be with out being zero.
Then do a mathematical assessment using what ever tools you have.
You don't have to .. you know... you can ignore the notion all together...but remember, it is your belief system that in concern .. not mine.

You haven't shown that such a number system exists and is sensibly compatible with Euclidean geometry which says if the length is not zero then half the length is smaller than the whole.
as I stated before this is not my province of concern. it is yours. [if the thought experiment upsets the mathematical fantasy... then so be it... I offer no apology]
so expand zero to it's greatest area or volume before gaining value and the same result occurs.
The infinitesimal is not a number. It is more a boundary condition between nothing and something. In saying so a solution is available...

Maybe it's worth offering the notion that one of the primary distinction between Math/science and Philosophy is that in science we work from something to nothing [proving a value using zero as a test] where as in philosophy we tend to go from nothing [ nihilo ] to something and presume nothing using "something" as a test...

Zero Represents Nothing/Non-occupied

0 is a non-counting number i.e. it has no value to pass on, except as a place holder for the first column.

Zero point in mathematics is dimensionless i.e. not only has no XY or Z values it is incosiderate of such dimensions.

A 2D( ex XY ) mathematically minimal point is a triangle, that considers and inside area seperate from and outside infinite area.

A 3D( XYZ ) mathematically minininalistic point is a tetrahedron, and again with no specified XYZ values, however, ZY and x are considered in definning an inside volume from an outside infinity value by those aspects of crossings/vertexes and edges/lines-of-relationship.

1.0 does not = 0.999...

Those alledged proofs given by alledge educated people, is no more than extremmely complicated mental masturbation illusions design to fufill their need to make sense of irrational infinities by and illusion that makes them rational finites. NOT. See renormalization, as best as I recall.

When they can actually use some kind of a simple explanation that is rational, logical and common sensical, then I will begin to give their illusions a more indepth consideration. They have no such simple, rational, logica and common sense explanations, and that is why they offer none, for the 80% of the less educated common peoples.

Bring those out-there, higher mathmatics back to Earth for the common people, and then you will have some thing significantly valid to crow about. imho

r6

Those alledged proofs given by alledge educated people, is no more than extremmely complicated mental masturbation illusions design to fufill their need to make sense of irrational infinities by and illusion that makes them rational finites.

As Woody Allen said about masturbation ... at least it's sex with somebody I love.

Infinite vs Finite #3, #7 and #9--Getting To Crux(?)......

Then objectors to 1 =0.99999 need not only to give extraordinary proof for their objection but also need to explain why one of the other infinite number of successes of this procedure fails to produce the rational fraction that is exactly equal to the infinite Repeating Decimal , RD.

Crux = .."A puzzling or apparently insoluble problem"...

Sorry I missed your above post Billy T. as once I began dividing #1 and other numbers by #'s 1-10 I see the calculator gives infinite values. Duhh!

I explained this in my most recent post with more simplicity-- see below --- involved and the obvious connundrum(?) weirdity/crux as soon as I see the calculator is giving infinite resulatant when dividing a finite number. Duhh, were immediately drawn into infinite values so all bets are off, i.e. via a calculator, we cannot use the numbers 9 7 and 3 to get a finite value. Simple introduction to understanding why there is two sides to this issue. It appears to me that calculators cannot handle some numerical operations with finite resultants and others it can.

Is this because of some cosmic thingie with some numbers? First I regive the simple introduction to the finite vs infinite connundrum(?) via a MS calculator.

1/10 = finite 0.1

1 / 9 = infinite 0.111...

1/8 = finite 0.125

1/7 = infinite 0.142857...
..here I would point out a significance differrence regarding #7 and #3( see bottom of page )...

1/6 = infinite(?) 0.666667...
...this one rounds off on the calculator. Why?....
...I believe #6 offers a clue since it is the only one the rounds off the seemingly infinite value....

1/5 = finite 0.2

1/4 = finite 0.25

1/3 = infinite 0.333...

1/2 = finite 0.5

Going back to geometry--- line segments + angle ---we find that it is only with the heptagon that we begin to get an irrrational(?) infinite value for the 7 internal angles, if not the also the external angles.

These are the kinds of cosmically numerical pattern differrentiations that I tend to look for with my less educated-- ergo more simple ---set of abilities.

A tri(3)angle = 3, finite, 60 degree parts or internal angles.

1/3 = 0.333....

So here is simple comparison that may be more complex than you 1D line segment explantions, but 2D polygons with their associated angles, may actually be more visually accessable/grasp-able to larger segment of humanity.

Nona(9)gon has nine, finite, 140 degree internal angles and nine, 40 degree radial angles.

This does not solve or renormalize the infinite vs finite crux but may does assist in seeing from another angle or perspective of viewpoint the crux of finite vs infinite that involves numbers, in a relatively simple way.

r6

Crux = .."A puzzling or apparently insoluble problem"...
Sorry I missed your above post Billy T. as once I began dividing #1 and other numbers by #'s 1-10 I see the calculator gives infinite values. Duhh!

I explained this in my most recent post with more simplicity-- see below --- involved and the obvious connundrum(?) weirdity/crux as soon as I see the calculator is giving infinite resulatant when dividing a finite number. Duhh, were immediately drawn into infinite values so all bets are off, i.e. via a calculator, we cannot use the numbers 9 7 and 3 to get a finite value. Simple introduction to understanding why there is two sides to this issue. It appears to me that calculators cannot handle some numerical operations with finite resultants and others it can.

Is this because of some cosmic thingie with some numbers? First I regive the simple introduction to the finite vs infinite connundrum(?) via a MS calculator.

1/10 = finite 0.1

1 / 9 = infinite 0.111...

1/8 = finite 0.125

1/7 = infinite 0.142857...
..here I would point out a significance differrence regarding #7 and #3( see bottom of page )...

1/6 = infinite(?) 0.666667...
...this one rounds off on the calculator. Why?....
...I believe #6 offers a clue since it is the only one the rounds off the seemingly infinite value....

1/5 = finite 0.2

1/4 = finite 0.25

1/3 = infinite 0.333...

1/2 = finite 0.5

Going back to geometry--- line segments + angle ---we find that it is only with the heptagon that we begin to get an irrrational(?) infinite value for the 7 internal angles, if not the also the external angles.

These are the kinds of cosmically numerical pattern differrentiations that I tend to look for with my less educated-- ergo more simple ---set of abilities.

A tri(3)angle = 3, finite, 60 degree parts or internal angles.

1/3 = 0.333....

So here is simple comparison that may be more complex than you 1D line segment explantions, but 2D polygons with their associated angles, may actually be more visually accessable/grasp-able to larger segment of humanity.

Nona(9)gon has nine, finite, 140 degree internal angles and nine, 40 degree radial angles.

This does not solve or renormalize the infinite vs finite crux but may does assist in seeing from another angle or perspective of viewpoint the crux of finite vs infinite that involves numbers, in a relatively simple way.

r6
You are ignoring the proof given that 1/1 =0.999... several times which does not used any of the things you discus in this post.
{from post 1492 in other thread about 1/1= 0.999.}... PS: Not only do I avoid use of all of general math's algorithms for operations including any limiting process, but I have no need to try to defined "infinity" in my proof! The closest to that I come is to use a line with no ends, called the number line, and do all movement on it by unity steps for some starting point on the number line. (My additions and subtractions are not general algorithms, but deal only with integer steps and the subtractions I use in the proof ALWAYS have positive results like 99, or 999 or 9999 etc. but not 999...)

Billty T Bite Me Becaus of Word Crux?

You are ignoring the proof given that 1/1 =0.999... several times which does not used any of the things you discus in this post.

Not. And your saying it is not a proof.

If and when you have a rational, logical and common sense, relatively simple explanation please share with me. I've not seen that from you and not from you directed at me specifically to explain.

There is a crux/conundrum(?)--- that you deny ---of this infinite vs finite and it apears to me to be related to some specific numbers( 9, 7 and 3 ) and/or how a calculator functions or as perhaps even a numerical base system. I dunno.

I have attempted to show geometry that also involves these numbers to better grasp the crux of this infinite vs finite issue and all you do is try to is bite me. Ouch!

Like, dont use any words Billy T don't like ex crux, or he may bite you.

If and when you want to address me comments as stated, insteading of biting me for not good reason, please come back and talk to me, Otherwise thx for no real information content in your reply to me, regarding this crux of this infinite( 0.999...) vs finite( 1 ) issue.

r6

... If and when you have a rational, logical and common sense, relatively simple explanation please share with me. I've not seen that from you ...r6
I and others, have PROVEN 0.9999... = 1/1 Perhaps mine is the most simple as does not use infinity or ANY of the general math operations as I define or derive every step used.
but the essence of it is:
... I first illustrate, several of the infinite numbers of examples, of true statements concerning terminating or infinitely Repeating Decimals. (And then a procedure for finding the Rational Fraction that equals to ANY given Repeating Decimal, RD):

1/3 =0.333333.... and 1/1 = 0.99999.... are rational fraction numbers with a "repeat length" of 1 in their equivalent decimal versions.

12/99 = 0.12121212... and 19/99 = 0.1919191919... and 34/99 = 0.343434... are rational numbers with a "repeat length" of 2 in their equivalent decimal versions.

In general, any integer less than 99 divided by (and not a factor of) 99 will produce a decimal repeating with length 2. Some of the factors will too. For example 3/99 = 03/99 = 0.03030303... does; but not 11 or 33. I.e. 11/99 =0.111... and 33/99 = 0.3333333333... In ALL cases with 99 as the denominator, the decimal repeats two numbers in blocks of two) is still true, as well as in "blocks" of one for some cases as bold type helps you see.

Likewise any integer less than 999 divided by 999 will be a decimal fraction with repeat length not more than 3 and always will repeat in blocks of 3, but for somecases, like 333 /999 = 1/3 the least long repeat block is less than 3. Check with your calculator if you like. Etc. For example, 678 /999 = 0.678,678,678, .... and that is slightly larger than 678 /1000, which equals 0.678 and should given you a hint of the proof to come.

However, any integer divided by a factor of the number base (1, 2 & 5 for base 10) or any product of these factors (like 4, 16, 2^n, 5 or 5^m, {2^n x 5^m} ) will terminate, not repeat. For example 17 /(1x4x5) = 0.85

The proof I and others have given that 1 = 9/9 = 0.99999.... is just particular case of the fact ALL rational fractions like a/b or a/9 (both a & b being integers and a < b) are equal to an infinitely repeating decimal (if they are not a finite decimal when b is a factor or product of factors of the base).

For example, the general proof of this goes like:
Rational Decimal, RD = 0.abcdefg abcdefg abcdefg .... Where each letter is one from the set (0,1,2...8,9) and the spaces are just to make it easier to see the repeat length in this case is 7.
Now for this repeat length 7 case, moving the decimal point 7 spaces to the right is not a multiply operation, but a notional change with the same effect on meaning as multiplying RD by 10,000,000. I. e. 10,000,000 RD = a,bcd,efg . abcdefg abcdefg ... Is a 2nd equation with comas for easy reading the integer part.

Now, after noting (10,000,000 - 1) = 9,999,999 and subtracting the first equation from the second, we have:
a,bcd,efg = 9,999,999 x RD. Note 9,999,999 certainly is not zero so we can divide by it to get: The Rational Fraction, RF = RD = a,bcd,efg / 9,999,999 I. e. the rational fraction of two integers exactly equal to the infinitely long repeating (with repeat length =7 in this case) decimal, RD.

Now lets become less general and consider just one of the repeat length = 7 cases. I. e. have a=b=c=d=e=f=g = 9 and recall RD was DEFINED as 0.abcdefg...so is now in this less general RD = 0.9,999,999,... and from green part of line above, The Rational Fraction which equals RD is 9,999,999 / 9,999,999, which reduces to the fraction 1/1 which is unity as the numerator is identical with the non-zero denominator. I.e. the least numerator rational fraction equal to 0.999,999... is 1/1.

By exactly the same procedure the RD = 0.123,123,123,.... is a case with repeat length of 3, can be shown be equal to the RF = 41/333 (as your calculator will show, as best as it can, if used to divide 41 by 333.
To prove this one moves the decimal point of the RD 3 places to the right get: RD' which is 1000 times larger than RD. I.e. RD' = 123.123123123... as the 2nd equation and then subtract the first from it to get after division by (1000 -1): RD = 123 /999, which reduces to 41/333 as the RF = 0.123123123...

The objectors to 1 =0.99999 need not only to give extraordinary proof for their objection but also need to explain why ONLY ONE of the other infinite number of successes of this procedure fails to produce the rational fraction that is exactly equal to the infinite Repeating Decimal , RD.
The objectors tend to fall into two classes: idiots and those not fully understanding the meaning of the "Bases, places and decimal point" notational system, so I will explain that below (with numbered logical steps from the starting definition for easy reference if you think there is any error in the logic or resulting proof than 1/1 = 0.999...):
-------------
A shorter summary of the logical development first:

I started with definition of "line segment" quickly specialized to one between 0 and 1, to establish a scale. Then joined two of these unit length line segments to make line segment of length twice as long and introduced the name "2" both as the name of the not-zero, new end of the longer new line segment and as the sum of two integers. I said 1 + 1 = 2 and then that 2 + 1 = 3. etc. up to 8 + 1 = 9

I did thus define addition of integers, and noted there was no largest one N as N+1 was larger. I also noted we have no symbol for what 9 +1 is equal too.* These two facts forced the introduction of a line (the "number line") with no ends to that 1D item list and the introduction of the place and decimal systems (with B as base) so I could tell that 9 + 1 = 10 in the system in common use by people, and begin to discuss the meaning of the "Base, place, and decimal point" notation systems.

* The hexadecimal (base =16) notation system does have a symbol for 9+1. It is "a" I.e. in that base system, 9+1 = a.

Note my proof, now well founded on logical extension of the defined "line segment" (with length 1), does NOT use ANY of the normal GENERAL math operation (+ , - , x or /). It needs no algorithms for these operation, except adding and subtracting lengths on the number line. I do write the notation commonly used for fractions, but even there only with integers for a & b in a/b but never describe how the indicated division would be done as the proof has no need of that.

Further more, even when in the proof I do use my limited well defined "subtraction" it is only using powers of 10 (like 1000, when the repeat length of the RD is 3) and one step back of magnitude unity towards 0 on the number line. That is the proof does need "integer subtraction" or reduction by 1. For example fact that 1000 - 1 = 999 is a location on the number line one unit closer to 0 than 1000 is. It does also use the algebraic fact that (x+a) - (y+a) = x-y. I.e. two points on the number line that are both greater than x or y by the same amount (a in this case), have length difference or separation of x-y where x > y as I never use or need negative numbers in the proof.

PS: Not only do I avoid use of all of general math's algorithms for operations including any limiting process, but I have no need to try to defined "infinity" in my proof! The closest to that I come is to use a line with no ends, called the number line, and do all movement on it by unity steps for some starting point on the number line. (My additions and subtractions are not general algorithms, but deal only with integer steps and the subtractions I use in the proof ALWAYS have positive results like 99, or 999 or 9999 etc. but not 999...)

Now the full text of the logical development:

(1) A “line segment” is a math defined term, which is 1 dimensional (1D etc here after) with two ends. They can be called “a” & “b” or “0” & “1” for more convenience as then one can say “length' of the line segment is 1.

(2) If a second equal to the 0 to 1 line segment is joined to the end called 1, the total length is twice the length of the first alone. This can conceptually be done with out end as there is no “largest number,” N. (N+1 is larger.)
To cope with this fact and yet retain the idea that any well defined number can correspond to one and only one point on a “line,” we define a line to be 1D and like a line segment, but with no ends. Zero has a special roll to play. By common convention, all the numbers of / on the line to the left of 0 are “negative” and to the right are positive, but I don't need to say more about negative numbers, so won't.

(3) All number system notations use a “base” I'll refer to as “B.” The most common in public use has ten different value symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9 and can be used to describe the length of the longer line segments produced by adding, end to end, more of the 0 to 1 line segments, but then two ends of line segments cease to exit. The pair of line segment of point (2) joined “end to end” have ends called, 0 & 2 and has length two, etc. as “2” = 1+1; 3 = 2 +1; etc. but when we get to 9 + 1 we lack a symbol to call it (except in higher than base ten systems, such as the b =16 system used in many computers. Then 9+1 = a; and a+1 = b etc. up to d + 1 = e, when this “hexadecimal” bases system also runs out of additional symbols. In general the number of symbols, #, is same as the base. # = B.

(4) Man used his fingers to count so and that is why B = ten is so commonly used. (Mental math would be easier if humans had twelve fingers, as division by any factor of the base is relatively easy and twelve has many more factors than ten does. Even dividing a base twelve number by 8 is easy as divide by 2 and then by 4, both factors of the base.) But I digress. Fact that transistors have two well defined states (conducting or not) has made B=2 a very commonly used number system for electronic machine made calculation.

(5) All base systems have the same meaning for 0 & 1 but that is all the symbols the “binary system" has, “places with values” in the notational system were invented. I'll count up to base ten 8 in binary to illustrate:
0; 1; 10; 11; 100; 101; 110; 111; 1000. The value of each place is a power of base ten 's 2. i.e. the green ones or even the green 0 last binary number I listed (the one with value 8 in the B =ten system) all are in a “notation place” one to the left of the first place and so have value, if not occupied by 0, of base ten's 2, which is 2^1. likewise the red 1 in the second place to the left of the first place has value of base ten's 2 squared or 2^2 = 4 of base ten. Etc. I. e. a 1 in the fifth place to left of the first place, has value of base ten's 2 raised to the fifth power. All base notation systems work this same way. The value of places, if not occupied by 0, is B^n where in is an integer telling how many places to the left of the first place the space is, if occupied. For example in the hexadecimal base system 1b is the base ten number 16 + 2 = 18 but I am already using the B = ten system when writing this “18.” i.e. the 8 is in what I have been calling the “first place” and the 1 of the 18 is in the first place to the left of the “first place.” So that one, in any base system has value = B^1 and the next to left of it has value = B^2, etc.

(6) We often mark with a decimal point where the “first place is. For example the hexadecimal number 1b is more clearly written as 1b. And the 1 as stated early is worth B^1 = 16^1 = 16 in the base ten system. The decimal point is essential if we want to write numbers less than 1 as they are written to the right of the decimal point. In the hexadecimal system base ten fraction 1/16 is written 0.1 or is not worried the reader will fail to see the decimal point, just as .1 and a value sixteen times smaller is written as 0.01 etc.

(7) Now with this general background, lets speak mainly of the base ten system:
Any of the ten symbol can be in the first place to the right or left of the decimal point. If for example we have 12. than mean twelve. 1.2 means 1 + 1/5 and 012 means 1/10 + 2/100 and this is BUILT INTO THE MEANING OF THE NOTATION SYSTEM.

It is true that if I multiply 1.2 by 10, I get 12, but exactly that same effect can be achieved by notational change ONLY. Just move the decimal point found in 1.2 to the other side of the 2. I. e. write 12.
Likewise 1.23456789 can be made ten times larger by move of decimal point one space to the right to get: 12.3456789 or a ten thousand (which is 10^4) times larger by move of decimal point 4 spaces to the right. I. e. 12345.6789 is ten times larger than 1.23456789 is. This is NOT an operation of multiplication, which I have not even defined any general algorithm for, but BUILT INTO THE MEANING OF THE base, place & point NOTATION SYSTEM.

For example as 16x16 = 256 then if I wanted to make the hexadecimal number 1b. larger by 256 times I just write it as 1b00. That hexadecimal number 1b. Had value in base 10 of 18 an if in base 10 I want the number which is 256 times larger I must multiply (or use my calculator) to get 4608.
Likewise I can increase the hexadecimal number 1b by 16^4 times easily. It is 1b0000 but to do that to base 10's 18 I will use my calculator to get: 1179648, but this only hints at the fact large values sre shorter to write in larger base number systems. Consider hexadecimal number ed. In decimal notation that is 16x15 + 13 = 253. I. e. take three places, not just two to write and even more to write in a binary system.

As I noted early: Challenge to those who don't think 1/1 = 1 = 0.99999 exactly, tell what is wrong with the general procedure I gave for finding the rational fraction equal to any repeating decimal. Or the logical foundation I describe for it above.

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Crux... 1/7 = infinite 0.142857...
..here I would point out a significance differrence regarding #7 and #3( see bottom of page )...

1/6 = infinite(?) 0.666667...
...this one rounds off on the calculator. Why?....
...I believe #6 offers a clue since it is the only one the rounds off the seemingly infinite value. ... r6
No 1/6 is not different. Your calculator is following the standard rule when it lack a display place for the next digit.
If that next digit it has no place to display is 5, 6, 7, 8 or 9 it rounds up. If it is some other "next" digit that cannot be display, it is just dropped.

In the case of 1/7 the next digit the calculator that you used could not display was 1, 2, or 4. A different calculator, with more display space may have the 8, 5 or 7 as the next digit it could not disply and rounded up as the display of 1/6 always will as the next not "displayable digit" is always 6.

Nothing strange here to discuss. (no crux).

1/6 = 0.1666... = 2/3 - 1/2 = 1/15 + 1/10 = 1/150 + 4/25 = 1/1500 + 83/500 = ...
1/6 rounds to 0.1666667 on an 8-digit calculator but 0.1666667 = 1/6 + 1/30000000 because finite precision decimal math cannot exactly represent all rational numbers between 0 and 1.

To convert the repeating decimal 0.1666... to a rational number, note that it has at least 1 non-repeating decimal digit to the right of the decimal point, all this n, and the period is 1, call this m. So the fraction must be an exact integer divided by $$10^{n+m}-10^m$$. Multiplying 0.1666... × (100 − 10) = 16.666... − 1.666... = 15. So 0.1666... = 15/90 = 1/6.

If you limit yourself to just calculators, you will believe lies that the calculator tells you. Lies like $$\color{red} \pi = \sqrt{\frac{30028}{22481}} e$$
$$\begin{eqnarray} \pi & = & 3.1415926535897932... \\ \sqrt{\frac{30028}{22481}} e & = & 3.1415926535742161... \end{eqnarray}$$

No Banana Billy T.--Sorry

As I noted early: Challenge to those who don't think 1/1 = 1 = 0.99999 exactly, tell what is wrong with the general procedure I gave for finding the rational fraction equal to any repeating decimal. Or the logical foundation I describe for it above.

Again, you do not address my given comments as stated and you certainly do not offer a rationally, logical, common sense and relatively simple explanation. That is all I've asked for, for the most part

Origin has offerred a simplest explanation via a few formula and very little actual step by step guides us through it.

Somebody else sent me to Wiki link involving "mapping" and unheard of terms like subjerctive(SP), injerctive(SP) etc..as tho even 5% humanity would even begin to have any idea what there talking about much less someone who has at least had 9th grade math and flunked.

Now you and repeat what I've actually skimmed through before addressing you specifically. Very complex you think line segments and the numerical base system are key to understanding how/why finite 1.0 = infinite 0.999.....

I don't see/understand any such proof that you claim nor anyone elses.

What I do get, is that the numbers 9, 7, 6 and 3 do infinite representation/process when used to divided some other numbers if not all numbers via MS calculator.

What I do get, is that the numbers 8, 5, 4 and 2 do NOT do this infinite representatio/process on some numbers, if not other numbers.

You use line segments and address numercial base system.

I use line segment and angles to address the issue as a way to clarify and hopefully resolve the crux of this controversial issue. It is partly controversial because no one has definitive shown in a rational, logical, common sense and relatively simple way that finite 1.0 = infinite 0.999...

If you think can respond to specifically as my comments as stated then please do so and please understand, that just quoting me is not really address my comments specifically as stated. That is what Origin did i.e. he did not address my comments specifically as stated, rather he just quoted me. Waste of bandwidth, that so many trolls land others like to complain about.

Is this because of some cosmic thingie with some numbers? Or is something inherent to calculators? Or is a combination of these?

1/10 = finite 0.1

1 / 9 = infinite 0.111...

1/8 = finite 0.125

1/7 = infinite 0.142857...
..here I would point out a significance differrence regarding #7 and #3( see bottom of page )...

1/6 = infinite(?) 0.666667...

1/5 = finite 0.2

1/4 = finite 0.25

1/3 = infinite 0.333...

1/2 = finite 0.5

Going back to geometry--- line segments + angle ---we find that it is only with the heptagon that we begin to get an irrrational(?) infinite value for the 7 internal angles, if not the also the external angles.

These are the kinds of cosmically numerical pattern differrentiations that I tend to look for with my less educated-- ergo more simple ---set of abilities.

A tri(3)angle = 3, finite, 60 degree parts or internal angles.

1/3 = 0.333....

So here is simple comparison that may be more complex than just a 1D line segment explanation, but 2D polygons with their associated angles, may actually be more visually accessable/grasp-able to larger segment of humanity.

Nona(9)gon has nine, finite, 140 degree internal angles and nine, 40 degree radial angles.

This does not solve or renormalize the infinite vs finite crux but may does assist in seeing from another angle or perspective of viewpoint the crux of finite vs infinite that involves numbers, in a relatively simple way.

It may be significan that the nonagon, which has rational internal angles, is sort of related to 1/4 = 0.25 because there also is a 4, and rational resultant. I dunno if geometry can be used to help clarify or resolve the connundrum?) crux of this infinite vs finite issue.

r6