1和0.999無限循環哪個大？小學生說1大，數學家說等於，高級數學家又說不等於 | 雅桑了嗎

下載99%不再繼續=死機

「問你個問題。1塊蛋糕切成等大小的3份，一份是1/3塊沒錯吧？」 「沒錯啊。」 「1/3相當於0.333……3，這也沒錯吧？」 「是啊。」 「那麼3份蛋糕合計就是0.999……9塊蛋糕，那剩下的那0.000…01塊蛋糕去哪裡了呢？」 「你看你的刀子上是不是沾了些奶油？」 「！！！謝謝，我懂了。」

我認為，無限循環是我們人類十進制的一個次元盲區，其實我們人類的數學本身建立在十進制的基礎上(因為人類十根手指頭)，導致很多數學家被困在十進制的一些數學難題裡，明明我們人類發明的電腦是運用2進制去運算，那麼，當遇到無限循環除不盡的狀況時，如果改用其他進制為基底去運算，你會發現原本除不盡的，變成整數了，我舉例 1/3=0.333333(循環) ， 如果大家試著用12進制去算就是1/3=0.4(12進制) ，當等式兩邊各乘以3 就回到1=1，不會有那個除不盡的問題，如果人類想進入宇宙變成更高等文明，就必須擺脫十進制的約束

教授A：59.9 = 60 教授B：59.9 = 59 看來我是遇到了不同派系的數學家

雅桑真的擁有把學問用哲學方式表達的能力與魅力！我真的很怕數學，我42歲了，可是我看到你的解析，我真的覺得如果我當時能遇到像您一樣的數學老師，我就不會只是個平凡的developer了…

這個問題 倒不如說「怎麼得到0.9999….」 其實「正常」的除法根本就得不到這東西 你可以1/1的時候故意不除好 得出來的就是0.99999….. 我覺得很多人都太把這個問題想的太哲學了 無限小為什麼等於零？無限又是什麼？ 其實都早有柯西所給出的極限嚴格定義 所以說穿了 不過是個根據定義所推論出來的東西罷了 每次想到這點總會有點對「無限」帶來的神秘感到破滅

雅桑總能把我怎麼也想不明白的事說清楚，只是我要看透徹這個清楚還需要99.999......次

(本馬鈴薯不知道a b是否必須是整數) 9:02 假設b=1 a=½ 雖然a÷b=½ 但是(b-a)÷b就會小於1÷b而不是≥ 假設b=¼ a=⅛ 也不會>1÷10^b 那1-a÷b就不會>1÷10^b 也就不會<1-1÷10^b 也不會<0.999... (有錯歡迎糾正)

高中時的數學老師說過一個很直觀的看法 2/2做長除法時從小數點後第一位開始除就是0.999… 那時候一聽完就沒有疑問了

利申，我並不是數學家，我這段內容基本都是消化了相關論證後再以自己的語言說出來，外加些少灰諧。所以有多少不乎合數學上的用詞標準或者有人覺得是冷笑話請見諒。 話說在前頭，我是認同0.999…=1的那一派。 要說「無窮小量」這個問題，我的理解是把微積分及離散數學一起來看就行了。微積分的下這個世界的「小」並沒存在下限，所以要解釋無窮小就要把思路逆轉過來，理解為無窮大的倒數，即「在概念上的所謂最小值」。言而這個字宙是有最小值的，這乎合離散數學的概念，「無窮小量」也就是就等於這個最小值了。 簡單點來講，「無窮小量」有多小就視乎系統允許的最小值有多「小」﹐還有系統允許的誤差值(這個很重要)。 首先說說微積分。因為「小」並沒有下限，任何一個非0值都可以被分割出更小的非0值，所以無窮小並沒有一個實值，只是一個概念，即是「沒有最小，只有更小」，只要一日未割至0仍然要割下去，但現有規側跟本不會讓你割剩0，只能繼續割，一直割，割到宇宙熱寂﹐但不代表這就成功割完，只是沒功夫繼續割，放棄不割了，也就出現所謂的餘值，不過這餘值基本上等同「無」。至於「無窮小」跟「無」之間的恩怨我就不用多說了。微積分的核心是「極值」，指有些情況下將某實值代入某公式並沒有答案(例如除以零)，但如果代入的並不完全等於該實值而是「有足夠靠近」那麼答案也會有足夠趨近某一個數值，其核心就是無窮小。有聰明的朋友也許會再提出這個「漏洞」：0.999…9跟1.000…0相差了0.000…1呀，怎會沒有實值？不能割不代表不存在吧。但我要說這些聰明的朋友搞錯了一個好關鍵的重點，就是0.999…首先有無限個9，然後才…抱歉「沒有然後」了。硬要找0.999…後面的東西﹐有都只會是繼續無限個9而不會只有一個，所以0.000…後面也頂多再來無限個0，這個「最後的1」的而且確不存在而非不能割(有得割不割罪大惡極阿朋友)。說到尾，也就回歸問題原點，變成了「遞歸」(recurrsion)的問題了。其實這個0.999…的問題不就雷同物理學四大神獸之一的芝諾烏龜的那個悖論﹐之不過將1/2(二進制0.111…)改為9/10(10進制)罷了，同樣用微積分就能破解了。 好，在離散數學下，那麼0.999…跟1的差別就的確有可能會是這個不可割的「系統允許的最小值」(但其實更有可能是0﹐真相只有一個﹐case closed，完結撒花，皆大歡喜…最後會提及原因)，但離散數學有一個很尷尬的難題﹐就是小數點後的位數總有個終點(那怕會有很多個，但總之就是會數完)，「無限循環小數」並不存在﹐更別說無理數，超越數等等並沒有規則可言的無窮小數了，也就是說1/3 =/= 0.333…、像無窮數列等等這些建基於無限循環小數的理論通通不能成立，圓周率及歐拉數等等也有絶對確值了(應該是「不被允許存在確值只能有約數」…但貌似反而是好事？)，但這明顥有違常理(雖然在微觀上反而是對的)。 要解決這一尷尬難題，首先要承認絕大部份宏觀系統都存在「可接受誤差值」的。舉個例子，我要你將769毫升水精準分成167等份(為何要769及167我待會再說明)。你會說，有多難？每份769/167即約4.60479041916毫升就是了。但我要提醒你「769毫升」「大概」有幾多個水分子…根據維基百科，標準情形下的水分子的摩爾數是約18.01528(g/mol)所以769毫升水有約42.6859865625摩爾(即約 257*10^23個)水分子﹐你真的打算要在精準到分子數的層面來分等份嗎？而且你能保證這個「約42.6859865625摩爾」能被19整除？不能整除的話何分成等份？再說在日常生活中要拿完全精確分子數的任何物質一個不差幾近不可能吧，別甭說視乎溫度壓力甚至同位數這個「約42.6859865625摩爾」根本就是一個浮動數值(所以這些數值前面全部有個「約」字)。所以系統需要有一個可接受誤差程度﹐不能少於這個浮動範圍，例如我會跟你說，「將769毫升水分成167等份，誤差值為1微升」，那麼你別理會水分子數的問題，也不用擔心769不能被167整除，總之將它分為167份介乎4.6047至4.6048毫升這個範圍就可以了(這誤差甚至明顯低於一微升)。好了﹐我選769和167的原因是因為剛好都是質數，能高機率出現不能整除的情況(除非摩爾剛好能被167整除)但又不會把問題搞得太複雜，雖然後來發現關係不太大…。 我就知道有人會質疑，數學就是要講求準確，在可行的情況下都要將誤差降到最低…「我現在就是要玩到對誤差零容忍，你奈我什麼何？」在離散數學下因為有明確的最小值，所以當然能做到表面上的零誤差，但也就等於先前的尷尬難題無法解決，這可是個更深層次的誤差處理，叫做「量化」。而且數學還講求「用於表達萬物」，明明現實世界就是充斥著誤差，更甚這些誤差大多明顯高於最小值，而你卻偏要把誤差的存在抹除掉，這還算是數學嗎？完善的數學理論是要將誤差考慮在內的。這裡跟大家說說一個小知識：我們身處的這個宇宙估算一共有約10的78至82次方個原子。是很多沒錯，但掉轉頭看1/(10^82)跟「無窮小」明顯有能夠理解的差距。單單是圓周率就已經被算出了小數點後上百萬個位或更多，可不是百萬分之幾喔…像我們的宇宙這一個概念上無限的實質有限集合，但在無窮小下都顯得何等渺小，「相比之下更加有限」的系統更就更甚，最簡單例子就是你跟我都在使用中的數字電腦中的浮點數(IEEE 754)，都有對應最小及誤差值的方針。有興趣的朋友可以去看看。這裡說個題外話：因為現今的宇宙正直青年，充滿著核反應，而且幾乎都是核融合，所以雖然幅度很少但這個原子數量是在持續滅少中。 總結，當一數值無可避免必需變得小於系統容許的非0最小值時﹐也就只能夠「歸邊」，要麼「塌縮」清零，要麼「擴散」成最小值，沒有中間可選，四捨五入的概念下是0的機會理應是大多數(因為怎想它也會比較接近0)，而這兩邊都同樣無可避免涉量化誤差的問題。所以到頭來歸那一邊最終其實沒分別。在微積分下0.999…毫無疑問=1，離散數學的話在允許「可接受誤差值」的絕大多數情況下等於基本上=1，不接受的話嘛…也絕對可以，但請先提出這個最小值不能被忽略的理由﹐而這個已經是純粹數學的範籌﹐玩味多於實用了… 後話，其實0.999…=1是在常理下(也就是0.999…的情況下了)被公認的事實。但數學就是這麼既有趣又煩人，可以在無限接近完全肯定的情況下提出那無窮小的否定的可能性來討論…

無窮小量只是一個函數f，而且有f(0)=0,f在x=0連續（連續性以epsilon-delta定義）

如果相信我們的手機 請輸入 1÷3+1÷3+1÷3= 我的手機告訴我是1 所以0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999=1

給任意不為0的整數n 複數(0.6 + 0.8•i)的n次方 與 複數平面上(1,0)這點的距離 永遠不會等於0 但 也永遠找不到最小值

其实我感觉 我们对数字的定义太狭隘了，可以把0.99循环 看成是一个动数，一个在数轴上运动的数。而1是一个静态数，在数轴上不运动。这样比较0.99循环 和 1就不应该比较。没有意义。 动数比较动数更有意义。

7:24 戴德金分割不是講過了嗎？ 5/3 的「一個數字竟會引起一場數學血案？直接爆發了第一次數學危機 | 雅桑了嗎」 11:04 的地方（傳送門： https://youtu.be/xZL5qDyAeWE?t=664 ） 突然發現 5/9 的「由芝諾的烏龜引出無窮小量等於0嗎？給牛頓擦屁股的第二次數學危機 | 雅桑了嗎」 0:49 的地方好像有先挖了畢氏學派 1 > 0.999... 的坑 （ https://youtu.be/Ox3ow_xMhCY?t=49 ）

分割Q（有理數）得到兩個區域 A|B及C|D B左邊端點對應的數=0.9循環，D則為1 則A{X|X屬於Q,且X<0.9循環} 則C{Y|Y屬於Q,且Y<1} （今想證明0.9...=1，即證明A集合=C集合） 在A中取一元素m，已知m<0.9....則m<1（必定）因此A是C的子集 在C中取一元素n，已知n<1又n屬於有理數，令n=p/q<1,所以pq>>>1/10^k<1/q 那麼1-n > 1/q > 1/10^k n<1-1/10^k. n<0.9.....9（小數點後面k個九，不是無限小數） 得n<0.99.......（這是無限循環的） 即C是A的子集 A屬於C,C屬於A，得證A集合=C集合 故1=0.9循環

談到無窮小量, 令我想起當年網遊的RO內的"零偷"理論一樣, 系統數位有限, 但存在發生的概率有可能發生的話, 系統會自動進位, 從而令概率增加成為事實. 而現實中會存在嗎? 物理存在很多最少的, 不可分割量. 這應該是個很好的議題.

我們教授曾說過一句話:「我們這些做數學研究的，就是把不該放進去的東西給強硬的塞進去，大家說1+1=2，數學家會思考三角形加正方形等於什麼」

數學就是根據一些公理建立的框架，如果公理變了，那麼框架自然就變了，用框架得出的結論自然也就可能改變

宇宙最小尺度也有類似的感覺，物理上可能有那個最小的尺度，但數學可能沒有