# Thread: The Most Amazing Number

1. Originally Posted by Renk
It seems to me that assuming that infinite objects mandatory have to share the same properties than finite one's and nothing else is overly presumptuous.
similar properties, not 'the same' properties
although, in another sense, 'the same' may apply as well: both appear in the mind as a result of thinking, therefore they necessarily have to share the same properties as all other things that appear that way

9*d=0
the solution of this equation is d=0
that means d=0, it does not mean d=0.0...1 or any other number

My argument only relies on some basic calculation rules, nothing more. If it is false, then at least one of these basic calculation rules fails in case of infinite sequences...
your calculation is erroneous, as already shown on several occasions - calculation rules did not fail here

in infinite case the part may be "as big" as the whole.
no, the part is still smaller than the whole, but nevertheless still infinite
it may be easier to think of them as 'countless': there are countless numbers between 0 and 1, but there are twice as much between 0 and 2

I have an other argument to try to convince you that 0.99999....=1
why complicate things further, when i already gave the correct value in the most simple way:

1 - 0.999... = 0.0...1

or written more elegantly:

1 - 0.9... = 0.0...1  Reply With Quote 2. no, the part is still smaller than the whole, but nevertheless still infinite
it may be easier to think of them as 'countless': there are countless numbers between 0 and 1, but there are twice as much between 0 and 2
Not always in case of infinite set: In your example, it is possible to match each number between 0 and 1 with an unique number between 0 and 2, and vice versa (mathematicians call that a bijection). In that case, you can mach e.g. every number x between 0 an 1 with the number 2*x, between 0 and 2, reciprocally you can match every number x between 0 and 2 with an unique number (in that case x/2) between 0 and 1. So there are exactly as many numbers in [0,1] than in [0,2] (and than in [0, 100000], and in fact than in the whole set R itself) This size (cardinality) is named aleph zero (I can't write it here).

the solution of this equation is d=0
that means d=0, it does not mean d=0.0...1 or any other number
Precisely. And as d=1-0.9999...., d=0 means 1=0.9999....

View differently, as we have 9*d=0, if d were non zero, then it would be possible to divide by d, leading to 9=0/d=0. Big contradiction again when assuming that 0.9999... is different from 1.

This calculation is only: 10*d = 10*(1-0.9999...) = 10*1 - 10*0.9999... = 10 - 9.9999... = 10 -9 -0.9999... = 1-0.9999 = d, so 10*d and 9*d=0. What is so erroneous in there?

why complicate things further
This second argument I presented is not complicated at all. It's only based on logic and on one fundamental property of the set of all "real numbers" (this set happens to be "archimedian"). this second proof is even (in some sense) simpler, more elementary than the first, as almost no calculation rules are implied here, only very basic and fundamental facts about numbers.

when i already gave the correct value in the most simple way:

1 - 0.999... = 0.0...1

or written more elegantly:

1 - 0.9... = 0.0...1
You are naming "0.0....1" what I named "d" (previously I defined d by the distance 1-0.9....). My choice is far more concise . Moreover, as the notation 0.0...1 may have no mathematical sense at all, I'm going to put it in quotes.

In any case, then again: 10*"0.0...1" = 10*(1-0.9...) = (see above) = 1-0.9... = "0.0...1".

So we have 10*"0.0...1" = "0.0...1", which leads to 9*"0.0...1"=0, and then, either "0.0...1"=0, either 9=0. What's your choice?  Reply With Quote 3. Originally Posted by Renk
In that case, you can mach e.g. every number x between 0 an 1 with the number 2*x, between 0 and 2,
So there are exactly as many numbers in [0,1] than in [0,2]
matching unique numbers is not the same as comparing them
it is obvious that the infinite set between 0 and 1 is smaller than the set between 0 and 2, simply because you can find numbers in the latter that aren't available in the former, while the opposite does not apply

(1) 9*d=0
(2) d=1-0.9999....
sorry, but there is no d that satisfies both of these equations simultaneously, because the solutions are as follows:
(1) d=0
(2) d=0.0...1

these two values are obviously different

if d were non zero, then it would be possible to divide by d, leading to 9=0/d=0
if d were non zero, the equation would be invalid, only d=0 satisfies the equation - isn't that the goal of solving equations?

10*d = 10*(1-0.9999...) = 10*1 - 10*0.9999... = 10 - 9.9999... = 10 -9 -0.9999... = 1-0.9999 = d, so 10*d and 9*d=0. What is so erroneous in there?
simplified, the error is: 0.99 * 10 = 9.99, whereas it should be: 0.99 * 10 = 9.9, you have 'one too many nines' after multiplying by 10

1*d = 0.0...1 <> 0
2*d = 0.0...2 <> 0
...
9*d = 0.0...9 <> 0
10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d

obviously, none of these equal zero and they are all different from each other  Reply With Quote 4. matching unique numbers is not the same as comparing them
I compare the cardinality of the sets [0,1] and [0,2] (ie the quantity of numbers they are containing) in matching every number of the 1st set with every number of the second set, leading to conclude that these two sets have the same cardinality (each element of one of the set can be "married" to only one of the other set).

When in some population you can marry each man with exactly a woman, and each woman with exactly one man, then in this population there are exactly as many men as women. This is even the very mathematical definition of "as many as".

it is obvious that the infinite set between 0 and 1 is smaller than the set between 0 and 2
"obviousness" is not an argument at all, particularly when considering infinite objects.

simply because you can find numbers in the latter that aren't available in the former, while the opposite does not apply
What you say means that [0,1] is a strict subset of [0,2]. That absolutely not implies that the cardinality of the second set is strictly greater than the cardinality of the first.

10*d = 0.0...10 <> 0 - this one has (n-1) leading zeroes, it is 10 times larger than d
This reasoning assume you are considering a (finite) integer n. But it does no more work here because in your "0.0...1", your "n" in fact equals to "infinite", and "infinite"-1="infinite".
(what I just said is more intuitive than rigorous, due to the fact that your "0.0...1" is only a symbol, and as no clear mathematical meaning).

sorry, but there is no d that satisfies both of these equations simultaneously, because the solutions are as follows:
(1) d=0
(2) d=0.0...1
The fallacy in your reasoning is that you implicitly suppose that what you name "0.0...1" differs from 0. For you it's kind of an axiom. You consider from the outset that="0.0...1" can in no way be equal to 0. But preliminary admitting an assertion is not a good way to prove it.

For my part, I don't presuppose anything concerning the value of "0.0...1". I just consider this object and, in applying very basics calculus rules (or some reasoning involving basic properties of numbers) , I come to the conclusion that your "0.0...1" is in fact 0.

You can refuse to accept it, but, nevertheless, your "0.0...1" has the property that 10*"0.0...1" = "0.0....1", and then equals to zero.

if d were non zero, the equation would be invalid
Exactly. And as the equation is valid, d can not be non zero, ie d=0.

0.99 * 10 = 9.99, whereas it should be: 0.99 * 10 = 9.9
I don't say that 0.99*10 = 9.99, I say that 0.999...... *10 = 9,999.....

What is then the result of the operation 10*0.999..... for you?

All that leads me to think to a 3rd proof that 0.999.... =1:

We have:
1/3=0.333.....

and similarly:
2/3=2*(1/3)=2*0.333... =0.666......

And we have too: 3*(1/3)= 3*0.333....= 0.999....

But on the other hand:
3*(1/3) = 3/3=1.

Hence, 0.999..... =1.

This proof is analogous to the one given by anon previously:

1/9=0.1111.....
Then: 9*(1/9)=0,9999......
But in the same time, 9*(1/9)= 9/9=1.
So 0.999...=1.

The point is, 0.999.... doesn't look like 1, it even doesn't look like an integer at all but it is an integer, and even it is the number 1.

In fact, every decimal number has exactly 2 decimal representations. For example, the two decimal representations of 1/2 are 0.5 and 0.49999.....

You can visit any forum about mathematics, any wikipedia page about these subjects, you will find that 0.999... =1, that there are as many odd integers as there are integers, and that the set [0,1] has as many elements as the set [0,2], with arguments probably close to the ones I presented here.

It can even be proved that there are as many odd (or even) integers as there are decimal numbers (and that, despite the fact that eg the set [0,1] does contain only one even integer, and an infinite quantity of decimal numbers).

Mathematics are about what mathematical objects are, not about what they look like...  Reply With Quote 5. Originally Posted by Renk
matching every number of the 1st set with every number of the second set, leading to conclude that these two sets have the same cardinality (each element of one of the set can be "married" to only one of the other set).
the matching works only for half of the numbers of the larger set: the smaller set [0,1] perfectly matches with the first half of the larger set [0,1], but the other 'half' of the larger set, namely <1,2] is left unmatched, which is obvious

"obviousness" is not an argument at all, particularly when considering infinite objects.
infinite objects are not immune to simple, obvious arguments about their nature
'obvious' means: easily understood by an average open minded person

This reasoning assume you are considering a (finite) integer n...."infinite"-1="infinite".
n is an infinite number of leading zeroes
n-1 is 'one zero less than' an infinite number of leading zeroes
it is a way to label the 9th neighbor of d, the 10*d, where d=0.0...1

in a way, the infinitely small number is also finite, we just don't know how small it is, other than saying 'it is the smallest' - nevertheless, it also has its immediate neighbors which are all equally far away from each other (this smallest distance is an infinitely small constant of 0.0...1)

you implicitly suppose that what you name "0.0...1" differs from 0. For you it's kind of an axiom.
if we were to apply your idea of 0.0...1 = 0, the whole calculation of infinite numbers collapses to zero, but an infinitely small (non-zero) distance keeps it mathematically consistent

Exactly. And as the equation is valid, d can not be non zero, ie d=0.
you missed the part where both equations (1) and (2) need to have a common solution, in order to both apply (to both be valid)

What is then the result of the operation 10*0.999..... for you?
the result would be:

0.999... -- n being the number of nines after the decimal point
10*0.999...=9.999...0 -- the result, n-1 being the number of nines after the decimal point

there are still 8 numbers between this result and your previous result:

9.999...0 -- the result, n-1 being the number of nines after the decimal point
9.999...1 -- a number in between, n-1 being the number of nines after the decimal point
...
9.999...8 -- a number in between, n-1 being the number of nines after the decimal point
9.999... -- your previous result, n being the number of nines after the decimal point

We have:
1/3=0.333.....
see, you just assumed that
the problem with 1/3 is that it cannot be precisely represented with decimal numbers (10 doesn't divide with 3 nicely)

In fact, every decimal number has exactly 2 decimal representations. For example, the two decimal representations of 1/2 are 0.5 and 0.49999.....
i can think of a third representation: 0.50...1
but only one of these (0.5) is equal to 1/2, while others are either approximations or nearest neighbors

and using your logic, we can use numerous other representations, like: 0.50...2, 0.50...3, etc. or 0.49...8, 0.49...7 etc. simply because 1*d, 2*d, 3*d, etc. all equal zero - you see the absurdity of such reasoning?

Mathematics are about what mathematical objects are, not about what they look like...
that is just new-math, which has the same validity as (orwellian) new-speak - it is simply a sign of the times we live in  Reply With Quote 6. My my, I've caused quite a racket here... but we can all agree that zero is even, right?   Reply With Quote amazing, com, golden, having, https, know, number, numbers, proper, properties, ratio, really, tps, video, watch, weird, youtube 