昨夜、0.999999999........ = 1の夢を見たというKids。学校数学が嫌いなKidsが数字の夢を見るとはたいしたものである。
こんなこと（0.999999999........equal to 1 or less than 1）が頭を駆け巡ったという。
1/3 = 0.3333....
0.3333.... × 3 = 0.9999....
Therefore, 0.9999.... = 1
0.3333.... × 3 = 0.9999....
Therefore, 0.9999.... = 1
一度はこの式の正当性を納得したはずなのであるが。実数の連続性・稠密性から１の両隣に数字はないということもかって理解してもらった。
hitrecord.org/records/1331011
しかし、今朝方の夢でまた議論を私に吹っかけてきた！Kidsの気持ちは充分に分かるが、Fortis君の議論を読んだらとsuggest。
Any fraction beginning zero-point-whatever is, so to speak, by definition less than unity.
Why? Where is it defined to be the case? Do you agree that the infinite series
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 ?
The difference between the truncated form of 0.999... and 1 tends to zero as the number of digits increases. So in the limit that the number of digits becomes infinite, the difference becomes zero. (As has been mentioned previously, "What number falls between 0.999... and 1?")
Those who propose that 0.99999… and 1 are equal are merely showing that they are EQUIVALENT FOR COMPUTATIONAL PURPOSES. Of course they are!
And to any number of decimal points of accuracy anyone is ever likely to require. I do not dispute it, in fact I still insist upon it. As I said before, the ONLY difference is that 1 is greater by an infinitely small amount.
It is a standard mathematical result that they are identical, not just equivalent for computational purposes. Thinking about infinite sums can seem tricky at first, but (as long as they are convergent) they are well defined beasts.
Until the thought police outlaw infinitesimal thoughts that will remain the case! :evil:
Have a thought about the following problem.
Have a thought about the following problem.
"What is the integral of x between 0 and 1?"
Well, it's just a triangle so it's obviously 1/2.
Now ask yourself what the integral of x^2 is between 0 and 1.
This is defined in terms of the limit of infinite sum, and the result is 1/3.
Not approximately equal to 1/3 (or equivalent to 1/3 for computational purposes), but equal to 1/3.
If limits of infinite series didn't exist, then an equals sign would be a rare sight in anything other than simple algebra. ;) :)
mathemusik.blogspot.jp/2010/07/09999999-1
polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it
VERY GOOD.
But there's a couple tricks
you missed.
you missed.
First, simple pattern
completion
completion
1/9 = .11111---
2/9 = .22222---
3/9 = .33333---
4/9 = .44444---
5/9 = .55555---
6/9 = .66666---
7/9 = .77777---
8/9 = .88888---
and therefore by logical
extension
9/9 = .99999---
but of course, 9/9 = 1.
And then there are the
SPIRITUAL implications
SPIRITUAL implications
.9 a soul
+ .09
+ .009 adding experience
+ .0009
+ .00009
!
! infinitely increasing
!
+ .09
+ .009 adding experience
+ .0009
+ .00009
!
! infinitely increasing
!
or the infinitely
repeating process
of growing greater
i.e. life
repeating process
of growing greater
i.e. life
--------
.99999---
.99999---
which EQUALS
1
the finished, static, unity
characteristic of a single
never-created being
characteristic of a single
never-created being
i.e. God
The equations show
IT MADE US TO BE ITS EQUAL
(the TRUTH behind all men
are created equal -
soul A = God,
soul B = God
transitive property -
A = God = B therefore A=B)
IT MADE US TO BE ITS EQUAL
(the TRUTH behind all men
are created equal -
soul A = God,
soul B = God
transitive property -
A = God = B therefore A=B)
but .999--- being = 1
and 1 being a rational
number, 1/1 is the ratio
of two integers, as a guy
named Hugh Richmond asked
me, where are the two
integers whose ratio is
.999---
THERE ARE NONE thus
I coined an equation to
go with .999--- = 1 which
demonstrates our equality
and 1 being a rational
number, 1/1 is the ratio
of two integers, as a guy
named Hugh Richmond asked
me, where are the two
integers whose ratio is
.999---
THERE ARE NONE thus
I coined an equation to
go with .999--- = 1 which
demonstrates our equality
x/y = .999---
where x and y are integers
where x and y are integers
THAT equation show the
DIFFERENCE between the
Creator and the Created -
only the Creator can make
it work.
DIFFERENCE between the
Creator and the Created -
only the Creator can make
it work.
Posted by: Fred Harry Wolnerman | June 17, 2006 at 11:53 PM