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

there are TWICE AS MANY points on a diameter as there are on the radius!
How many points are there on the radius?
Can you show, rather than just insist, that there are twice as many points as this on the diameter?

How many points are there on the radius?
Can you show, rather than just insist, that there are twice as many points as this on the diameter?

You can divide a line segment into 2 parts of .5.
You can divide a line segment into 5 parts of .2.
You can divide a line segment into 10 parts of .1.
You can divide a line segment into 100 parts of .01.
You can divide a line segment into 1,000 parts of .001.
You can divide a line segment into 1,000,000,000,000,000,000,000,000 parts of .000000000000000000000001.

You see the pattern? The more pieces you divide the line into the smaller the pieces are. Is there a limit to the amount of pieces that you can divide a line into? NO! You can always divide the line into more, smaller, pieces. So we say that there are an infinite quantity of infinitely small pieces, always in exact proportions!

So the answer to your question is that there are an infinite quantity of infinitely small points on the radius, but there are TWICE AS MANY infinitely small points on the diameter as there are on the radius!!!

What does (1-0.999...) equal?

(1 - 0.999... ) = (1.000... - 0.999... ) = 0.000...1 . This value is something non-zero, infinitesimal which is tending towards zero but not equal to zero.

If no third point exists (can be found) between two points, then the two points cannot be consecutive, they must be the same point in that case.

What is your idea of 'consecutive points'?

You're confusing the integers with the reals: there is no integer between two consecutive integers and a finite number between any two distinct integers, the reals have an infinite number of points between any two points which are distinct.

Here i am not talking about integers but real numbers. I am talking about two consecutive real numbers and their corresponding two points in the axis/line of real numbers.

(1 - 0.999... ) = (1.000... - 0.999... ) = 0.000...1 . This value is something non-zero, infinitesimal which is tending towards zero but not equal to zero.

What is your concept of 0.000...1? How do you pronounce 0.000...1? Like, I pronounce 0.01 as "One Hundredth." One hundredth to me means that if I divide a line segment of 1 meter in length into 100 parts, .01 meters means exactly 1 of those parts. Exactly one of those parts. Exactly one of those parts!!!

How many parts does the meter long line segment have to be divided into in order for each part to equal 0.000...1?

What is your concept of 0.000...1?

'0.000...1' means between the '.'(point) and '1' there are infinity numbers of '0's(zeros).

How do you pronounce 0.000...1? Like, I pronounce 0.01 as "1 Hundredth." One hundredth to me means that if I divide a line segment of 1 meter in length into 100 parts, .01 meters means exactly 1 of those parts. Exactly one of those parts. Exactly one of those parts!!!

This is an 'infinitesimal', which has a limiting value towards '0'(zero). Its value is tending towards zero but not equal to zero.

How many parts does the meter long line segment have to be divided into in order for the parts to equal 0.000...1?

You have to put infinite numbers of '0' before 1 after the point('.'). So, it is similar to dividing the line segment by infinity. You can consider it as 1/(1000...infinity times '0') .

'0.000...1' means between the '.'(point) and '1' there are infinity numbers of '0's(zeros).

So 0.999...+0.000...1=1.0? So it's really BS that .999...=1 because it's really 0.999...+0.000...1 that equals 1? Will you please speak out of just 1 side of your neck? Thanks.

This is an 'infinitesimal', which has a limiting value towards '0'(zero). Its value is tending towards zero but not equal to zero.

So adding that to .999... doesn't quite bring the grand total to 1.0? Is that what you are saying? Are you saying that 0.999...+0.000...1 is tending towards 1 but not equal to 1? Speak more clearly so that I too can understand your intended meaning instead of receiving what I perceive to be a smoke screen. I'm choking...help. (rolls eyes)

You have to put infinite numbers of '0' before 1 after the point('.'). So, it is similar to dividing the line segment by infinity. You can consider it as 1/(1000...infinity times '0') .

So after an infinite amount of 0's there is a one at the end? I take it there are zeros after that 1, or is it not allowed to put another number after the 1 in 0.000...1???

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So the answer to your question is that there are an infinite quantity of infinitely small points on the radius, but there are TWICE AS MANY infinitely small points on the diameter as there are on the radius!!!
What's 2 x infinity?
What you seem to have missed is that you need to count all the (infinite) points to show that the diameter contains twice as many as the radius. Then you will find that the radius and the diameter have an uncountable number of points, so they have the same number of points.

What's 2 x infinity?
What you seem to have missed is that you need to count all the (infinite) points to show that the diameter contains twice as many as the radius. Then you will find that the radius and the diameter have an uncountable number of points, so they have the same number of points.

Bingo!!! Well put!

What's 2 x infinity?

What you seem to have missed is that you need to count all the (infinite) points to show that the diameter contains twice as many as the radius. Then you will find that the radius and the diameter have an uncountable number of points, so they have the same number of points.

What you seem to have missed is that all the red points are matched to a blue point, but only half of the blue points are matched to a red point. Now run along like your buddy, and be smart enough to realize when you've been bamboozled into thinking BS!

What's 2 x infinity?
What you seem to have missed is that you need to count all the (infinite) points to show that the diameter contains twice as many as the radius. Then you will find that the radius and the diameter have an uncountable number of points, so they have the same number of points.

Quite a weak mathematical argument. The real numbers and the power set of the real numbers are two uncountable sets, but there is no bijection between them.

Hi Motor Daddy, someguy1, handsa, arfa brane. Been reading you all. Can't stay long. Will post this for your consideration in the context of your recent side-discussion/exchanges which led to arfa's question...

What's 2 x infinity?

I would like to point out something which could assist in prevention of more cross-purpose argument among you on certain aspect of 'infinities' highlighted by Motor Daddy's observations and arfa's question in response as above...

Consider: There are effectively THREE separate 'infinities' involved in the 'radius' set, 'diameter' set and any 'correspondence' set' between the radius and diameter sets respective 'points' content.

Unless you realize this, then your arguments/logics will be 'mixed' and so confuse which infinity is which when coming to conclusions about any one or more or all of them. The mixing is among the following THREE infinities...

1) The radius 'line' is one entity containing its particular 'infinity of points'.

2) The diameter 'line' is another entity containing its particular 'infinity of points'.

3) And THEN we have one MORE entity containing NOT 'points' but 'correspondences' drawn between the 'points content' of the first two infinities.

In effect what is happening when you start arguing about the three above infinities is that in 1) and 2) you have TWO fundamentally physical sets, each with its OWN CONTEXTUAL CONDITIONS and QUALIFIERS informing what exactly their RESPECTIVE 'infinity' consists of; whereas in 3) you have an ARBITRARY CONSTRUCT based on COMPARISON between any one point in 1) and any other point in 2).

I say 3) is an ARBITRARY infinity because we cannot do the comparison IN REVERSE ORDER like we could do with a finite set, starting from two opposite ends of a finite string to construct the third 'correspondence' set.

So, before rushing ahead on the same arguments, all 'sides' should take a moment and consider the implications of the above observations for your respective 'takes' on infinities and their nature/generation/existence and their axiomatic/reality treatment according to context, fundamentality and arbitrariness etc.

Sorry, no more time. Back to read-only mode and log out! Bye all.

PS: Anyhow, please do rethink for yourself all that you have been 'trained/told' to do/think with/about 'infinities', especially as I will soon be publishing my book and in it will be a special section 'demolishing' the Hilbert Hotel arguments/explanations (and other related 'devices' which do not stand up against closer scrutiny in the above vein that I have given only a hint of here). I can't say any more at this juncture lest I let all the cats out of the bag prematurely! Cheers; and enjoy your further discussions of all those outstanding subtleties which have been missed all this time. Good luck; and DO PLEASE have a safe Winter/Summer break, everyone!

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'0.000...1' means between the '.'(point) and '1' there are infinity numbers of '0's(zeros).

There is no such thing as .000...1. What decimal notation means is that you are assigning a numeral to each of the decimal positions 1, 2, 3, 4, ... So it makes no sense to say there's a "1 at the end" because there is no end. Your notation is not defined.

This is an 'infinitesimal', which has a limiting value towards '0'(zero). Its value is tending towards zero but not equal to zero.

There are no infinitesimals in the real numbers. What you say makes no sense. Does the value if 3 "tend towards" something else? A number has a value, it doesn't "tend towards" anything.

Even in Nonstandard Analysis, where they have a way of making infinitesimals legit, it's still a theorem that .999... = 1.

You have to put infinite numbers of '0' before 1 after the point('.'). So, it is similar to dividing the line segment by infinity. You can consider it as 1/(1000...infinity times '0') .

It's totally false to say that you can "consider" .000...1 to be any such thing, or anything at all. The notation has no standard definition, and you haven't provided one.

someguy said:
Quite a weak mathematical argument. The real numbers and the power set of the real numbers are two uncountable sets, but there is no bijection between them.
But probably a lot stronger than Motor Daddy's.

Try explaining cardinality, or what you mean by measure, of the intervals (0,1), and (0,2). See if he gets the idea of set union, say.

So 0.999...+0.000...1=1.0?

YES.

So it's really BS that .999...=1 because it's really 0.999...+0.000...1 that equals 1? Will you please speak out of just 1 side of your neck? Thanks.

You see that the number "0.999..." is ever increasing towards '1'; whereas the number "0.000...1" is ever decreasing towards '0'.

So adding that to .999... doesn't quite bring the grand total to 1.0?

It will.

Is that what you are saying? Are you saying that 0.999...+0.000...1 is tending towards 1 but not equal to 1? Speak more clearly so that I too can understand your intended meaning instead of receiving what I perceive to be a smoke screen. I'm choking...help. (rolls eyes)

0.999... + 0.000...1 = 1.0; this is always true. I am saying that 'o.999...' is tending towards 1 but not equal to 1. Similarly, '0.000...1' is tending towards 0 but not equal to 0.

So after an infinite amount of 0's there is a one at the end? I take it there are zeros after that 1, or is it not allowed to put another number after the 1 in 0.000...1???

Adding 0's after 1 in a decimal number does not change its value. If you put another number after 1 other than 0 but keep on adding 0's before 1, this number will still tend towards 0 but not equal to 0.

There is no such thing as .000...1. What decimal notation means is that you are assigning a numeral to each of the decimal positions 1, 2, 3, 4, ... So it makes no sense to say there's a "1 at the end" because there is no end. Your notation is not defined.

Consider three numbers x, y and z; such that z = 0.999...( infinite times 9). x = 1000...(infinite times 0). The number y is such that x*y = 1. So, y = 1/x. Now, how you will denote y? I have only tried to denote y as '0.000...1'.

There are no infinitesimals in the real numbers. What you say makes no sense. Does the value if 3 "tend towards" something else? A number has a value, it doesn't "tend towards" anything.

If you consider the axis of real numbers, it will become a continuous straight line, where all the points correspond to some real number. An infinitesimal is something non-zero. Do you think that the infinitesimal will not correspond to any point in this axis of real numbers?

Even in Nonstandard Analysis, where they have a way of making infinitesimals legit, it's still a theorem that .999... = 1.
May be it is right but what i have said is also right.

It's totally false to say that you can "consider" .000...1 to be any such thing, or anything at all. The notation has no standard definition, and you haven't provided one.
In my example above this is same as y.

NO!

What you don't realize is that when you put a "1" in a decimal position it represents a specific quantity. As I explained before, 0.01 means there is a "1" in the "hundredths" decimal position. So in order to place a "1" in a decimal position you have to define which position it is that you are going to position that "1." You have no idea which position it is, and if you did then you would have to eliminate the ... because then there wouldn't be an infinite series of 0's, there would be a FINITE quantity of 0's, then a 1. You then would know which decimal position the "1" was in. But again, you can't do that!

Adding 0's after 1 in a decimal number does not change its value. If you put another number after 1 other than 0 but keep on adding 0's before 1, this number will still tend towards 0 but not equal to 0.

Maybe you don't understand what you are saying when you type 0.000...1. You are saying there are an infinite amount of zeroes after a decimal point, and then at the end of the infinite amount of zeroes there is a one. So the string ends in a 1, which means there is a finite amount of decimal places that the 1 is after the decimal point. That means there is a finite quantity of zeroes before the one. So you are contradicting yourself. On one hand you are saying there are an infinite quantity of zeroes and then a one at the end, and on the other hand you are saying there are a finite quantity of zeroes before a one.

So which is it? Does a one come after a finite quantity of zeroes, or does a one come after an infinite quantity of zeroes? Are the zeroes infinite or finite?

If you consider the axis of real numbers, it will become a continuous straight line, where all the points correspond to some real number. An infinitesimal is something non-zero. Do you think that the infinitesimal will not correspond to any point in this axis of real numbers?

6 is nonzero. Is 6 an infinitesimal?

If you think there are infinitesimals in the standard real numbers then you are worse than Motor Daddy, because you are here pretending to know better. And you clearly don't. I prefer an honest crank to a an ignorant poseur.

NO!

What you don't realize is that when you put a "1" in a decimal position it represents a specific quantity. As I explained before, 0.01 means there is a "1" in the "hundredths" decimal position. So in order to place a "1" in a decimal position you have to define which position it is that you are going to position that "1." You have no idea which position it is, and if you did then you would have to eliminate the ... because then there wouldn't be an infinite series of 0's, there would be a FINITE quantity of 0's, then a 1. You then would know which decimal position the "1" was in. But again, you can't do that!

Maybe you don't understand what you are saying when you type 0.000...1. You are saying there are an infinite amount of zeroes after a decimal point, and then at the end of the infinite amount of zeroes there is a one. So the string ends in a 1, which means there is a finite amount of decimal places that the 1 is after the decimal point. That means there is a finite quantity of zeroes before the one. So you are contradicting yourself. On one hand you are saying there are an infinite quantity of zeroes and then a one at the end, and on the other hand you are saying there are a finite quantity of zeroes before a one.

So which is it? Does a one come after a finite quantity of zeroes, or does a one come after an infinite quantity of zeroes? Are the zeroes infinite or finite?

I will explain this in a different way.

Consider the following equations.

1 - 0.9 = 0.1;---(1)
1 - 0.99 = 0.01;---(2)
1 - 0.999 = 0.001;---(3)
1 - 0.9999 = 0.0001;---(4)
1 - 0.99999 = 0.00001;---(5)
.
.
.
1 - 0.9999999999 = 0.0000000001;---(6)
.
.
.

From the above equations we can make a general equation as:" 1 - 0.9...n times 9 = 0.0...(n-1) times 0 then 1; where n is a positive integer greater than 0".---(7)

In the equation (7)
if n = 1, it becomes equation (1);
if n = 2, it becomes equation (2);
if n = 3, it becomes equation (3);
.
.
if n = 10, it becomes equation (6);

Like this it will continue.

In the equation (7) if n becomes infinity, it becomes "1 - 0.999... = 0.000...1" .

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I will explain this in a different way.

Consider the following equations.

1 - 0.9 = 0.1;---(1)
1 - 0.99 = 0.01;---(2)
1 - 0.999 = 0.001;---(3)
1 - 0.9999 = 0.0001;---(4)
1 - 0.99999 = 0.00001;---(5)
.
.
.
1 - 0.9999999999 = 0.0000000001;---(6)
.
.
.

From the above equations we can make a general rule as:" 1 - 0.9...n times 9 = 0.0...(n-1) times 0 then 1; where n is a positive integer greater than 0".---(7)

In the equation (7)
if n = 1, it becomes equation (1);
if n = 2, it becomes equation (2);
if n = 3, it becomes equation (3);
.
.
if n = 10, it becomes equation (6);

Like this it will continue.

In the equation (7) if n becomes infinity, it becomes "1 - 0.999... = 0.000...1" .

1-0.9=.1

"One hundred percent minus ninety percent equals 10 percent."

1-.99999=.00001

"One hundred percent minus ninety nine point nine nine nine (99.999) percent equals one thousandth of a percent (.00001)."

If you truly believe you are correct prove it to yourself and fill in the blanks with your 0.000...1 concept:

"One hundred percent minus "blankity blank blank blank" percent equals "blank" percent."

If you fail to tell me what you fill in the blanks with, I will take that as the signal that you don't know what you are talking about.