You surely already know Pi, e, the golden ratio... each of them having weird and marvelous mathematical properties.
But this one is the most amazing of all. It really deserves to be considered as the first of all numbers:
You surely already know Pi, e, the golden ratio... each of them having weird and marvelous mathematical properties.
But this one is the most amazing of all. It really deserves to be considered as the first of all numbers:
Last edited by Renk; 10.02.19 at 22:33.
Originally Posted by https://en.wikipedia.org/wiki/Wau_(letter)seems all other numbers can be derived from that one, using various mathematical operationsOriginally Posted by Renk
∞
Parable of the Two Birds
Two birds, beautiful of wings, close companions, cling to one common tree: of the two one eats the sweet fruit of that tree; the other eats not but watches his companion. The self is the bird that sits immersed on the common tree; but because he is not lord he is bewildered and has sorrow. But when he sees that other who is the Lord and the beloved, he knows that all is His greatness and his sorrow passes away from him...
...@ en.wikipedia.org Paramatman
∞
Joke or not, I'd never heard of it before, whereas my pewblic skool edekasion at least taught me that 0.999... = 1
"Come visit sometime, okay? We'll always be here for you. We... we all love you."
if we wanted to be precise, 0.999... can never reach 1, not even in infinity
so one could write: 0.999...≈1 or 0.999...≐1
∞
Parable of the Two Birds
Two birds, beautiful of wings, close companions, cling to one common tree: of the two one eats the sweet fruit of that tree; the other eats not but watches his companion. The self is the bird that sits immersed on the common tree; but because he is not lord he is bewildered and has sorrow. But when he sees that other who is the Lord and the beloved, he knows that all is His greatness and his sorrow passes away from him...
...@ en.wikipedia.org Paramatman
∞
No joke here. If 1/9 = 0.111... then 9/9 = 0.999..., but of course, any number divided by itself yields 1, therefore 0.999... = 1. Or at least that's what my math teacher told me, and my mind was blown at the time, but eventually it made sense. The trick is realizing that 0.999... is just an alternate representation of 9/9 and not a different number.
"Come visit sometime, okay? We'll always be here for you. We... we all love you."
4,195,835 / 3,145,727 = 1.333739068902037589
There's a good floating point joke, one that cost millions of dollars.
"Come visit sometime, okay? We'll always be here for you. We... we all love you."
I tnink you are refering to a kind of "asymptote's phenomena": A curve never reaches it's asymptote. It's true, but the point is here that 0.9999...(infinite expansion) is the asymptote itself, or more precisely a coding of the asymptote (the number 1).
It's a pretty explanation. The sole point that remains: Is 0.1111111... x 9 really equals to 0.999999....? There are plenty of paradoxes when calculations involving an infinite number of terms are performed.
Last edited by Renk; 14.02.19 at 12:49.
thats contradictory, 0.999... doesn't magically become 1, there is a difference involved, even if an infinitely small differenceOriginally Posted by Renk
hence the approximation symbolOriginally Posted by Renk
admit it, you used a calculator with the '=' symbolOriginally Posted by anon
other math teachers would say to leave the fraction as is, if it cannot be further reduced, so you would have a nice, clean calculation like this:Originally Posted by anon
1/9*9=9/9=1
0.111...*9 does equal 0.999...Originally Posted by Renk
but 0.111... does not equal 1/9, nor does 0.999... equal 1
the paradox here is assuming the ability to do precise calculations involving imprecise infinite numbers - so you start with an error (erroneous assumption) and keep making errors, just like IRLOriginally Posted by Renk
∞
Parable of the Two Birds
Two birds, beautiful of wings, close companions, cling to one common tree: of the two one eats the sweet fruit of that tree; the other eats not but watches his companion. The self is the bird that sits immersed on the common tree; but because he is not lord he is bewildered and has sorrow. But when he sees that other who is the Lord and the beloved, he knows that all is His greatness and his sorrow passes away from him...
...@ en.wikipedia.org Paramatman
∞
Mathematically it does: The "magic" here is the infinite. For every finite sequence of 9's, the number 0.999...999 is not egal to 1. But "0.999...." doesn't represents any finite sequence of 9. It represents (ie is a notation for) the limit value of such a (finite) sequence of 9 (more precisely, the limit value of the infinite sequence of finite sequences of 9). And this limit value is exactly 1. "1 = 0.9999..." in the same sense that "1/3 = 0.3333....." In fact, every decimal number (and only decimal numbers) has 2 decimal expansions: one expansion ending with 0 (named the "proper decimal representation", and an other expansion ending with an infinite sequences of 9 (named the improper one). Example: 0.25 = 0.2500000... = 0.24999999.... The two are equal to 1/4.
This is related to the formal meaning of the sequence "0.11111...". Formally: Let Sn be the sum 10^(-1)+...+10^(-n): Sn=0,111..11 ("1" n times). The notation "0.1111..." doesn't represents any Sn's. It represents the limit value of the Sn's.
0.111...*9 does equal 0.999...
but 0.111... does not equal 1/9, nor does 0.999... equal 1
By direct calculation (geometrical sum) we have Sn= (10^(-1) - 10^(-n-1))/(1 - 10^(-1))
The limit value of 10^(-n-1) is 0. Then the limit value of the Sn's is (10^(-1) )/(1 - 10^(-1)), which is exactly 1/9.
We start with an error, and there always be an error (for every finite expansion), but these errors are going smaller and smaller and the limit value of theses errors is nothing else than 0, and so in considering limit value, we have equality.the paradox here is assuming the ability to do precise calculations involving imprecise infinite numbers - so you start with an error (erroneous assumption) and keep making errors, just like IRL
Saying that "0.999...=1", or that "0.111... = 1/9", is exactly the same as saying that the limit value of, say, 1 + 1/n, is exactly 1.
No 1 + 1/n is exactly equal to 1. But the limit value is.
Last edited by Renk; 15.02.19 at 05:10.
you don't understand infinity - the point is simple: 'it never ends', therefore it never gets equal to 1, because that would mean an end to infinityOriginally Posted by Renk
and you said it yourself: no finite sequence equals one, no matter how lengthy the sequence, therefore it would be illogical to assume some special behavior far far away in infinity
nothing special happens in numerical infinity, other than more of the same, so in the example above, you simply get an infinite number of nines, and not a zero or one, ever
such notation is flawed, because all these are different from each other and different from one:It represents (ie is a notation for) the limit value
0.9 (0.1 difference)
0.99 (0.01 difference)
0.999 (0.001 difference)
0.999... (0.000...1 difference, infinitely small difference, but a difference nevertheless and a never-ending difference as well)
. <---inconvenient insurmountable gap for the sequence, the tiny elephant in the room
1 (no difference) <---limit value
so it would be inappropriate to use the same notation for the sequence and for its limit, which are two different things, as shown above
∞
Parable of the Two Birds
Two birds, beautiful of wings, close companions, cling to one common tree: of the two one eats the sweet fruit of that tree; the other eats not but watches his companion. The self is the bird that sits immersed on the common tree; but because he is not lord he is bewildered and has sorrow. But when he sees that other who is the Lord and the beloved, he knows that all is His greatness and his sorrow passes away from him...
...@ en.wikipedia.org Paramatman
∞
"Philosophically", you are contradicting yourself in saying that it would be "illogical" to "assume some special behavior" concerning infinite sequences, just after having stated that (by definition) infinite sequences, unlike the finite ones, "never end". Ending vs never ending is a big, big qualitative difference, rendering illogical to consider that infinite sequence could in no way have specific properties that the finite ones don't have.you don't understand infinity - the point is simple: 'it never ends', therefore it never gets equal to 1, because that would mean an end to infinity
and you said it yourself: no finite sequence equals one, no matter how lengthy the sequence, therefore it would be illogical to assume some special behavior far far away in infinity
I your case, let me say that a=b if a-b=0. Let notice d=1-0.9999..... the distance between 1 and the (number represented by) infinite sequence 0.999.....
We have: 10*d = 10 -9,999... = 10-9 -0.9999... = 1-0.999.... So 10*d=d. So 9*d=0, and then d=0, which implies 1=0.9999......
The trick here is that I'm considering that 10*(1-0.999...) = 10- 10*0.999..., and that 10*0.999.... = 9.999... These properties are well known for finite sequences, but for the infinite ones???
Considering that these classic calculus rules remain true with infinite sequences can be seen as a daring gamble, and a questionable claim. But this can be mathematically derived (in a totally logical in rigorous way) from the very (rigorous) definition of the number represented by the infinite sequence 0.999.... . That's where the problem lies: What is the precise, exact, rigorous signification of "0.999...."???? Different choices can be made. One of these choices implies no contradiction, and lead to the conclusion that 0.9999.... = 1.
Last edited by Renk; 31.07.20 at 17:24.
on the contrary, it is you who are assuming special behavior, wherein 0.999... somehow magically becomes 1, even though not one segment of the infinite sequence, no matter how lengthy, can confirm this assumptionOriginally Posted by Renk
again, it is you who are assuming a magical loss of difference in infinity, but as said before, there is an infinitely small difference between an infinite sequence and its finite limit, which in the above mentioned example would be:Ending vs never ending is a big, big qualitative difference, rendering illogical to consider that infinite sequence could in no way have specific properties that the finite ones don't have.
1 - 0.999... = 0.0...1
it could be said that this infinite sequence is as close as you can get to the finite number 1 (its limit) - but still, the limit is never actually reached, as can be seen in the previous equationWhat is the precise, exact, rigorous signification of "0.999...."????
in an infinite division of numbers between 0 and 1, the closest to number 1 is an infinite sequence 0.999...
on the other hand, in a finite division of numbers between 0 and 1, the closest to number 1 is:
0.9 if divided into 10 segments
0.99 if divided into 100 segments
0.999 if divided into 1000 segments
however, both in the finite and infinite division, there is always a segment left between the limiting number (limit) and its closest neighbor - how close the neighbor is depends on the division involved:
- infinite division produces an infinitely close neighbor
- finite division produces a finitely close neighbor
- no division produces no neighbor - this is the only case when the neighbor and the limit are the same number, as can be seen below for d=0
10*d=d only works for 0 (d=0):So 10*d=d. So 9*d=0, and then d=0, which implies 1=0.9999......
for d=0 we have 10*0=0 OK
for d=0.1 we have 10*0.1=1<>0.1
for d=1 we have 10*1=10<>1
similarly, it doesn't work for d=0.999...
this can be used for practical purposes, to simplify calculations, but it is still only an approximation or simplification or assumed equality, not actual equality, hence it would be appropriate to use the approximation symbolDifferent choices can be made. One of these choices implies no contradiction, and lead to the conclusion that 0.9999.... = 1.
∞
Parable of the Two Birds
Two birds, beautiful of wings, close companions, cling to one common tree: of the two one eats the sweet fruit of that tree; the other eats not but watches his companion. The self is the bird that sits immersed on the common tree; but because he is not lord he is bewildered and has sorrow. But when he sees that other who is the Lord and the beloved, he knows that all is His greatness and his sorrow passes away from him...
...@ en.wikipedia.org Paramatman
∞
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