周易《繫辭上》

一陰一陽之謂道，繼之者善也，成之者性也。仁者見之謂之仁，知者見之謂之知。百姓日用而不知，故君子之道鮮矣。顯諸仁，藏諸用，鼓萬物而不與聖人同懮，盛德大業至矣哉。富有之謂大業，日新之謂盛德。生生之謂易，成象之謂乾，效法之為坤，極數知來之謂占，通變之謂事，陰陽不測之謂神。

若問易經能解『夏農熵』嗎？

《易經》是中國最古老的文獻之一，並被儒家尊為「五經」之首；一般說上古三大奇書包括《黃帝內經》、《易經》、《山海經》，但它們成書都較晚。《易經》以一套符號系統來描述狀態的簡易、變易、不易，表現了中國古典文化的哲學和宇宙觀。它的中心思想，是以陰陽的交替變化描述世間萬物。《易經》最初用於占卜，但它的影響遍及中國的哲學、宗教、醫學、天文、算術、文學、音樂、藝術、軍事和武術等各方面。自從17世紀開始，《易經》也被介紹到西方。在四庫全書中為經部，十三經中未經秦始皇焚書之害，它是最早哲學書。

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『三畫』則八卦成列，『六爻』現六十四卦圖象，此『三畫六爻』儼然是『比特值』耶！所謂『陰陽不測之謂神』，實寫『真隨機』乎！！？？

Randomness

Randomness is the lack of pattern or predictability in events.^[1] A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or “trials”) is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.

Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science.^[2] By analogy, quasi-Monte Carlo methods use quasirandom number generators.

RandomBitmap

A pseudorandomly generated bitmap.

今人尚且難以捉摸 $\pi$ 是否為『正規數』

Normal number

In mathematics, a normal number is a real number whose infinite sequence of digits in every base b^[1] is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b² pairs of digits are equally likely with density b⁻², all b³ triplets of digits equally likely with density b⁻³, etc.

Intuitively this means that no digit, or (finite) combination of digits, occurs more frequently than any other, and this is true whether the number is written in base 10, binary, or any other base. A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is “favored”.

While a general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal. For example, Chaitin’s constant is normal. It is widely believed that the numbers √2, π, and e are normal, but a proof remains elusive.

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$\pi$ 之訝異公式已現身︰

貝利-波爾溫-普勞夫公式

貝利-波爾溫-普勞夫公式（BBP公式）提供了一個計算圓周率π的第n位二進位數的spigot算法（spigot algorithm）。這個求和公式是在1995年由西蒙·普勞夫提出的，並以公布這個公式的論文作者大衛 ·貝利（David H. Bailey）、皮特·波爾溫（Peter Borwein）和普勞夫的名字命名。在論文發表之前，普勞夫已將此公式在他的網站上公布^[1]^[2]。這個公式是：

\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right]

這個公式的發現曾震驚學界。數百年來，求出π的第n位小數而不求出它的前n-1位曾被認為是不可能的。

自從這個發現以來，發現了更多的無理數常數的類似公式，它們都有一個類似的形式：

\alpha = \sum_{k = 0}^{\infty}\left[ \frac{1}{b^k} \frac{p(k)}{q(k)} \right]

其中α是目標常數，p和q是整係數多項式，b ≥ 2是整數的數制。

這種形式的公式被稱為BBP式公式（BBP-type formulas）^[3]。由特定的p,q和b可組合出一些著名的常數。但至今尚未找出一種系統的算法來尋找合適的組合，而已知的公式多是通過實驗數學得出的。

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此事是否能當真 ─── 十六進位制之 $\pi$ ！！？？

『The Quest for Pi』給證明︰

TheQuestForPi

……

BBP-Pi-1

BBP-Pi-2

※ 註

$\int \limits_0^1 \frac{4y}{y^2 - 2} dy = \int \limits_0^1 \frac{-4y}{2 - y^2} dy = 2 \ln (2 - y^2) \Big |_0^1$

$\int \limits_0^1 \frac{4y - 8}{y^2 - 2y + 2} dy = \int \limits_0^1 \frac{4y - 4}{y^2 - 2y + 2} dy - \int \limits_0^1 \frac{4}{{(y-1)}^2 + 1} dy = 2 \ln (y^2-2y + 2) \Big |_0^1 - 4 \arctan(y - 1) \Big |_0^1$

$\therefore \int \limits_0^1 \frac{4y}{y^2 - 2} dy - \int \limits_0^1 \frac{4y - 8}{y^2 - 2y + 2} dy = \pi$

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如是『亂數』是什麼乙事豈可不深思熟慮也？？？

『創世紀』第十三章『亞伯蘭』以起先『築壇的地方』分別了『左‧右』，讓『羅得』來『選擇』。這就是今天稱之為『一分‧一擇』 I cut, you choose 的『公平分享』規範。舉例說，一人以他所認為的『公平』切蛋糕，讓另一人先作『選擇』。這也就是 Bruno de Finetti 所講的︰由此方來設定輸贏『前提』之『賠率』和『賭注』，讓彼方決定購買『前提』之『正反方』一樣。

那麼 Bruno de Finetti 所堅持的『主觀機率論』是什麼呢？他認為『機率』就是一個人對某『事件』發生之『相信度』評估，這是由那個人的『知識』、『經驗』以及『資訊』等等來決定。比方講，假使問多個人『印象派大師莫內的生日是十一月十五號的機率是多少？』。不知道『莫內是誰』的，可能認為是 $\frac{1}{365}$ ；過去聽說

克勞德‧莫內

印象‧日出

自己跟『莫內同星座』的也許以為 $\frac{1}{30}$ ；還有一個上網『谷歌』 Google 的說『機率是零』。所謂的『客觀機率』真的是存在的嗎？因此 Bruno de Finetti 的論點，自有不可忽視的重要性，更由於『量子力學』的『量測理論』將『觀察者』放進了『不確定性』框架中，這個『主‧客觀』的爭論，目前勢將持續進行下去的吧！在此僅用『亂數產生器』的『擬似』 Pseudo 與『真實』 Real 之說來看，人們真的有『判準』來『區分』這兩者的嗎？比方講，現今所相信的『真實』之『亂數產生器』來自於那些『隨機性』的『物理現象』；常用之『擬似』的『亂數產生器』可以從某種『計算式』 $X_{n+1} = (a X_n + b) \ \textrm{mod} \ m$ 裡得到。雖然說『已知』之『演算法』在『夠長的產生序列』後，難免於『重複再現』，要是真存在一個『演算法』，它的『再現所需時間』是『千百億年』的呢？那我們能『發現』它是有『公式』的嗎？？