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中華民國第四十六屆中小學科學展覽會作品說明書國中組數學科 0010 有 " 機 " 可呈 ~ 揭開無理數的神秘面紗學校名稱 : 彰化縣立和美國民中學作者 : 國二邱太乙指導老師 : 粘憲昌國三姚宏治國三柯奕帆國二楊政龍關鍵詞 : 無理數亂數與隨機實驗解析幾何

率不率來率行不 EXCEL 亂數落了 10 9 亂數數數.116070 理數 1.16008 1.599866 1.189180 1.71951 1.608 1.160 度度 1 r 立 r r π 索 1. 數 Sinθ. 數. 理數度 1

六便年了裡率不率來率了便了了了 60cm60cm 見不易了 18cm 數落數 500 行 500 數 (,) (07,9)(9,108)(66,1)(79,11) 論了率 π 600 落率 600 15 數 000 數 07+9+66+79=15 落率 000 率理論來數落率落率 π 15 8.577778 600 000? 數率來率行不不率來率率來理數

參率來率行了率料數數 (http://www.math.tku. edu.tw/chinese/index.htm) 率率率臘來數拉率歷說來流度量了率利來利量數盧了數不類了利利數連來切數數

tan 1 x = x x + x 5 5 x 7 7 + 利數數年利切數數數利切數數數年數了不更錄年倫數年數年年錄了度數度數留數錄數例數數年利數率兩切劉率數數率利數異兩識不更不理易易來不 EXCEL

了更行陸了數數來數了數不落領不數便不了率理論不落了率兩數兩了不落了 5cm50cm 10cm 行落便了六數 (,) (17,71)(56,68)(80,668)(5,66) (15,680)(98,71) 100π 17 + 56 + 80 + 5 + 15 + 98 1750 17 + 71+ 56 + 68 + 80 + 668 + 5 + 66 + 15 + 680 + 98 + 71 166 = 767.57071 了不不落亂數立落靈不來數錄亂數 5

亂數來 EXCEL 來 EXCEL 數 RAND 亂數亂數 0 1 便亂數 AVERAGE 數 SUM 數 COUNTIF 數 PERCENTILE 數列料 K 數列行 ( 見 ()) 1 RAND 令亂數數 (Xi,Yi) 0<Xi<10<Yi<1 數 (Xi,Yi) 落量度數 100000 i 1 i 100000 數 (Xi,Yi)() 離 COUNTIF 令落 A 數數 SUMAVERAGEPERCENTILE 令率數來數數 10000 100000 6

亂數數了 10 9 亂數數數 50% 落 5 數 75 數數 K(a,b)a K 數數列 5 數度 b-a b K 數數列 75 數若度數數數數行數數度了 10 5 10000 料 10 5 (.18,.1500 ) 10 6 1000 料 10 6 (.1050,.1716 ) 10 7 100 料 10 7 ( 10 8 10 料 10 8 ( 度 10 9 亂數數數.116070 數數了率理數不理數來列行來 AC = ( 見 ()) 0 A C B 1 RAND 令亂數 (Xi) 0<Xi< 數 (X) 落數 AB 量度數 100000 i 1 i 100000 7

數 (Xi) COUNTIF 令落 AC 數 5 數 SUMAVERAGE PERCENTILE 令 1000 了 10 8 亂數數 10 5 10 6 10 7 10 5 (,)10 6 (,) 10 7 (,) 度 1.16008 數數理數 10 5 (1.57855, 1.616)10 6 (1.59191, 1.6057) 10 7 (, ) 度 0.0005,0.00181, 0.0007 1.599866 數數 10 5 (,)10 6 (, ) 10 7 (, ) 度,, 1.189180 數數 10 5 (,)10 6 (,) 10 7 (,) 度,, 1.71951 數數 10 5 (,)10 6 (,) 10 7 (,) 度,, 1.608 數數 8

10 5 (,)10 6 (,) 10 7 (,) 度,, 1.160 數數陸率了率亂數數 K( 5 數, 75 數 ), 來 K 50% 率落數數數數了 10 9 亂數數數.116070 理數 1.16008 數數 1.599866 數數 1.189180 數數 1.71951 數數 1.608 數數 1.160 數數不理數利數數利不理數臨來 EXCEL 度若力來 9

論理數了拉若索理數了理數索了更理數見理數數 log 六數數 e ( 度 ) 理數更 1 不零數 m nm + ne 理數不 e, e, ( 理數兩理數理數 1 =6 1 = 理數不不靈來來行了理數 =m ( m 數 ) 數來便了 ( x m) a ( y n) b + =1 10

C(0,b) A(-a,0) O(m,n) B(a,0) D(0,-b) O AB 度 acd 度 b ( x m) a ( y n) b + 1 列行 ( 見 ()) 1 RAND 令亂數數 (Xi,Yi) 0<Xi<10<Yi<1 數 (Xi,Yi) 落量度數 100000 i 1 i 100000 數 (Xi,Yi)() 離 * Xi +Yi COUNTIF 令落亂數數數 SUMAVERAGEPERCENTILE 令亂數數數不 (100000 亂數 ) 落亂數數落亂數數落亂數數略略 1 1 XX=1 X X 數 0X- 0 X 11

C(0,1) B(1,1) O(0,0) D(1/,0) A(1,0) 了 10 9 亂數數 X- X 1 105 10 6 10 7 10 8 10 5 (,)10 6 (,) 10 7 (,)10 8 (,) 度 0.00017 X- X 1 數 0 說 1 π, 若 OC OA OC OD 1 1 度度來 1

C O(0,0) O D A E OD OA OE =::6 OD DA AE =:1: OD DA AE =:1: :1: 類了 *10 8 亂數累數亂數落亂數累數數 10 5 618 105 969 10 6 616 19757 96 10 7 618518 10778 96959 10 8 *10 8 55516 度度 1 立 1 度 1 1 度切立 D D 理數 1

D 立行亂數落落亂數數亂數數 XY 亂數 (Z 數 )(,, )(,, )(,, ) 亂數立 Z (0,1,0) 亂數 ( 見 ()) 立 A=(X,Y,Z)0X10Y10Z B=(X,Y,Z)(X,Y,Z) A X Y 1 C=(X,Y,Z)(X,Y,Z) A X Y ( Z 1) D=(X,Y,Z)(X,Y,Z) A X Y Z 1 1 0 了 *10 8 亂數落累數亂數落亂數累數亂數數數 10 5 7861 576 611 10 6 785817 517 61988 10 7 78510 588 618006 10 8 785156 558167 6178781 *10 8 11576 095976 10776765 1

令落立亂數數切 1 若立立 r π 立 r π 切 r π r π 理數率 r r π 理數率 1 若來亂數亂數兩 θ θ 若落數 Sinθ 10 0 =110 1 =1010 x = x 亂數 x10 x 數來便數率理度率什理數度論 EXCEL 亂數行理數論 15

理數.116070 數數 1.16008 數數 1.599866 數數 1.189180 數數 1.71951 數數 1.608 數數 1.160 數數度 1 r 立 r r π 1. 數 Sinθ. 數. 理數度參料參料數數拉理數 http://zh.wikipedia.org/wiki/%e6%97%a0%e7%90%86%e6%95%b0 16

亂數 A=(X,Y)0X10Y1 (X,Y)(X,Y) A X Y 1 A=X0X XX A X A=X0X XX A X A=X0X XX A X A=X0X XX A X A=X0X XX A X A=X0X XX A X A=(X,Y)0X10Y1 (X,Y)(X,Y) A X Y 1 A=(X,Y,Z)0X10Y10Z (X,Y,Z)(X,Y,Z) A X Y 1 A=(X,Y,Z)0X10Y10Z (X,Y,Z)(X,Y,Z) A X A=(X,Y,Z)0X10Y10Z (X,Y,Z)(X,Y,Z) A X Y Y ( Z 1) Z 1 1 0 EXCEL 亂數數 A B C D E =rand() =rand() =A1^+B1^ =countif(d1:d100000, <1 ) =rand()* =A1^ =countif(d1:d100000, < ) =rand()* =A1^ =countif(d1:d100000, < ) =rand()* =A1^ =countif(d1:d100000, < ) =rand()* =A1^ =countif(d1:d100000, < ) =rand()* =A1^ =countif(d1:d100000, < ) =rand()* =A1^ =countif(d1:d100000, < ) =rand() =rand() =*A1^+B1^ =countif(d1:d100000, <1 ) =rand() =rand() =rand()* =A1^+B1^ =countif(d1:d100000, <1 ) =rand() =rand() =rand()* = A1^+B1^ +(C1-1)^ =countif(d1:d100000, <1 ) =rand() =rand() =rand()* = A1^+B1^ -(1-C1/)^ =countif(d1:d100000, <0 ) 17

評語 0010 有機可呈 ~ 揭開無理數的神秘面紗利用幾何機率的觀點來估算無理數的近似值, 作為數學教學的題材或許不錯, 然而若以此為科展的主題, 內容上其實沒有太多新的想法, 且數學性稍嫌不足