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4 CHAPTER 0. http://www.jaist.ac.jp/~t-yama/k116 0.3 50% ( Wikipedia ) ( ) 0.4! 2006

0.4. 5 MIT OCW ( ) MIT Open Courseware MIT (Massachusetts Institute of Technology) Open Courseware MIT Creative Commons Lisence MIT OCW http://ocw.mit.edu (transcription)

Chapter 1 1 1 1m 1 1 2 1.1 ( 1 ) 1 60 2 7

8 CHAPTER 1. 0-1/3=0.3333... π,e, 2 = 1.4142135..., 3 = 1.7320508.. i = 1 ( ) ( ) 2 : 1 + 2i ( ) ( )

1.1. 9 1.1.1 1 ( ) Figure 1.1: = 1 1/10 π = 3.141592... 3.1 1/10 π 7 a < b( a 0) a < b 6 < 7 < 8 6 = 2 3 8 = 2 2 2 = 2 2 ( 2 2 = 2 2 = 2) 2 = 1.41 3 = 1.73 6 = 2.44, 8 = 2.82 7 7 [ 6, 8] = [2.44, 2.82] 2.44 2 = 6 2 0 < a < b a 2 < b 2 2.6 2 = 6.76, 2.7 2 = 7.29 2.65 2.67 2 7

10 CHAPTER 1. octave-3.0.0:16> 2.65^2 ans = 7.0225 octave-3.0.0:17> 2.62^2 ans = 6.8644 octave-3.0.0:18> 2.63^2 ans = 6.9169 octave-3.0.0:19> 2.64^2 ans = 6.9696 octave-3.0.0:20> octave-3.0.0:20> octave-3.0.0:20> 2.65^2 ans = 7.0225 octave-3.0.0:21> 2.63^2 ans = 6.9169 octave-3.0.0:22> 2.64^2 ans = 6.9696 octave-3.0.0:23> 2.645^2 ans = 6.9960 octave-3.0.0:24> 2.648^2 ans = 7.0119 octave-3.0.0:25> 2.646^2 ans = 7.0013 octave-3.0.0:26> 2.6455^2 ans = 6.9987 octave-3.0.0:27> 2.6458^2 ans = 7.0003 octave-3.0.0:28> 2.6457^2 ans = 6.9997 octave-3.0.0:29> 2.6458^2 ans = 7.0003 3 2.646 PC Octave

1.1. 11 1.1.2 Octave Octave Octave octave-3.0.0:> (3+4+5)/100 ans = 0.12000 octave-3.0.0:> (3+4)+5/100 ans = 7.0500 octave-3.0.0:> sqrt(3) ans = 1.7321 () sqrt(x) x 1. 5 2. 3. 4. 10 3

12 CHAPTER 1. 1.1.3 (number theory) 3 n x n + y n = z n (x, y, z) (n = 2 3 2 + 4 2 = 5 2 ) 2000 :Simon Singh, Fermat s Enigma: The Epic Quest to Solve the World s Greatest Mathematical Problem Anchor Books 1998 (paperback) 2008 : David Wells Prime Numbers The Most Mysterious Figures in Math John Wiley & Sons 2005 ( ) :Simon Singh, The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography Anchor Books 2000 (paperback)

1.1. 13 1.1.4 a b c d 1. a = b, a = c b = c 2. a = b, c = d a + c = b + d 3. a = b, c = d a c = b d ( ) ( ) 4. a = b, c = d ac = bd ( ) 5. a = b, c = d c 0 a c = b d 6. a + (b + c) = (a + b) + c 7. a (b c) = (a b) c 8. a (b + c) = a b + a c 1. 2 2. 2

14 CHAPTER 1. 3. (a + b) 2 4. 1 ax + b = 0 x 5. () 0

1.2. 15 1.2 1. 1.2.1 1 2 2 2 3 5 ( 5 ) 3 1l 2

16 CHAPTER 1. 1.3 1.3.1 m kg ( ) s A Table 1.1: 4 SI ( MKSA ) k (1000 ) g( ) SI 4 ([K]) ([mol]) ([cd]) [ ] ( ) SI N m kg/s 2 Pa N/m 2 m 1 kg /s 2 J N m m 2 kg/s 2 J N m m 2 kg/s 2 J N m m 2 kg/s 2 W J/s m 2 kg/s 2 V W/A m 2 kg /(s 3 A 1 ) Table 1.2: SI ( ) 1.3 (9.8 [m/s 2 ])

1.3. 17 / km/h / m/s m 2 bar 10 5 Pa kgw 9.8 N 9.8m kg/s 2 Table 1.3: ( ) P 10 15 peta T 10 12 tera G 10 9 giga M 10 6 mega k 10 3 kilo h 10 2 hecto da 10 1 deca d 10 1 deci c 10 2 centi m 10 3 mili µ 10 6 micro n 10 9 nano p 10 12 pico f 10 15 femto Table 1.4: SI ( ) hpa ( ) h mbar ( ) SI 1 1013 hpa

18 CHAPTER 1. 1.3.2 ( ) 100% 1.3.3 1[g] 100[kg] 100001[g] ( ) 100[kg] 0.1[kg] 99.9[kg] 100.1[kg] 1[g] 1.000 10 2 [kg] 1 10 4 9.9995 10 1 = 1.000 10 2 = 1.0004 10 2 5 1.3.4 SI ( 1.2) ( = x ) F = m a = kg m/s 2 ( 1.2 ) = (kg) x (m/s 2 ) = kg m/s 2 =

1.4. 1 19 E = mc 2 1.0 10 3 [hpa] 10[cm] 1.4 1 3 1. 2 = 1.414, 3 = 1.732, 5 = 2.236 10 30 ( 10, 11,... 30) 3 25 = 5 12 = 2 3 ( : ) 2. 3. π 4. SI 5. 2 ax 2 + bx + c = 0 x pp.13 ( )

20 CHAPTER 1. 1.5 (= 1 ) xx 200m (= ) () ( ) (1,3), (-1,1), ( 3/2,1/2), (x,y) 2 (1,3,2), (-1,0,1), ( 3/2,1/2, 2), (x,y,z) 3 () 2 2 Figure 1.2:

1.5. 21 2 ( ) (a,b) a 2 + b 2 (a,b,c) a 2 + b 2 + c 2 1 1 e x = (1, 0) (1.1) e y = (0, 1) (1.2) 3 3 x, a, b x, a, b 1. 3 3 1.5.1 X = (x 1, x 2 ) = (3, 2) X = ( x 1 x 2 ) ( ) 3 = 2 Octave

22 CHAPTER 1. octave:> A=[1,2,3] A = 1 2 3 octave:> A=[1;2;3] A = 1 2 3 octave:> (,) (;) 1.5.2 ( ) 1.5.3 (cf. ) 1. 2.

1.6. 23 1.6 1.6.1 ( 1.3) (x 1, x 2 ) + (y 1, y 2 ) = (x 1 + y 1, x 2 + y 2 ) (1.3) (x 1, x 2 ) (y 1, y 2 ) = (x 1 y 1, x 2 y 2 ) (1.4) 2 (x 1, x 2 ) ± (y 1, y 2 ) = (x 1 ± y 1, x 2 ± y 2 ) (1.5) Figure 1.3: 1. 3 Octave 2. 2 A=(1;3;-1) B=(2;5;1)

24 CHAPTER 1. 1.6.2 2 2 2 1.6.3 m (a 1, a 2 ) = (m a 1, m a 2 ) (1.6) m ( ) 1.6.4 2 2 2 (x 1, x 2 ) (y 1, y 2 ) = x 1 y 1 + x 2 y 2 (1.7) 2 x = (x 1, x 2,...x n ), y = (y 1, y 2,...y n ) n x y = x i y i (1.8) i=1 ( x ) x y = x y cos θ (1.9) x, y 2 cos θ = x y x y (1.10) cos(π/2) = 0 ( )2 0

1.6. 25 Octave Octave 2 A,B A*B (JIS Shift+7) ctave-3.0.0:38> X=[1;2;3;4] X = 1 2 3 4 octave-3.0.0:39> Y=[9;8;7;6] Y = 9 8 7 6 octave-3.0.0:40> X*Y error: operator *: nonconformant arguments (op1 is 4x1, op2 is 4x1) error: evaluating binary operator * near line 40, column 2 # octave-3.0.0:40> X ans = 1 2 3 4 # octave-3.0.0:41> X *Y ans = 70

26 CHAPTER 1. 1. 2. ( ) a b = b a (a + b) c = a c + b c 2 3 x = (x 1, x 2, x 3 ), y = (y 1, y 2, y 3 ) θ z = x y (1.11) z = (x 2 y 3 x 3 y 2, x 3 y 1 x 1 y 3, x 1 y 2 x 2 y 1 ) (1.12) z = x y sin θ (1.13) 1.7 ( ) ( ) a b c 2 1, d e f 3 1 2 3 (2x3) 2 2 (2x2) (, ) (1,2) b 1

1.7. 27 A 3 1 3 2 A = 1 2 1 0 (1.14) 3 5 4 7 1. A 2. A (1,4) (3,2) 3. A 2 4. A 3 1.7.1 Octave Octave (,) (;) ( ) (, ) A (2,1) A(2,1) (:) A n A(:,n) m A(m,:) A,m 1..3 A(m,1:3) (:) Octave 1:n 1...n... octave:1> A=[3,1,-3,2;1,2,1,0; 2,5,4,7] A = 3 1-3 2 1 2 1 0 2 5 4 7 octave:2> A(1,4)

28 CHAPTER 1. ans = 2 octave:3> A(3,2) ans = 5 octave:4> A(:,2) ans = 1 2 5 octave:5> A(2,:) ans = 1 2 1 0 octave:6> A(3,:) ans = 2 5 4 7 1.7.2 A,B Octave 3 1 3 2 1 1 3 0 A = 1 2 1 0, B = 1 1 4 5 (1.15) 3 5 4 7 2 1 1 2 1. A+B 2. A-B 3. 2A-B 4. t A ( )

1.7. 29 5. t A+ t B ( ) A B AB A B ( ) t t (transpose) Octave ( : A ) shift-7 shift-@ ( ) a a b c A =, t A = b d e f c d e (1.16) f A B A tb (1.17) A(m n ) B(n p ) C C (m p ) c ij = n a ik b kj, i = 1 m, j = 1 p (1.18) k=1 a ij, b ij, c ij A,B,C (i j ) Figure 1.4:

30 CHAPTER 1. A,B A ta A tb Octave Octave.*./ ( ) N, P Octave:> N = [1,2,10; 3,0, 5; 4, 0, 1]; Octave:> P = [ 30, 20, 99; 40, 0, 112; 25, 0, 80]; octave:> Q = N.* P #.* Q = 30 40 990 120 0 560 100 0 80 octave:> sum(q) ans = 250 40 1630 # ( ) octave:> sum(q ) ans = 1060 680 180 # ( ) octave-3.0.0:80> sum(sum(q)) ans = 1920

1.7. 31 # 1.7.3 1 3 ( ) 1 0 E 1 0 1 E =... 0 1 (1.19) 0 4x1 m 1 0 0 0 0 1 0 0 0 0 m 0 0 0 0 1 (1.20) 1.7.4 2 1( ) 5 1/5 x 1/x = x 1 0 1 0 A AX = E (1.21) X A 1 Octave inv(a) 0 A 3

32 CHAPTER 1. A 0 A Octave det(a) 1. 3 1 3 1 1 3 A = 1 2 1, B = 1 1 4 3 5 4 2 1 1 2. 3. 2x2 X = ( a c ) b d det(x) = ad-bc R = ( cos θ sin θ ) sin θ cos θ 1.7.5 Ax = b (1.22) a b c x j d e f y = k (1.23) g h i z l ax + by + cz = j (1.24) dx + ey + fz = k (1.25) gx + hy + iz = l (1.26)

1.8. 2 33 A A 1 Ax = A 1 b (1.27) A A 1 = I x = A 1 b (1.28) x 2 x 2 1.8 2 3 1. x 1 OAB Figure 1.5: OAB OA, OB, AB, BA ( ) 4 2.

34 CHAPTER 1. (a) ( ) ( ) ( ) 1 1 x 9 = 4 2 y 24 (b) 9 24 (c) 3. 4. () 5. () 6. ( Excel)

1.9. 35 1.9 1 1. θ ( ) cos θ sin θ sin θ cos θ (1.29) 2. a ( ) a 0 0 a function y=rot(x) y = [cos(x),-sin(x);sin(x),cos(x)]; endfunction x0=[1;0]; step=0.1*pi; for(i=1:21) x0=rot(step)*x0; X(i,:)=x0 ; endfor plot(x(:,1),x(:,2), + ); (1.30) 1.9.1 Octave function y=rot(x) x = x*pi; y = [cos(x),-sin(x);sin(x),cos(x)];

36 CHAPTER 1. endfunction function T=trans(x,y) for(i=1:5) T(i,:)=[x,y]; endfor endfunction function plotr(x) plot(x(:,1),x(:,2)) endfunction function Y=rotation(X,a) Y=(rot(a)*X ) ; endfunction Bigbox=[-3,-3;5,-3;5,4;-3,4;-3,-3]; hold off; plotr(bigbox); hold on; R1 = [0,0;2,0;2,1;0,1;0,0]; (5,2) R1 R2 = rotation(r1, θ) R1 θ[rad] R2 (plot ) plotr(r1) Octave:> plotr(r1) Octave:> R2 = rot(0.3)*r1; Octave:> plotr(r2) Octave:> R3 = trans(1,2)+r1; Octave:> plotr(r3) Octave:> for(u=0.1:0.3:1.0) plotr(rotation(r3,u)) endfor