[數學]0.999...= 1? @ 學術文化綜合研討區

堂我流 · 10-21-06, 12:46 PM

大家點睇?
0.999... 即係 0.9999999(無限不斷重複個 9)......

BeBop · 10-21-06, 12:49 PM

make x=0.999...
10x=9.999...
10x-x=9.999... - 0.999...
9x=9
x=1

二伍 · 10-21-06, 01:29 PM

似乎有d數學上錯誤

masaki · 10-21-06, 01:59 PM

引用:

作者: BeBop

make x=0.999...
10x=9.999...
10x-x=9.999... - 0.999...
9x=9
x=1

似乎唔係幾arm wor...
數學上你去極都唔到就即係limit啦~咁即係永遠都唔會到達果個位

0.[9](我當[]係循環小數)應該係
　 lim x
　x->1

咁 lim (10x-x )
　x->1

應該=8.[9]1

堂我流 · 10-21-06, 02:03 PM

#2位人兄係用algebra 去證明....
我睇過其他forum o既討論.........有人話 1- 0.999... =0.000.....0001 <---有個相差,所以0.999...唔= 1
亦有人話因為四捨五入,所以 0.999...= 1
OTL

IamO · 10-21-06, 02:13 PM

引用:

作者: 堂我流

大家點睇?
0.999... 即係 0.9999999(無限不斷重複個 9)......

我覺得

等於一與非常接近一
是二件事

二伍 · 10-21-06, 02:37 PM

引用:

作者: BeBop

10x-x=9.999... - 0.999...

應該由lee句開始錯
10x-x=9.999....-0.999...
(10-1)x=(10-1)0.999...
9x=9(0.999...)
x=0.999...

BeBop · 10-21-06, 02:44 PM

引用:

作者: masaki

似乎唔係幾arm wor...
數學上你去極都唔到就即係limit啦~咁即係永遠都唔會到達果個位

0.[9](我當[]係循環小數)應該係
　 lim x
　x->1

咁 lim (10x-x )
　x->1

應該=8.[9]1

　 lim x
　x->1
=1

lim (10x-x )
　x->1
=9

That's how you take limits.

sunnycard · 10-21-06, 02:57 PM

引用:

作者: BeBop

10x-x=9.999... - 0.999...
9x=9

呢Part好似係米有d問題?

當1個數乘大10倍，已經少左一個位(無限數唔知)
假設唔係無限9

0.999
0.999 * 10 - 0.999 = 8.991 已經唔係9

(光速逃~~~~)

GUSTAV · 10-21-06, 03:11 PM

對手係小學生時: 你唔識ga 啦算吧啦
對手係初中生時: 1/9= 0.1111111..... => 0.111111....*9 = 0.999.......= 1/9*9=1
對手係ce考生時: set x lim x->1
對手係al 考生時: as 0.9999 is very near to 1, let's assume it=1 \

/

Jamie · 10-21-06, 03:28 PM

用infinity的concept去理解, 0.999..... =1

BeBop · 10-21-06, 03:30 PM

引用:

作者: GUSTAV

set x lim x->1

That's the point, to realize that 0.999...=(limx as x->1)=1

BTW, nice put

堂我流 · 10-21-06, 03:30 PM

引用:

作者: GUSTAV

對手係al 考生時: as 0.9999 is very near to 1, let's assume it=1 \

/

我無考過AL, CE亦係麻麻地......你唔好恰我好WO

GUSTAV · 10-21-06, 05:57 PM

引用:

作者: BeBop

That's the point, to realize that 0.999...=(limx as x->1)=1

BTW, nice put

基本上lim 都係計數用既姐
同埋lim x->1 係唔同既數式好多時唔可以代x=1...係會有分別的
其實應該x_->1 tim 因為0.9999... 係細過1

不過我幫人補習時係唔係都係咁頹解\

/
對手係初中生時: 1/9= 0.1111111..... => 0.111111....*9 = 0.999.......= 1/9*9=1
唔明點解9成人會好相信我

雖然條式真係好似好有道理....好似...

引用:

作者: 堂我流

我無考過AL, CE亦係麻麻地......你唔好恰我好WO

呢個問題係唔同既case會有唔同既答案
基本上正常情況下都會當佢係1就得
但係某d case 唔得就會要用到limit
例如有時d式分子分母都有x....sub完x=1之後會變左0/0 o個d

GUSTAV · 10-21-06, 06:04 PM

引用:

作者: Jamie

用infinity的concept去理解, 0.999..... =1

小弟天生愚笨，後天又沒有好好努力進修數理，什麼理論概念全都忘了
所以完全不明白無限概念和0.999... =1 有什麼大關係
大大可以解釋一下嗎?

Cucu · 10-21-06, 08:30 PM

0.99999.....!=1

masaki · 10-22-06, 12:51 AM

引用:

作者: BeBop

　 lim x
　x->1
=1

lim (10x-x )
　x->1
=9

That's how you take limits.

lim (10x-x )
　x->1

lim (9x )
　x->1
=9

係就係既...但take得都唔當係9既其實
咁已經解釋左果度有問題@@

李某人 · 10-22-06, 12:58 AM

0.999... != 1, but it tends to 1.

limit 是接近, 不是等於...

newagedreams · 10-22-06, 03:45 PM

0.999... is actually a limit, not an exact number.

0.999...... = lim(n->inf)(.9 + .9*0.1 + .9*0.1^2 + ...+ .9*.1^n)
= .9 * lim(n->inf){(1 - .1^n)/(1 - .1)} = .9 / (1 - .1) = 1

This result means the sum of the sequence {0.9*0.1^n} converges to 1 as n tends to infinity.

newagedreams · 10-22-06, 03:46 PM

引用:

作者: 李某人

0.999... != 1, but it tends to 1.

limit 是接近, 不是等於...

0.999... is already a limit, so the limit is 1 means the limit equals 1.

PTX-003C · 10-22-06, 04:23 PM

引用:

作者: GUSTAV

對手係小學生時: 你唔識ga 啦算吧啦
對手係初中生時: 1/9= 0.1111111..... => 0.111111....*9 = 0.999.......= 1/9*9=1
對手係ce考生時: set x lim x->1
對手係al 考生時: as 0.9999 is very near to 1, let's assume it=1 \/

that's just cuz the error will be unlimitedly small...

0.9999...... is unlimitedly close to 1, but can never be equil to 1.
you can assume it to be 1 because like i said earlier, the error will be unlimitedly small, so the result will be basically the same.

so in conclusion...as an ammount, it is unlimitedly close to 1, but as in calculation....just assume it to be one...cuz the answer won't really make a difference anyways..../

\

李某人 · 10-22-06, 05:05 PM

引用:

作者: newagedreams

0.999... is already a limit, so the limit is 1 means the limit equals 1.

那是不是所有循環小數都可以化成 limit 再找它相應的分數...?
例如 0.343434...:

0.343434...
=lim(n->INF) (0.31 + 0.31*0.1 + 0.31 *0.1^2 + ... + 0.31*0.1^n)
=0.31*lim(n->INF) (1 + 0.1 + 0.1^2 + 0.1^n)
=0.31*lim(n->INF) (1 - 0.1^n)/(1 - 0.1)
=0.31/0.9 = 31/90

不過 31/90 = 0.3444...

我的計算有什麼問題...?

GUSTAV · 10-22-06, 05:30 PM

引用:

作者: 李某人

那是不是所有循環小數都可以化成 limit 再找它相應的分數...?
例如 0.343434...:

0.343434...
=lim(n->INF) (0.31 + 0.31*0.1 + 0.31 *0.1^2 + ... + 0.31*0.1^n)
=0.31*lim(n->INF) (1 + 0.1 + 0.1^2 + 0.1^n)
=0.31*lim(n->INF) (1 - 0.1^n)/(1 - 0.1)
=0.31/0.9 = 31/90

不過 31/90 = 0.3444...

我的計算有什麼問題...?

lim x->1_=1 的意思並不等於 x=1 耶....

李某人 · 10-22-06, 05:42 PM

引用:

作者: GUSTAV

lim x->1_=1 的意思並不等於 x=1 耶....

我知道, 不過 0.3444... 和 0.343434... 差別太大, 我怕自己計錯而已...

Shing · 10-22-06, 05:55 PM

引用:

作者: 李某人

那是不是所有循環小數都可以化成 limit 再找它相應的分數...?
例如 0.343434...:

0.343434...
=lim(n->INF) (0.31 + 0.31*0.1 + 0.31 *0.1^2 + ... + 0.31*0.1^n)
=0.31*lim(n->INF) (1 + 0.1 + 0.1^2 + 0.1^n)
=0.31*lim(n->INF) (1 - 0.1^n)/(1 - 0.1)
=0.31/0.9 = 31/90

不過 31/90 = 0.3444...

我的計算有什麼問題...?

0.31+0.031+0.0031.............
=0.3444444444...........4441
=/=0.34343434...

10-21-06, 12:46 PM	#1
堂我流 Insane Gamer -廚術- 註冊日期: Sep 2005 文章: 777	[數學]0.999...= 1? 大家點睇? 0.999... 即係 0.9999999(無限不斷重複個 9)......

10-21-06, 03:11 PM	#10
GUSTAV The One poor 註冊日期: Oct 2003 文章: 29,578 邊有錢傻的嗎	對手係小學生時: 你唔識ga 啦算吧啦對手係初中生時: 1/9= 0.1111111..... => 0.111111....9 = 0.999.......= 1/99=1 對手係ce考生時: set x lim x->1 對手係al 考生時: as 0.9999 is very near to 1, let's assume it=1 \/ __________________ この袁紹おもしろい事は求めぬ！小よく大を制するような奇計も求めぬ！制覇は当然のごとく成すべし！それが王者の戦いというものだ窮

10-21-06, 03:28 PM	#11
Jamie God of Gamer バンブラ小職人註冊日期: Apr 2003 文章: 15,224 JaMieGAF	用infinity的concept去理解, 0.999..... =1 __________________ ありがどう。。。スキ。。。さようなら男は＿＿で勝負！！ [NICO] 實況 KY 彈幕外語自重 [/NICO]

10-21-06, 08:30 PM	#16
Cucu Game Master 註冊日期: Feb 2003 文章: 3,035 cululu	0.99999.....!=1 __________________ RX-0 UNICORN GUNDAM CONSTRUCTOR: ANAHEIM ELECTRONICS FULL PSYCHO FRAME PROTOTYPE MOBILE SUIT U.C.0096 PROJECT UC / La+

10-22-06, 12:58 AM	#18
李某人 Game Master 初心者註冊日期: Jun 2004 文章: 2,822	0.999... != 1, but it tends to 1. limit 是接近, 不是等於... __________________ Notes - I know nothing. GAF★MHFO M群: group126866@xiaoi.com -- 人類因為別人幸福而自己也感到幸福這種事，只不過是望梅止渴而已。

10-21-06, 12:49 PM	#2
BeBop Ultimate Gamer 註冊日期: Jun 2003 文章: 3,733	make x=0.999... 10x=9.999... 10x-x=9.999... - 0.999... 9x=9 x=1

10-21-06, 01:29 PM	#3
二伍 God of Gamer 註冊日期: Sep 2002 文章: 16,906	似乎有d數學上錯誤

10-21-06, 02:03 PM	#5
堂我流 Insane Gamer -廚術- 註冊日期: Sep 2005 文章: 777	#2位人兄係用algebra 去證明.... 我睇過其他forum o既討論.........有人話 1- 0.999... =0.000.....0001 <---有個相差,所以0.999...唔= 1 亦有人話因為四捨五入,所以 0.999...= 1 OTL

10-22-06, 03:45 PM	#19
newagedreams Crazy Gamer 註冊日期: Jan 2003 文章: 1,466	0.999... is actually a limit, not an exact number. 0.999...... = lim(n->inf)(.9 + .90.1 + .90.1^2 + ...+ .9.1^n) = .9 lim(n->inf){(1 - .1^n)/(1 - .1)} = .9 / (1 - .1) = 1 This result means the sum of the sequence {0.9*0.1^n} converges to 1 as n tends to infinity.

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