e

ME Too.......

 

9999999999999999999999999999999999999999999999

 

What?

 

e=2.71828182845...

 

本帖最後由 Mathematics tutor 1 於 2015-7-19 21:59 編輯

自然對數的底數e是個無理數，就是說它不能被表示成有限小數，也不能被表示成循環小數。e的前1000位小數是

2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952605956307381323286279434907632338298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537423454424371075390777449920695517027618386062613313845830007520449338265602976067371132007093287091274437470472306969772093101416928368190255151086574637721112523897844250569536967707854499699679468644549059879316368892300987931277361782154249992295763514822082698951936680331825288693984964651058209392398294887933203625094431173012381970684161403970198376793206832823764648042953118023287825098194558153017567173613320698112509961818815930416903515988885193458072738667385894228792284998920868058257492796104841984443634632449684875602336248270419786232090021609902353043699418491463140934317381436405462531520961836908887070167683964243781405927145635490613031072085103837505101157477041718986106873969655212671546889570350354

有一款著名的符號代數運算軟件，它是開源免費的，叫做maxima。上面的數字是用這個軟件計算出來的。命令是
fpprec:2000;
fpprintprec:1001;
bfloat(exp(1));

maxima的維基百科頁面是https://zh.wikipedia.org/wiki/Maxima
你可以調整上面的命令來得到更多位的e的值。第一個參數fpprec是計算精度，第二個參數ffprintprec設定顯示數位。例如你可以設定頭一個參數是10100，第二個參數是10001，來得到e的首10000位小數。當然，因爲這個數字是無限不循環小數，你無法期望得到它的循環節。

如果你升入大學，學習了微積分(calculus)，你甚至只需要十幾分鐘，一張白紙和一隻鉛筆，就可以輕鬆算出e的首10位小數。你需要用到的數學工具叫做Taylor series.你平常使用的科學計算器，它們計算sin, cos, tan, e^,等等這些函數的時候，也用到了這種數學工具，所以你也可以使用鉛筆和白紙，來計算這些函數。你可以查看維基百科的頁面來瞭解這些方法的細節。如果你讀了微積分，你就能明白爲什麼使用這些方法，只消使用加減乘除，就能算出這些複雜的函數的值。

--by Guo Xin

 

so what are we going to do with the Taylor series and what are the steps to do it with calculus?

 

本帖最後由 Mathematics tutor 1 於 2015-8-3 13:51 編輯

回覆 6# Au Yeung Yat


It usually takes three whole semesters to study the details of what you asked. To make the story short, you may take a look at the wikipedia page https://en.wikipedia.org/wiki/Taylor_series to have some feeling on the Taylor series, and to fetch some useful formulas for computing the values of some basic functions like exponential, sine, tangent, or arcsine. There is an online handbook http://www.math.hkbu.edu.hk/support/aands/page_880.htm which you may also feel useful. If you want to find more details, a general university textbook, like http://www-math.mit.edu/~djk/calculus_beginners/ or http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf would definitely help you.

-- by Guo Xin