Quod Erat Demonstrandum

2013/06/27

[YouTube] Vertex centered 24-Cell

Filed under: Fun — johnmayhk @ 11:14 上午

2013/06/26

藤球

Filed under: Fun — johnmayhk @ 10:50 上午

johnmayhk-Sepak-Takraw-Ball1

藤球起源於西元15世紀 (more…)

2013/06/25

[YouTube] Is Math a Feature of the Universe or a Feature of Human Creation?

Filed under: Fun — johnmayhk @ 11:55 上午

數學之神啊,數學是研究…數學的一科,重言即是重言,阿門。

2013/06/24

Fun site: Archimedes’ Laboratory

Filed under: Fun — johnmayhk @ 10:53 上午

johnmayhk-magic-math-tricks

要練習

More at
http://www.archimedes-lab.org/index.html

[TED] 用溼軟電路動手做科學

Filed under: Fun,Physics — johnmayhk @ 10:37 上午

2013/06/20

just-a-q

Filed under: NSS — johnmayhk @ 3:10 下午
Tags:

Just set a so-called senior form core mathematics question:

johnmayhk-hexgonal-numbers

Q.1 Find the general term T_n.

Solution (more…)

2013/06/19

CH

Filed under: NSS,Pure Mathematics — johnmayhk @ 3:30 下午
Tags:

偶爾找回 2006 年擬的一份中六考卷,

johnmayhk-ch

當時我寫:

“Just for your reference, the result follows immediately by applying Cayley–Hamilton theorem; students, you may study it if you are interested."

不知那時有沒有學生真的 (more…)

2013/06/18

小遊戲

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 12:47 下午

網上遊戲盛行,幸好仍有學生喜歡在現實中和朋友一起玩遊戲。學期初,一些中四學生偶爾找以下遊戲向我「挑機」(點擊下圖玩玩吧。玩法:有三堆「包子」,每次在同一堆包子取去若干個,取到最後一個包子者勝。)

那是古老的數學遊戲「捻」(Nim)(注:不一定用 3 堆包子,但為簡單起見,我以 3 堆為例)可視為二進制「應用」之例,隨便談談。

設 3 堆包子,分別有 7,6,2 個包,見下:

好了,以數學人眼光看,可能會看到「色彩」如下:

其實 (more…)

2013/06/17

無聊寫篇

Filed under: mathematics — johnmayhk @ 4:06 下午

(只是草稿,不必認真)

《大象無形》

我們身處的「世界」是甚麼形狀? (more…)

2013/06/16

錯在哪裡之 1 不大於 1/2

Filed under: Fun — johnmayhk @ 11:12 下午

易知

1

=\frac{1\times 2}{2}

=\frac{1\times 2\times 3}{2\times 3}

=\frac{1\times 2\times 3\times 4\times\dots}{2\times 3\times 4\times 5\times\dots}

\frac{1\times 2\times 3\times 4\times\dots}{2\times 3\times 4\times 5\times\dots}

=\frac{1}{2}(\frac{2}{3})(\frac{3}{4})(\frac{4}{5})\dots

\le \frac{1}{2}

1\le \frac{1}{2}

剪報:中文數學

Filed under: Fun — johnmayhk @ 7:31 下午
Tags:

偶爾找到《東方日報》副刊「豪筆起」的幾篇,作者歐陽偉豪。短文應和中文數學用語有關,貼來參考。

2013-05-24
johnmayhk-by01 (more…)

f4 m2 revision

Filed under: NSS — johnmayhk @ 6:39 下午
Tags:

(免插聲明:純為中四同學溫習用,高手見諒。)

同學應該頗掌握乘冪法則(Power rule):

\frac{d}{dx}(x^n)=nx^{n-1}

留意,x^n 當中的基(base)是變量 x,而冪(power)是常數 n

一旦遇上冪或/和基是變量時,比如

\frac{d}{dx}(n^x)

\frac{d}{dx}((\ln x)^x)

就不能以乘冪法則處理。

記著一招: (more…)

2013/06/15

平行

Filed under: Fun — johnmayhk @ 2:03 下午

Parallel lines are usually defined as lines with no points in common. Parallelism is clearly symmetric. If line 1 has no points in common with line 2, then line 2 also has no points in common with line 1. Is parallelism reflexive? In other words, is a line parallel to itself? This appears to be a matter of convention. Since the advantage of a positive answer far outweighs the alternative, let’s modify the definition of parallel lines to be lines which do not have exactly one point in common. Finally, is parallelism transitive? Suppose line 1 is parallel to line 2 and line 2 is parallel to line 3, but line 3 is not parallel to line 1. Then line 1 and line 3 intersect at exactly one point P, which cannot be on line 2. Otherwise, being parallel to both line 1 and line 3, line 2 cannot have only one point in common with them and must coincide with both. However, line 1 and line 3, not being parallel to each other, are distinct. Through a point P not on line 2, we now have two lines parallel to line 2, which contradicts Playfair’s Axiom. So parallelism is transitive and, in turn, this implies Playfair’s Axiom. Let P be a point not on line 2. Suppose both line 1 and line 3 pass through P and are parallel to line 2. By transitivity, they are parallel to each other, and hence they cannot have exactly P in common. It follows that they are the same line, which is Playfair’s Axiom. Thus we have possibly the shortest statement of the parallel axiom. (Three words!) “Parallelism is transitive." The five-word version is: “Parallelism is an equivalence relation," and the answer to our question is: “Yes, if the geometry is Euclidean."

(Andy Liu, University of Alberta, from the College Mathematics Journal, Volume 42, Number 5, November 2011, p. 372)

2013/06/06

[FW][YouTube] 120-cell

Filed under: Fun — johnmayhk @ 12:09 下午

Beautiful!

2013/06/05

[FW] One of the most abstract fields in math finds application in the ‘real’ world

Filed under: Fun,Report — johnmayhk @ 2:07 下午

最抽象的數學在現實生活中的應用!Category theory (竟)可被應用在電腦,量子物理,音樂,物流…

見 ScienceNews

http://www.sciencenews.org/view/generic/id/350567/description/One_of_the_most_abstract_fields_in_math_finds_application_in_the_real_world

Introduction to Category Theory 1: Course Overview
https://johnmayhk.wordpress.com/2013/01/05/ct/

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