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

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

　 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@@

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...

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~~~~)

循環小數計法, 冇錯丫

岩岩教緊sum to infinity of a geometric series... 試試看

0.99999
= 0.9 + 0.09 + 0.009 + 0.0009 +......

S (INFINITY) = 0.9 / 1 - 0.1
= 0.9 / 0.9
= 1

中五生的理解程度

似乎有d數學上錯誤

#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

我覺得

等於一 與非常接近一
是二件事

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 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...

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

我無考過AL, CE亦係麻麻地......你唔好恰我好WO

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..../\

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?

0.99999.....!=1

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.

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....

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

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