ii th-note
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1 4 I alpha α nu N ν beta B β i Ξ ξ gamma Γ γ omicron o delta δ pi Π π, ϖ epsilon E ϵ, ε rho P ρ, ϱ zeta Z ζ sigma Σ σ, ς eta H η tau T τ theta Θ θ, ϑ upsilon Υ υ iota I ι phi Φ ϕ, φ kappa K κ chi X χ lambda Λ λ psi Ψ ψ mu M µ omega Ω ω th-note th-note kutomi@collegium.or.jp FTEXT Ver Ver4.00 Ver4.00 Ver4.00 Ver4.00
2 ii th-note
3 th-note 4 th-note () () FTEXT FTEXT FTEXT TEX TEX Donald E. Knuth SCII Corporation th-note iii
4 L TEX emath th-note.. th-note 4 5. iv th-note
5 iii iv B 9 B B B th-note v
6 B C (tan) (cos) (sin) vi th-note
7 D D D D D th-note vii
8
9 ... *,,, 4, 5, (natural number) * th-note
10 B C IV5X0L50C00D500M000 * * 0,,,, 4, 5, 6, 7, 8, 9. VIII + XIII. XXII + XXVIII XVIIIIII VIIIII X XXI. XXXXVIIIII VIIIII X XXXXX L. 4.. D. 0 (integral number) 568,,, 0, 4, 57 E. (number line) X X a X(a) X X() X 4 5 * * (4.00) th-note
11 = 6 (ratio) 5 0 < < 5 5 a b a b B. (rational number) *4 8,, 0, 9, 8 9, 6 (reduction) (irreducible fraction) (p.99) = 7 5 = 5. 9 = 4 C *4 ratio rational number th-note.
12 D. 7 7 = 4 < 4 4 a b, a b = ad bd < ad bc ad+bc bd < 4 4 = ( c a d b < c d < bc bd = c d ) *5 6 5, = 48 00, 4 = E. (decimal number) 5 =.5 (finite decimal) 4 5 = (infinite decimal) = = (circulating decimal) 0.4 = = = = *6 000 = ) = = 46.5 = = = ) ) *5 (densit) *6 III 4 (4.00) th-note
13 4 () 9 () () 0.65 (4) () () = 0. 5 () 0.65 = = 5 8 (4) = = = ) = = 49 = = 4.. (irrational number) *7 *8 p.8 (p.8) B. (real number) *9 * 0 π , 5,, π =.4596, * e =.7888 ab *7 ir-rational ir irrational rational *8 (p.5) *9 *0 (continuit) III * e III th-note. 5
14 0 } 5,, 0, 5, 5,,. 5, 6 6, ( ) 6, 5, π () () () (4) 6 (), 6, ( ) = ( 6 ) 5 = 5 6 (),, 0, 6, ( ) 6, 5 6 = 4 (),, 0, 5, 5,. 5, 6 6, ( ) 6, 5. 5 = 5 99 (4), π (p.5 ) 4.. (a) a (absolute value) a * =, 4 = = ( ). + = + = 5. 5 = 8 = = ( ) 4. * a a 6 (4.00) th-note
15 7 5, 4, = 5 = 5, 4 = 4 = 5 0 = 5 0 = a = { a (a 0 ) a (a < 0 ) a a a 0, a = a B. (a) B(b) B B = b a b a a b b a a B a b b B b a 8 ( 4), B( ), C(), D(5) CD, BC, D, C CD = 5 =, BC = ( ) =, D = 5 ( 4) = 9, C = 4 = 6 = 6 9. = = =. + = + =. 5 =. + 5 = 0.7 < 0 = + 5 = ( + 5 ) = 5 C. 0. =. =. = 4. = 4 5. < 6.. =. = th-note. 7
16 . = 4. = 5. = ( ) 6. 0 = (), (4), (6) () = 5 () = () = a 4 a (4) = a < a < 4 ()= + = + = ()= + 4 = + 4 = 7 () 4 a a 4 0 a 4 = a 4 4 a a + 0 a + = a + = (a 4) + (a + ) = a (4) < a < 4 a 4 < 0 a 4 = (a 4) < a < 4 a + > 0 a + = a + = (a 4) + (a + ) = a + 6 a = 5 () a = () D. ab () b \= 0 () a = a () ab = a b () a b = a b p.84 () a = () a = b = 4 () a = 5b = = 9, ( ) = 9 ( ) 4 =, 4 = 5 = 5, 5 = 5 () () () 8 (4.00) th-note
17 (set) * (element) * U J M U J (Venn diagram) U U (universal set). M. J M. M 4. J M M., 5, 6, 7, 9. 7.,, 4, 8, 0 4., 4, 0 B. p.9 J 470 * 5 J = {, 4, 7, 0} * *4, B, C,, Y, Z a, b, c,,, z *5 (etensional definition) th-note. 9
18 C. J, M J M J M (sum of sets) J, M J M J M J, M (common part) J M = {,, 4, 5, 6, 7, 9, 0}, J M = {7} (empt set) * 6, B B = J M J M J M = {,, 4, 5, 8}B = {, 5, 7}C = {, 6} B, B, B C, B C B = {,,, 4, 5, 7, 8} {,, 4, 5, 8} {, 5, 7} B = {5} B {,, 4, 5, 8} {, 5, 7} B B C = {,, 5, 6, 7} B C B C = D. U J J p.9 J = {,, 5, 6, 8, 9} J (complement) U J M 4 U = {,,, 4, 5, 6, 7, 8, 9}.,. B B, B. = {, 4, 6, 8}, = {,, 5, 7, 9}. B = {, 6, 9} B = {6}, B = {, 9}. *6 0 0 (4.00) th-note
19 E. a X a X (in) p.9 J J J J * 7 J J,,, F., B B B (subset) B (contain) B B * 8 B, B,, * 9 B B (equal) = B \= B B B = B 5 {,,, 4} {,, }, {,, } {, }, {, } {,, 5}, {} {,,, 4} {,, }, {,, } {, }, {} 6 {,, } {,, }, {, }, {, }, {, }, {}, {}, {}, {,, } {,, } (p.0) 7 U, U = U = U = *7 *8 B B B *9 B B (proper subset) B B \= B th-note.
20 8 U = {,,, 4, 5, 6, 7, 8, 9, 0,, } 0 B () (), B, B () B, B, B 4, B, B, B U B () U B () = {5, 7, 8, 9, 0, } B = {,,, 4, 5, 6, 0, } B = {, } () B, B 4, B, B, B B, B B B, B B, G. B, B B B B B B H. B B 9 () B, B () B, B, B () ()() B B () B B B B B B () B B B B B (4.00) th-note
21 B B B B B B B B B B () () () B = B() () B = B B (law of de Morgan) B = B B = B 0 () B () U = { } n n = {,, 4, 8}, B = {,, 5, 7, 9} B, B, B, B () B B B B () B = {, 5, 7, 9} B = {,, 4, 6, 8} B = {} B = B = {,, 4, 5, 6, 7, 8, 9} B = {,,, 4, 5, 7, 8, 9} B = B = {6} B U B () B B = {, 5, 7, 9} {, 5, 6, 7, 9} {,, 5, 7, 9} B = {,, 4, 6, 8} {,, 4, 8} {, 4, 6, 8} th-note.
22 .. X = {,, 5, 7, 9} * 0 X = { } a a 0 = { { } a a 8 }, B = p p 0 X , B. 6, 6 B { }. Y = {,,, 4, 6, } =. = {,,, 6, 9, 8}B = {,, 5, 7,,, 7, 9} (prime number). 6 6 B 6 B. B. = { } 00 = {,, 5,, 99} * n n n = = n = = n = = 5. n = = 99 = { } n n 50n = {,,, * B = { z z } = { n n } = {, 6, 9, } = {,,, } 5 99, 50 }. n + n = n = n = n = 4 C = { n + n =,,, 4 } C =. n + n = 0 D = { } n + n 0 D = *0 (intensional definition) * * (finite set) (finite set) 4 (4.00) th-note
23 . : + = 4 : 7 : 0 : : {4, 7, 0, }. : 9 : {4, 7, 0,, 9} () = {k k =,,, 4, 5} () B = { } n n () C = { n n n 5 } (4) D = { k k k 50 } (5) E = { n n n 6 } (6) F = { } a 0 a a () = {, 4, 6, 8, 0} () B = {,, 5, } () C = {,, 5, 7, 9} (4) D = {, 4, 6,, 00} (5) E = {, 4, 8, 6,, 64} (6) F = {,,, 5} = {,,, 4, 5, 6} C. = { } <,, 0,,, P = { < } = { < } Q = { } P Q <, >, 4. a b B c C d. = { < }, 0, 0.8,,,, D 4 4.a = { } bb = { < } cc = { < < } dd = { 4 < 4}. 0, 0.8,,,, th-note. 5
24 5 X = { }Y () X Y () X Y, X Y () X (a) { < } (b) { } (c) {, } (d) { <, < } (4) Y X () Y () X Y = { < } X Y = { < } X = { < < } () (d) ± X ± X (4) Y = {, } 6 B = { < 4}, B = { } B, B B = B = { < } { } = { < } B = B = { <, 4 }.. n() = {, } n() = n( ) = 0 B. B n( B) B = B + B B B n( B) = n() + n(b) n( B) } {{ } B (principle of inclusion and eclusion) B = n( B) = n() + n(b) 6 (4.00) th-note
25 n() = a, n(b) = b, n( B) = p n( B) = a p, n( B) = b p U a a p p B b b p B n() =, n(b) = 8, n( B) = B n( B) = n() + n(b) n( B) = + 8 = 7 7. B n() = 7, n(b) = 0, n( B) = 40 B n( B) = n() + n(b) n( B) 40 = n( B) n( B) = 7 7 C. = = 75 U n() = n(u) n() U = U U U th-note. 7
26 8 U = { } 00 = { } B = { } 5. n(). n(b). n( B) 4. n( B) 5. n() 6. n(b). 00 = = {,,, } n() = = 0 B = {5, 5,, 5 0} n(b) = 0. B = 6 0 B = {5, 5,, 5 6} n( B) = n( B) = n() + n(b) n( B) = = 47 (p.6) 5. n() = n(u) n() = 00 = n(b) = n(u) n(b) = 00 0 = 80 9 U, B n(u) = 50, n() = 0, n( B) = 4, n( B) = 6 U B () () ) n( B) ) n( B) ) n( B) () : n(u) = 50 : n() = 0 : n(b) n( B) = n() + n(b) n( B) 4 = 0 + n(b) 6 n(b) = 8 : 0 6 = 4 : n( B) = 6 : 8 6 = : n( B) 50 4 = 8 () ) n( B) = 4 ) n( B) = n( B) = 50 4 = 8 ( Ip.) ) 50 = 8 8 (4.00) th-note
27 B 08 () () B U (89) BB n(u) = 9, n( B) = 70, n() = 89, n(b) = 08 () n( B) n( B) = n(u) n( B) = 9 70 = () n( B) n( B) = n() + n(b) n( B) 70 = n( B) 7 U(9) B(08) B (89) U(9) B(08) 4.. a, b, c, p, q, r, s a, p, r, s. ( B) C. (B C). ( B) C 4. (B C) 5. (B C) 6. (B C) B b p a s q r c C 7. ( B) ( C) 8. ( B) ( C). a, b, p, q, r, sc, q, r, sa, b, c, p, q, r, s. a, p, r, sb, c, q, r, p, sa, b, c, p, q, r, s. p, sc, q, r, ss 4. a, p, r, sq, ss 5. a, p, r, sq, sa, q, r, p, s 6. a, p, r, sb, c, p, q, r, sp, r, s 7. a, b, p, q, r, sa, c, p, q, r, sa, p, q, r, s 8. p, sp, rp, r, s th-note. 9
28 BC i) ( B) C = (B C) B C ii) ( B) C = (B C) B C iii) (B C) = ( B) ( C), (B C) = ( B) ( C) iii) (B + C) = B + C B. B C = ( B) C B = B C B C (p.) = B C B = B B C = B C B C = ( B) C B = B C B C = B C C. n( B C) (p.6) n( B C) = n() + n(b) + n(c) } {{ } p, q, r s n( B) n(b C) n(c ) } {{ }} {{ }} {{ } p, s q, s r, s + n( B C) } {{ } s B b p a s q r c C BC n( B C) = n() + n(b) + n(c) n( B) n(b C) n(c ) + n( B C) U000 B5 C7 n( B C) 0 (4.00) th-note
29 000 = n() = B = 00 n(b) = 00 C = 4 6 n(c) = 4 B = 66 0 n( B) = 66 B C = 8 0 n(b C) = 8 C = 47 n(c ) = 47 B C = 9 55 n( B) = 9 n( B C) = n() + n(b) + n(c) n( B) n(b C) n(c ) + n( B C) = = 54 p.8 U(000) B(00) (66) () 9 (8) (47) C(4) (p.0) BC 0 B 5 C 60 B 4 BC 6 C 0 U(00) U (0) BB CC n( B C) n() = 0, n(b) = 5, n(c) = 60 (4) B(5) () C(60) n( B) = 4, n(b C) = 6, n(c ) = B C = B C 0 (6) 0 n ( B C ) = n(u) n( B C) 0 = 00 n( B C) (p.7) n( B C) = 90 n( B C) = n() + n(b) + n(c) n( B) n(b C) n(c ) + n( B C) 90 = n( B C) 90 = 78 + n( B C) n( B C) = (p.0) th-note.
30 ... * 0 * 4 (proposition) * cm m. B. (true) (false) (countereample) = = = 6. m 40 cm m.. 4. () () () (4) + 5 = 8 * *4 *5 (4.00) th-note
31 .. ab 0 i) a = 0 ab 0 ii) a, b ab 0 (assumption) (conclusion) (condition) i) a = 0 ab 0 ii) a, b ab 0 7. a, b ab. a + b ab. a, b ab. a + b ab a = b = a =, b = B. p q p > 0 q + 0 > 0 p q > > 0 p q p, q 8. p :a = bq :a = b p q. p :ac = bcq :a = bp q.. (a, b, c) = (,, 0) c = 0 th-note.
32 .. a a + a + a a a = * 6 B. p p p (negation) p m pm p m a qa 0 q a 0 0 < a 9 a n. pn 0 p. qn q. r a r 4. s4 < a s. pn 0. qn. ra < 4. sa 4 C. a > 0 b > 0a > 0 b > 0 a > 0 b > 0a > 0 b > 0 * 7 p qp q p q p q p q i) ii) iii) iv) 40 a, b p :a > 0q :b > 0. a =, b = pp qp q. a =, b = pp qp q. a = 0, b = 0 pp qp q. pp q p q. pp q p q. p p qp q a > 0 a 0 *6 *7, 4 (4.00) th-note
33 4. a =, b = a + b = a = b = a + b =. a, b a b. = =, = = =... D. U p U p p p p q p q p p p q p q pq (law of de Morgan) p q p q p q p q p q p qa = 0 b = 0 a \= 0, b \= 0 a \= 0 b \= 0 4 a, b m, n. a = b =. a = b =. a \= b = 4. m, n 5. m n 5 6. a > 0 b < 0. a \= b \=. a \= b \=. a = b \= 4. m n m, n m 5. m n 5 6. a 0 b 0 n th-note. 5
34 4 p q p q p q p q p q i) ii) iii) p q p q p q p q p q i) ii) iii) p q p q 44 a, b () a, b ab () a b ab () a a (4) a = b a b () a = k +, b = l + ab = 4kl + k + l + = (kl + k + l) + () a =, b = a b () a = k ± a = 6k ± (4) a a b b E. * 8 * 9 45 n (n + )(n ) 4 + = 0 n (n + )(n ) 4 (n + )(n ) 4 + \= 0 n *8 *9 6 (4.00) th-note
35 4.. p q p q p q p, q p q p q p q p q p q B. (converse) a = a = a = a = p qq p 46. = 0 = 0., +. = 0 = 0. +, =, = 47 P : p q p, q P p, q : p q : q p : q p th-note. 7
36 C. p q p q p q p q (necessar condition) q p p q (sufficient condition) p qq pp q (necessar and sufficient condition) p q (equivalence) 48 a, b p :a, b q :ab r :a + b. p q q p p q. q r r q r q. r p p r r p 4. p q q r r p p q q p q p q p. : : : :. : : : :. : : : : 4. : : : 8 (4.00) th-note
37 p q p q, q p a = b ac = bc. = 4 =. a 4 a 6 4. a = b = 0 a + b = 0 4. a = b ac = bc p q p ac = bc a = b. = 4 = = = = 4. a 4 a 6 a = 8 a 6 a 4 a = a = b = 0 a + b = 0 a + b = 0 a = b = 0 D. q p p q p q q p q p p q * 0 q p *0 "equivalence" th-note. 9
38 50 4 () < < () BCD B//DC () a <, b < ab < 4 () < < < < = () BCD B//DC B//DC BCD D\//BC () a <, b < ab < a =, b = ab < a <, b < 4 a =, b = 5.. p q p q (converse of contraposition) a = a = a \= a \= 5. = 0 = 0., +. \= 0 \= 0. + =, = 0 (4.00) th-note
39 B. q p p q (contraposition) a = a = a \= a \= 5. = 0 = 0., +. \= 0 \= 0. + C. p.7. p q p q q p (p.86) D. pq pq q p p q q p p q p q p q q p q p 5 = = 4 P P P, P, P P, P, P P = 4 = = P \= \= 4 = = \= P \= 4 \= = 4 th-note.
40 54 55 (4) BC = PR BC PQR () = = 0 (), () + = 5 = = (4) BC = PR BC PR () = 0 = = 0 \= \= 0 = 0 \= 0 \= (), = = = = () = = + = 5 + \= 5 \= \= \= \= + \= 5 = 0, = 5 (4) BC PR BC = PR BC \= PR BC PR BC PR BC \= PR,.4 (4.00) th-note
41 .. p q r > > > + > > > + > > > + > p q q r p r 56. = 0 =, =, = 0. a = b = c k a = k, b = k, c = 5k 5 k a = k, b = k, c = 5k a : b : c = : : 5 > + > p q r. = 0 = 0. a = b = c a : b : c = : : 5 5 B. > > > > 4 > 4 > > 4 > p r q p q q r (sllogism) > > p r > 4 > p q r 57 a a k a = k a = k a = a : 4k : q th-note.4
42 C. p q p qq p p qq pp q 58 a, b a b a + b a b a + b a b a b = mm a b a + b a + b = m + = ( ) a + b a + b a b a + b a + b = nn a b = n = ( ) a b a b a + b : b : m + b : n b 59 a, b a + b 4 a b 4 a + b 4 a b 4 a + b 4 a + b = 4mm a b = (a + b) 4b = 4m 4b = 4(m b) m b a b 4 a b 4 a + b 4 a b 4 a b = 4nn a + b = (a b) + 4b = 4n + 4b = 4(n + b) n + b a b 4 a + b 4 a b 4 4 (4.00) th-note
43 . (p.) p q q p 60 a a a a a a = k ka = 4k a 6 BP PB \= 90 B P : B P PB = 90 : 6 = a = b, ( ) + ( ) = 0 = = \= \= ( ) + ( ) \= 0 \= ( ) > 0 ( ) 0 ( ) + ( ) > 0 ( ) + ( ) \= 0 \= ( ) > 0 ( ) 0 ( ) + ( ) > 0 ( ) + ( ) \= 0 \=,, a, b = a = b ( a) + ( b) = 0 p.86 th-note.4 5
44 .. p q i. p q ii. i. (contradiction) iii. q q (reduction to absurdit) * 6 a + b = a b a b a b a < b < a + b < a + b = a b p q q p p q 64 = a = b, ( )( ) = 0 = = = = \= \= \= 0 \= 0 ( )( ) \= 0 ( )( ) = 0 = = (p.5),, a, b = a = b ( a)( b) = 0 p.87 * p qp q p qp q 6 (4.00) th-note
45 B. * * 65 a = a = a + = a + = a a = + a = a = + 4a = 4a + 4 *, th-note.4 7
46 a = + a = + a = ( + ) a = a 5 = 6 a C. 68 = a b a b 0 (p.) = a b b = a a a a = a a b = (a ) b = 4a b = a b b ab a a 8 (4.00) th-note
47 B B. p.46p.5.. ab (monomial) (degree) 0 * 4 (coefficient) a, b, ab 4 4 ab 4 69 b, 5, 6, z... b z. 5 6 B. ab ab = (ab) ab a ab = (b )a b { }} { ab 70 ka 4 b 5. a. b. a, b. a 4 kb 5 ka 4 b 5 = (kb 5 )a 4. b 5 ka 4 ka 4 b 5 = (ka 4 )b 5. a b 9 k *4 0 m n m + n 0 ab }{{} z }{{} = 6abz } {{ } 5 (= + ) th-note B. 9
48 7 [ ] () 4 5 [], [], [ ] () ab [], [], [ ] () i) = ( 5 ) 4 ii) = ( 4 ) 5 iii) 9 () i) ab ab = (ab ) ii) ab ab = (ab) iii) ab ab = (ab) C. a n n n { }} { a a a a n a n a n (eponent) a a (square) a a (cube) a, a, a, a (power) (eponential law) * 5 mn } 6 6 {{ 6 } 6 = ( ) = } {{ } i) a m a n = a m+n ii) (a m ) n = a mn iii) (ab) n = a n b n 4 = 8 i) a a 4 = (a }{{} a) (a } a {{ a } a) = a 6 (= a +4 ) ii) (a ) 4 = (a }{{} a) (a }{{} a) (a }{{} a) (a }{{} a) = a 8 (= a 4 ) 4 iii) (a b) 4 = (a b) (a b) (a b) (a b) } {{ } a b 4 = a 4 b ( ). ( ) 5 4. ( ) 5. (a ) 6. ( a). = + = 5 i). ( ) = = 6. ( ) 5 = 5 = 5 ii) 4. ( ) = ( ) = 6 5. (a ) = (a ) = 4a 6 iii) ii) 6. ( a) = ( ) a = a iii) *5 II 40 (4.00) th-note
49 .. a b + ab (polnomial) (integral epression) * 6 (term) 0 (constant term) a b 4 + ab a, b, 4, ab +ab 4 * 7 B. (similar term) = + B = + 7 5a b + ab + a b + ab = (5a b a b) + (ab + ab) + = 4a b + 5ab + }{{} + B = ( + ) + ( + 7 ) B = ( + ) ( + 7 ) = = = ( + ) + ( + 7) + ( ) = ( ) + ( 7) + ( + ) = = B = = B = = ab + a c c a c. X = a + a 5, Y = a + a + 5 X + Y, X Y. ab + a c c a c = ab a c c ab, a c, c. X + Y = a + a 5 + a + a + 5 = a + 6a X Y = a + a 5 a a 5 = a 0 *6 *7 th-note B. 4
50 74 () a b (a ) () (4 ) () ( ) (4) a a ()= a b a 4 = a 7 b ()= 6 4 = 5 ()= 9 6 = 8 (4) a a (a ) = a 6 C. n n (epression of degree n) 4a b + 5ab a b 4a b + 5ab } {{ } D. (descending order of power) * } {{ } = } {{ } * : 4 + : 4 : 4,, : +. : = + 5 = 5 : :,, 5 : 5 *8 (ascending order of power) = *9 ab + bc + ca 4 (4.00) th-note
51 E. b a + + = b a {}}{ a + b {}}{ + ( + ) 0 a b + = a + b = a + + b 0 { }} { + ( a + ) + b 0 a + b a = + ( ) + ( + 4 ) = = ( + ) + ( ) = + ( ) = ( + ) + ( ) + (4 + 5) = 6 + ( ) ( + ) ( ) th-note B. 4
52 77 () 4a + a + a () [ ] ) cb a c a [c] ) k + k + 4k + 4k [] () 4a + a + a = 5a + a 4 = a + 5a 4 () ) cb a c a = c a + cb a = ac + bc a a ac ) k + k + 4k + 4k = k + (k + 4k) + 4k 4k k F. (B + C) = B + C( + B)C = C + BC B = B ( + )( 4 + 5) ( + )( 4 + 5) = ( + ) = + ( + B)C = C + BC = ( 4 + 5) + ( 4 + 5) = (B + C) = C + BC = ( + ) ( 4 + 5) = = * ,,,, (epansion) 0 m n m + n *40 II 44 (4.00) th-note
53 78 ( + ) () = + () = + 5 () = 6 + () ( + ) = + ( + )( + ) = () ( + ) = ( + )( + 5) = = = 6 + ( 9) + ( + 5) + 5 () ( + ) = ( + )( 6 + ) = = 6 + ( 6 + ) = +, B = () B () B () B = ( + )( ) = = () = = 5 = 6 + ( ) th-note B. 45
54 B... i) (a + b) = a + ab + b, (a b) = a ab + b i) ( + ) = 9 + () + 4 } {{ } = ii) ( + ) = ( + )( + ) = = (a + b)(a b) = a b i) (5 + )(5 ) = (5) () } {{ } = 5 4 ii) (5 + )(5 ) = = 5 4 ( + b)( + d) = + (b + d) + bd i) ( + )( 4) = + ( 4) + () ( 4) } {{ } = ii) ( + )( 4) = 4 + = 46 (4.00) th-note
55 80 4. =, B = +, C =. (a + 4). ( + )( ). (p + )(p 4) B 6. C. (a + 4) = a + 8a + 6 (p.46). ( + )( ) = 4 (p.46). (p + )(p 4) = p p 8 (p.46) 4. = ( ) = (p.46) 5. B = ( )( + ) = 9 (p.46) 6. C = ( )( ) = 4 + (p.46) B. (a + b)(c + d) (a + b) (c + d) = ac + 4 ad + bc + 4 bd = ac + (ad + bc) + bd } {{ } c d a ac ad b bc bd ( + )(5 4) i) ( + )(5 4) = 0 + ( 8 + 5) + () ( 4) } {{ } = (a + b)(c + d) = ac + (ad + bc) + bd ii) ( + )(5 4) = = (ad + bc) ad+bc 8. ( + )( + ). ( + )( ). (5 )( ) 4. ( )( + ). + = 5( + )( + ) = ( + ) ( + ). ( ) + = 5( + )( ) = ( + ) ( ). 5 ( ) + ( ) = (5 )( ) = 0 + (5 ) ( ) 4. () + ( ) = 7 ( )( + ) = th-note B. 47
56 8 () ( + )( + ) () ( + 4)( ) () (4 + )( ) (4) ( )( ) (5) ( + )( ) (6) ( + )(4 ) (7) ( + 5)( ) (8) ( )(5 + ) () + + () + 5 () (4) 0 + (5) 6 (6) + (7) (8) 0 C. 8 () ( 5)( + 5) () ( + 5)( 8) () ( + )( ) ( (4) + ) (5) (a )(4a + ) (6) (a 4)(a + ) ( (7) (a )(a + 7) (8) a ) b (9) ( ab + c)(ab + c) ()= () (5) = 4 5 ()= + (5 8) + 5 ( 8) = 40 (p.46) (p.46) ()= + { ( ) + } = 5 ( ) (4)= ( ) + + = (p.46) (5)= a + { + ( ) 4} = a 5a (p.47) (p.47) (6)= (a 4)(a + 4) = (a 6) = a 48 (p.47) (p.46) (7)= (a ) + ( + 7)a + ( ) 7 = a 4 + 4a (p.46) (8)= (a) a (9)= (c ab)(c + ab) b + ( b ) = 9a ab + 4 b (p.46) = 9c 4a b (p.46) 48 (4.00) th-note
57 D. (rationalization) * 4 = ( + ) ( ) ( + ) ( + ) = ( + ) ( ) ( ) = + (p.46) = = = = 4 ( 6 ) ( 6 + ) ( 6 ) = 4 ( 6 ) 4 = 6 (p.46) ( 7 ) ( ) ( ) = ( 7 ) ( 6 + ) ( ) 4 = 7 ( ) ( ) = ( ) ( ) ( 5 ) ( 5 + ) ( 5 + ) (p.46) 5. = ( ) ( ) = ( ) ( ) = ( ) ( ) = = ( ) 6 + = (p.46) (p.46) (p.46) = (p.46) * (.7.44) = ( ) =.46 th-note B. 49
58 ... ( + + )( + ) = (M + )(M ) M = + = M + M 6 (p.46) = ( + ) + ( + ) 6 M + = (p.46) ( + z)( + z) + z = ( z) ( + z)( + z) = { + ( z)} { ( z)} + z = ( z) = ( + )( ) = z = (p.46) = ( z) z = ( z + z ) (p.46) = + z z ( ) 85. ( + 5)( + + ). ( + + z)( + z). (a + a )(a a ). ( + ) ( + 5)( + + ) = ( + ) ( + ) 5 (p.46) = (p.46). ( + ) ( + + z)( + z) = ( + ) z (p.46) = + + z (p.46). (a ) (a + a )(a a ) = (a ) a (p.46) = a 4 a + a (p.46) = a 4 a + X 50 (4.00) th-note
59 B (6 5) = 4 80 (a + b)(a + b )(a b) = (a + b)(a b)(a + b ) (a b) (a + b) = (a b )(a + b ) (p.46) = a 4 b 4 (p.46) p.40 B = () (BB) = (B) (B) = (B) ( + ) ( ) ( + )( ) = {( + )( )} = ( ) (p.46) = 4 + (p.46) ( + )( + )( )( 4) = {( + )( )} {( + )( 4)} = ( )( ) = { ( ) } { ( ) } = ( ) 4( ) + 4 = ( 4 + ) ( ) = ( )( )( + )( + ). (a + b) (a b). (a )(a )(a )(a 4). ( ) ( + ) ( ) ( + ) ( )( )( + )( + ) = ( )( + )( )( + ). (a + b)(a b) = {(a + b)(a b)} = { (a b ) } = ( )( 9) (p.46) = (p.46) (p.40) (p.46) = a 4 a b + b 4 (p.46) (p.40). (a )(a 4) (a )(a ) a 5a (a )(a )(a )(a 4) = {(a )(a 4)} {(a )(a )} = (a 5a + 4)(a 5a + 6) = (a 5a) + 0(a 5a) + 4 a 5a ( + 4)( + 6) = th-note B. 5
60 = (a 4 0a + 5a ) + 0a 50a + 4 = a 4 0a + 5a 50a + 4 C. (a + b + c) (a + b + c) = {(a + b) + c} = (a + b) + (a + b)c + c a + b (p.46) = a + ab + b + ca + bc + c (p.46) = a + b + c + ab + bc + ca ( + ) i) ( + ) = () ( ) + ( ) } {{ } = ii) ( + ) = ( + )( + ) = = (a + b + c) = a + b + c + ab + bc + ca 87. (a b + c). (a + a ). (a b + c) = 9a + b + 9c 6ab 6bc + 8ca (p.5). (a + a ) = (a ) + a + + a a + a ( ) + ( ) a = a 4 + a a a + 88 (a )(a + )(a + 4)(a 4 + 6) (a b + c)(a + b + c) ( + + z + w)( + z w) 4 ( 4) ( + 5) 5 (a + b) (a b) (a + b ) 6 ( )( + )( )( + 4) (a )(a + )(a + 4)(a 4 + 6) = (a 4)(a + 4)(a 4 + 6) (a )(a+) = (a 4 6)(a 4 + 6) (p.46) = a 8 56 (p.46) = {(a + c) b} {(a + c) + b} (a + c) = (a + c) b (p.46) 5 (4.00) th-note
61 = 4a + 4ac + c b (p.46) = {( + ) + (z + w)}{( + ) (z + w)} z w = (z + w) = ( + ) (z + w) (p.46) = + + z zw w (p.46) 4 ( 4)( + 5) = {( 4)( + 5)} = ( + 0) = ( ) + + ( 0) + + ( 0) + ( 0) (p.5) = (a + b)(a b)(a + b ) = { (a + b)(a b)(a + b ) } = { (a b )(a + b ) } = (a 4 b 4 ) (p.40) (p.46) = a 8 a 4 b 4 + b 8 (p.46) (p.40) (a 4 ) = a 8 6 = ( + )( + ) ( )( + ) ( )( + 4) = ( + ) 4( + ) = ( ) = ( )( ) = II p.88 B... BC B C (factor) * 4 (factorization) * 4 a 4ab = (a 4ab) = (a ab) = a = a (a b) (a b) a 4ab *4 *4 th-note B. 5
62 B. (common factor) * = () + () = (z + + ) + () a( + ) + b( + ) = (a + b)( + ) 89. p q + pq pq. a( + ) b( + ). p( ) + q( ). p q pq pq pq pq p q + pq pq = pq(p + q ) p q pq p q pq pq. a( + ) b( + ) + + = X X a( + ) b( + ) = ax bx + + = X = (a b)x X = (a b)( + ) X +. = X = ( ) = X X p( ) + q( ) = px qx = (p q)x X = (p q)( ) X C ( 7 ) = ( )( ) 5 + ( )( ) * 46 * 45 * 45 ( )( ) ( )( ) *44 *45 I II *46 I II 54 (4.00) th-note
63 i) ii) = () + () + = ( + ) ( + ) = () + () + = (p.46) a + ab + b = (a + b), a ab + b = (a b) 6a b i) ii) 6a b = (4a) b = (4a + b)(4a b) (4a + b)(4a b) = (4a) b = 6a b a b = (a + b)(a b) (p.46) i) ii) = 6 = 6 7( ) + ( + ) + 6 = 5() = ( + )( + ) ( + )( + ) = + ( + ) + = (p.46) + (b + d) + bd = ( + b)( + d) 90 () 6a b + 4ab ab () (s + t) (s + t) () a( ) + 6b( ) + 9c( ) (4) (5) (6) a 9 (7) 4 5 (8) (9) a + ab 8b (0) a 4 + 4a + 4 () a 4 () (a b) () 4 9(a b) (4) (a b) + 0(a b) + th-note B. 55
64 () 6a b + 4ab ab = ab(a + b ) ab () s + t = (s + t) (s + t) = s + t = = ( ) = ( )(s + t) s + t () = X = X X a( ) + 6b( ) + 9c( ) = ax + 6bX 9cX = X(a + b c) a( ) + 6b( ) + 9c( ) = a( ) + 6b( ) 9c( ) = ( )(a + b c) = ( )(a + b c) a b c (4)= + + = ( + ) ( ) a( ) 6b( ) 9c( ) (5)= () ()() + () = ( ) (6)= a = (a + )(a ) (7)= () (5) = ( + 5)( 5) (8)= ( )( 4) (9)= (a b)(a + 6b) (0) a = a 4 = = = ( + ) = (a + ) () a = a 4 = = = ( + )( ) = (a + )(a ) = (a + )(a + )(a ) a () a b = X = = ( + )( ) = { + (a b)}{( (a b)} = { + (a b)} { (a b)} = ( + a b)( a + b) = ( + a b)( a + b) = a b +, () a b = 4 9(a b) = 4 9 = () {(a b)} = () () = { + (a b)} { (a b)} = ( + a b)( a + b) = ( + )( ) = { + (a b)} { (a b)} = ( + a b)( a + b) (4) a b = = = ( + )( + 7) (a b) + 0(a b) + = {(a b) + }{(a b) + 7} = (a b + )(a b + 7) 56 (4.00) th-note
65 = (a b + )(a b + 7) B. (p.47) i) ii) = ( ) + ( + 4 ) + 4 = ( + 4)( + ) ( + 4)( + ) = ( ) + ( + 4 ) + 4 = ( + 4) ( + ) = ( + 4) ( + ) } {{ }} {{ } ( ) ( 8) a + 7ab + b b 5b 5 b b 7b ( + )( + ) ( + )(4 + ) (a + b)(5a + b) b 0b 5 b b b th-note B. 57
66 6 + 6 * * = ( + )( 4) a + 7a ( )( + 5). 4 6 (a + 4)(4a ) (4 )( + 6) (4 )( ) ac + (ad + bc) + bd = (a + b)(c + d) = 6 a () () 6 5 () (4) 9a b ab b () () (5 + 6)( + ) () ( 5)( + ) () (7 )( ) (4) = b(a 4a 4) () 5 0 = b(a + )(a ) 9 () * (4.00) th-note
67 C. 94 () a 4ab + 49b () () a 5a + a (4) b 7c (5) 8 (6) a 4 (7) 8 (8) 5( + ) 8( + ) 4 (9) (a + b) + 0c(a + b) + 5c ()= (a 7b) ()= ( )( + ) ()= a( 5 + ) = a( )( ) (p.55) (p.58) (4)= (b 9c ) = (b + c)(b c) (5)= ( 8 ) = ( + )( ) (6)= (a 4 6) (p.55) = (a + 4)(a 4) (p.55) = (a + 4)(a + )(a ) (p.55) (7)= ( 4 + )( 4 ) (p.55) = ( 4 + )( + )( ) (p.55) = ( 4 + )( + )( + )( ) (p.55) (8) + = X = 5X 8X 4 = (5X + )(X ) = ( )( + ) (9) a + b = X = X + 0cX + 5c = (X + 5c) = (a + b + 5c) (p.58) (p.55) II p.89 D (double radical sign) = 5 + ( 5 + ) = = th-note B. 59
68 , a. 5 + b. + c. 5 + d. 5 + ( 5 + ) ( ) ( ) = = = ( ) 5 + = c b. a > 0, b > 0 ( a + b ) = a + b + ab a + b > 0 a + b = a + b + ab a, b = (5 + ) + ( 5 ) 5 = + = 5 + a > b > 0 ( a b ) = a + b ab a b > 0 a + b ( ) ab = a b = a b ± ( 5 ) 0 = + = ( 7 ) = + = ( 7 ) 4 = = ( 5 ) 5 = = ( ) > = 7 + ± ( 4 ) = + 7 = 4 + = + 60 (4.00) th-note
69 6 5 5 = = = = 6 5 ( 5 ) 5 = a + a a + a = a( + ) a = a( + ) ( + ) + = (a )( + ) ( + ) = ( ) ( + ) m + m n n = (m n ) + m n = (m + n)(m n) + (m n) m n = (m + n + )(m n) m n = X {(m + n) + } X 984 () ab + ac + b + c () mn + m n () a 5a + 5b b () ab + ac + b + c = (b + c)a + (b + c) b c = (a + )(b + c) () mn + m n = (n + )m (n + ) = (m )(n + ) (m )n + (m ) () a 5a + 5b b = (a b ) 5(a b) a b a b = (a b)(a + b) 5(a b) = (a b)(a + b 5) a b = X X(a + b) 5X = {(a + b) 5} X th-note B. 6
70 B. * 48 a + ab a + b 4 a b = (a + )b + a a 4 b = (a + )b + (a 4)(a + ) a + = (a + )(a + b 4) b + a 4 99 () a + ab + bc + ca () + () () b c = (a + c)b + (a + ca) = (a + c)b + (a + c)a = (a + b)(a + c) () = ( + ) + ( ) = ( ) ( ) + ( )( + ) = + ( ) = ( )( + ) = { + ( )} () = ( + 4) + ( + + ) = ( + ) + ( + )( + ) + = + ( + ) = ( + )( + + ) = { + ( + )} C. (p.58) (5 + ) (5 + ) + ( )( + ) (p.58) *48 6 (4.00) th-note
71 5 + { ( )} { ( + )} ( + )( + + ) = + (4 + ) = + (4 + ) + ( )( + ) + + = ( + + )( + ) 4 +. = + ( + ) ( + ) = + ( + ) ( )( ) ( ) + 4 = { ( )}{ + ( )} + = ( + )( + ) D. a 4 + b + c (biquadratic epression) a \= 0 i) = X X X + 6 = (X 4)(X 9) = ( 4)( 9) ii) = ( + )( )( + )( ) = X X + X = = 6 4 = ( + ) } {{ } () = { ( + ) + } { ( + ) } = ( + + )( + ) th-note B. 6
72 a 4 + b + c i) = X ii) a 4 c b i) ii) = ( + 8)( ) = ( + 8)( )( + ) = X = = ( + ) = ( + + )( + ) = X + 7X 8 = (X + 8)(X ) = ( + 8)( ) = ( + 8)( )( + ) = ( + + )( + ) E. () () () 0 (). +. a + b + ac bc ab. a 4 + a b + b (). ( z) + (z ) + z ( ). ab(a b) + bc(b c) + ca(c a). a ( )( 6) ().= ( ) ( ) = ( )( ) 64 (4.00) th-note
73 . c c = (a b)c + a + b ab = (a b)c + (a b) = (a b)c + (a b)(a b) = (a b)(a b + c). a 4 b 4 (p.6) = a 4 + a b + b 4 a b = (a + b ) (ab) = (a + b + ab)(a + b ab) = (a + ab + b )(a ab + b ) 4. (p.6) = + ( + 5) ( 6) = + ( + 5) ( + )(4 ) (4 ) 4 + = { (4 )}{( + ( + )} = ( 4 + )( + + ) (). = ( z) ( z ) + z z = ( z) ( + z)( z) + z( z) = ( z) { ( + z) + z } = ( z)( )( z). a = (b c)a (b c )a + b c bc a = (b c)a (b + c)(b c)a + bc(b c) = (b c) { a (b + c)a + bc } = (b c)(a b)(a c). a 4 64 (p.6) = (a + 8) 6a = (a a)(a + 8 4a) = (a + 4a + 8)(a 4a + 8) 4. (p.6) = 6 + ( 5 + 4) (6 7 + ) = 6 + ( 5 + 4) ( )( ) = { ( )}{ + ( )} = ( + )( + ) ( ) = ( ) 8( ) + = ( )( 8) = ( )( + )( 4) = th-note B. 65
74 4.. +,, ( + ) = = ( + ), = +, = + = 4, =, = ( ) = +, + 0 = ( + ) = 4 = 4 = ( ) = ( + )( ) = 4 = 8. = 6 +, = 6 ) + ) ) 4) + 5) =, = ) + ) ) 4) 5) ) + = ( 6 + ) + ( 6 ) = 6 ) = ( 6 + ) ( 6 ) = ( 6 ) ( ) = ) = ( 6 + ) ( 6 ) = 4) + = ( + ) = ( 6 ) = 8 + = 6, = 5) 4 4 = ( ) = () ( + )( ) = ( 6 ) ( ) = 08, +,. ( ) 7 + = ( ) ( ) = = 0 + = = ( 7 ) ( 7 + ) ( 7 ) = = 0 4 = 5 ) + = = 5 ( ) ( ) ) = = 5 ( ) = 4 ) = = 4) = ( + )( ) = 5 + = 5, = 5) = ( + ) ( + ) = = {( + ) } () + = ( + ) = () = (5 ) = = 57 iii) p (4.00) th-note
75 B. 04 F = ab a + b 6 i. F ii. F = 6 (a, b) mn + m n = (m, n) i. F = a(b ) + (b ) = (a + )(b ) ii. i. (a + )(b ) = 6 ab 6 6 = 6,,, 6 a + { { a + = 6 a + = b = b = (a, b) = (4, 4), (, 5) mn + m n = m(n + ) n = m(n + ) (n + ) = (m )(n + ) = =, ( ) ( ) { m = { m = n + = n + = (m, n) = (, ), (0, ) th-note B. 67
76 B.4 = >.. (a sign of inequalit) < * 49 a < b a b a b a b a b a < b a = b a > b a b a b a b a > b a = b B. a 4 a + > 4 (inequalit) (left side) (right side) (both sides) a a b. ( ). a b a + b B B } {{ }} {{ } a + b a + b. ( ) } {{ }} {{ } } {{ } < B < B *49 a < ba ba ba ba > ba ba ba b 68 (4.00) th-note
77 C. i) a < b a + < b + a b a < b a < b a b a + b + a b ii) a < b a < b a b a b a < b a < b a b a b iii) a < b a > b a b b a a < b a > b a b b a 06. a > b i. a + 4 b + 4 ii. a b iii. a b iv. a b v. a b vi. a b vii. 4a 4b viii. a b. i. v. a > b, a < b, a b, a b i. 5a < 5b ii. a < b iii. a 4 < b 4 iv. a 4 b v. a 4 4 b 4. i. > ii. > iii. > iv. > v. > vi. < vii. > viii. <. i. a < b ii. a > b iii. a < b iv. a b v. a b i) c a < b a + c < b + c, a c < b c ii) 0 < c a < b ac < bc, a c < b c iii) c < 0 a < b ac > bc, a c > b c p.7 th-note B.4 69
78 07 () + < 5 + < 5 () < 8 < 8 () 5 ( ) 5 < < () : : () : : 4 () : : 5.. (linear inequalit) + > 5, 5 + 4, < 7 (solution) + > = 5 = ( ) + = 7 7 = 5 ( ) = 9 = < +. = =. = =. = 4 = 4.= 5,= 0.= 5,= 5.= 7,= 6 70 (4.00) th-note
79 B. (solve) p.69 (transposition) + > 5 5 > > 6 < < < <, >, < 4. < < > < 5 <. 4 8 > > 8 > th-note B.4 7
80 () 8 () ( ) > (4 ) + 4 () 5 > + () () ( ) > (4 ) > > 0 > 4 () 5 > + 9 (5 ) > > 6 + > 7 4 < 7 7 () < 7 = = 5 () = = 5 () = 5 7 = () = = 5 C. (simultaneous inequalities) < < < 8 4 < (4.00) th-note
81 < < 0. < 0 0. < < < 4 < 4 4 < D. + 6 < < < < 4 { + 6 < < < < < < < < 4 6 < < < > < th-note B.4 7
82 6 () 4 > () () 4 > 5 6 > 8 0 > 4 > < () (4.00) th-note
83 E. 7 () 5 km B 5 km km 4 km () 5 % 800 g 8 % g 6 % 8 % g () km 5km km B (5 ) km km (5 ) km = + 5(5 ) = km () 8% g 5% ( ) 800 g g 8% ( ) g 8 00 g 6 % ( ) 00 (800 + ) 6 00 (g) (g) 5% % = (800 + ) (800 + ) % 400 g th-note B.4 75
84 F. 8 < < 4 () + () () (4) 5 (5) () < < 4 + < + < 4 + < + < 7 () < < 4 < < 4 4 < < () < < 4 ( ) < < 4 4 < < 8 (4) < < 4 4 < < < 5 < < 5 < (5) < < 4 ( ) > > 4 4 > > 8 8 < < 4 9 a 4 b 6 a, b a + b a b.5 a < 4.5, 5.5 b < a < a < b < a <.5 +) 5.5 b < a + b < 0 6 a + b < b < < b a < 4.5 +) 6.5 < b 5.5 < a + ( b) < < a b < 76 (4.00) th-note
85 .. (p.7) = a = ± a < a a < < a > a < a a < < a a > a 0 a a > 0 * 50 a 0 a 0 () = () = 6 () + > 4 (4) 5 4 ()= > 0 = ± =, 4 ()= 6 > 0 = ±6 = ±6 = 4, 8 = 4, 8 ()= 4 > 0 + < 4 4 < + < 5 < (4)= 4 > = = = = 4 i)(p.69) 5 ii)(p.69) *50 0 a = 0 a < 0 < > 0 0 th-note B.4 77
86 B. = = 4 = + 4 = + 4 = + ( 4) = 4 < 4 = ( 4) = + 4 = ( 4) = = 4 4 = = () = + () = 4 () i) 0 = + ( ) = + = = ii) < 0 < = + = ( ) = + i)ii) < () 4 = 4 4 i) = 4 = 4 8 ii) 4 < 0 < = ( 4) = i)ii) 4 = < 4 () = 4 < 0 > 0 78 (4.00) th-note
87 + = 4 = + 8 = 4 i) = = ii) + < 0 < = = = i) ii) = i) = + 8 = = 6 4 ii) 4 < 0 < = = 4 = 4 i) ii) =, 6 i) 0 5 = 4 = 5 ii) < 0 < 6 + = 4 = 6 = 6 i), ii) = = 4 th-note B.4 79
88 + 6 > + i) > < 6 < 6 < 6 ii) + 6 < 0 < 6 6 > 4 < 6 < < 6 i) ii) < 6 i) 0 + ii) < 0 < < i) ii) 80 (4.00) th-note
89 C () 8.69 () 8 8 () = = = 4 (4) 4??4? 48 8 = 84 4 < 49 9 = 44? 8 (5) (6) 56??969? = < 568 8? = = () 8.69 () 8.69 () (4) (5) (6) th-note C. 8
90 B. (etraction of square root) * 5 C * ?4? (4.00) th-note
91 = = =.6 05 = =.0 = (m/s ) π th-note C. 8
92 . 5 ab () b \= 0 () a = a () ab = a b () a b = a b () i) a 0 a = a = a = a = ii) a < 0 a = a = a = ( a) = a = i)ii) a = a () 4 b 0 i) a 0b 0 b < 0 ab 0 a = a b = b = ab = ab, = a b = ab a 0 a < 0 i) iii) ii) iv) ii) a 0b < 0 ab 0 a = a b = b = ab = ab, = a b = a( b) = ab b b iii) ii) a b iii) iv) a < 0b < 0 ab > 0 a = a b = b = ab = ab, = a b = ( a)( b) = ab a b () a b ab = a b () b = b i) b > 0 b > 0, b = b = b = b, = b = b ii) b < 0 b < 0, b = b = b = b, = b = b = b 84 (4.00) th-note
93 i)ii) a b = a b = a b = a b = a b () a b = a b.. p q p q p q p.7 p q p q p q B. (p.) p p p (law of ecluded middle) p p * 5 C. p q q p (I). p q p q (II) B. p p p (III) p, q p q q p *5 p p q r p q p r p pp pp p p p p p p p th-note C. 85
94 pq p qq p p q p q (I) q p (III) q p (II) q p (I) 4. q rq r. q r q r q r B. = a = b p.5 = a = b( a) + ( b) = 0 = a = b( a) + ( b) = 0 = = z ( ) + ( z) + (z ) = 0 6 () n = k + k n n n 8 () + = + = = = () + + z = + z + z = = z () n n n n = (k + ) (k + ) = 4k + k = k(k + ) n n = (k + ) = 4k + 4k = 4k(k + ) n n = n(n ) n n k k + k(k + ) 4k(k + ) = n 8 () ( ) + ( ) = 0 ( p.78) ( ) + ( ) = = ( + ) ( + ) + = 4 + = 0 = + = + ( ) = 4 + = 0 ( ) = 0 = = = 86 (4.00) th-note
95 () ( ) + ( z) + (z ) = 0 ( ) + ( z) + (z ) = ( + + z ) z z = ( + z + z) z z = 0 C. q r q r q r * 5 7 () ac = bc c = 0 a = b () ab = 0 a = 0 b = 0 () c 0 a = b c 0 ac = bc c a = b () a 0 b = 0 a 0 ab = 0 a b = 0 D. = a = b p.6 = a = b( a)( b) = 0 = a = b( a)( b) = = + = = ( )( ) = + = ( + ) = + = 0 = = z = z, + z + z =,, z,, z ( )( )(z ) = 0 ( )( )(z ) = z z z z = z ( + z + z) + ( + + z) = z ( ) z = 0,, z *5 r q th-note C. 87
96 5.. (a + b) (a + b) = (a + b)(a + b) = (a + b) (a + ab + b ) = a a b + 6 ab + = a + a b + ab + b 4 ba + 5 ab + 6 b a ab b a a a b ab b ba ab b ( + ) i) ( + ) = () + () + () + } {{ } = ii) ( + ) (a b) = a a b + ab b = ( + )( + ) 5 (a + b) = a + a b + ab + b, (a b) = a a b + ab b = ( + )( ) = = a = 5, b = a b, ab. (a) ( + ) (b) ( + 4) (c) ( + ) (d) ( + ). a b = (5) = 50, ab = 5 = 60. (a) ( + ) = (p.88) = (b) ( + 4) = = (c) ( + ) = () + () + () + = (d) ( + ) = () + () + () + = (4.00) th-note
97 (a b) = a a b + ab b (a + b) = a + a b + ab + b (a b) (a b) = { a + ( b) } b ( b) = a + a ( b) + a( b) + ( b) = a 6a b + ab 8b (a + b) n II (a + b) 4 = a 4 + 4a b + 6a b + 4ab + b 4 (a + b) 5 = a 5 + 5a 4 b + 0a b + 0a b + 5ab 4 + b 5 () (a 4) () (a ) () (a + 5) + (a 5) () (a 4) = a + a ( 4) + a ( 4) + ( 4) = a a + 48a 64 () (a ) = (a) + (a) ( ) + (a) ( ) + ( ) = 7a 54a + 6a 8 () (a + 5) + (a 5) (a 4) = { a + ( 4) } (a ) = { a + ( ) } = (a) + (a) 5 + (a) (a) + (a) ( 5) + (a) ( 5) + ( 5) = 8a + 50a + 8a + 50a = 6a + 00a B. (a + b)(a ab + b ) (a + b) (a ab + b ) = a = a + b a b + ab + 4 ba 5 ab + 6 b a ab b a a a b ab b ba ab b ( + )(9 + ) i) ii) ( + )(9 + ) = ( + ) { () () + } } {{ } = 7 + (a b)(a + ab + b ) = a b ( + )(9 + ) = = 7 + th-note C. 89
98 6 (a + b)(a ab + b ) = a + b, (a b)(a + ab + b ) = a b a ± b a ± b p.90. ( + )( + 4), (ab )(a b + ab + 9) a) ( + )( ) b) ( + )( ) c) ( + )(4 6 9) d) ( )( ) e) ( )( ) f) ( )(4 6 9). ( + )( + 4) = + = + 8 (p.89) (ab )(a b + ab + 9) = (ab) = a b 7. ( + )( ) = () + ( )( ) = () b)8 7 d) C. (p.89) 8 + i) ii) 8 + = () + = ( + ) { () + } = ( + )(4 + ) ( + )(4 + ) = ( + ) { () + } = () + = 8 + (p.90) 7 a + b = (a + b)(a ab + b ), a b = (a b)(a + ab + b ) ± a ± b a ± b a a 5b. + 7 = (4.00) th-note
99 = ( + )( + 9). 8a + = (a) + = (a + )(4a a + ). 8 7 = () () = ( )( ) 4. 64a 5b = (4a) (5b) = (4a 5b)(6a + 0ab + 5b ) 4a a a 8a 4a a 4a a a 0ab 5b 4a 64a 80a b 00ab 5b 80a b 00ab 5b D. 4 ( ( ) a + ) b (p + q)(p pq + q ) 4 ( + 4) 5 (a + b) (a b) = + ( ) + ( ) + ( ) = (p.88) = a + a ( ) b + a ( ) b + b = a + a b + 4 ab + 8 b (p.88) = (p + q)(p pq + q ) = (p + q ) = p + q (p.90) 4 = {( + )} = ( + ) iii)(p.40) = 8( ) = (p.88) 5 (a + b)(a b) = {(a + b)(a b)} = { (a b ) } (p.40) (p.46) = (a ) + (a ) ( b ) + (a ) ( b ) + ( b ) (p.88) = a 6 a 4 b + a b 4 b 6 (p.46) (p.40) (a 4 ) = a 8 5 4a 8b + 8 a 6 b 6 4 a + ab + b + = (7a) (b) = (7a b)(49a + 4ab + 4b ) (p.90) = ( + 7 ) = ( + )( + 9 ) (p.90) th-note C. 9
100 = (a + b )(a b ) (p.55) = { (a + b)(a ab + b ) } { (a b)(a + ab + b ) } (p.90) = (a + b)(a b)(a ab+b )(a +ab+b ) 4 b a b a + ab + b + = (a + )b + (a + ) = (a + )b + (a + )(a a + ) = (a + )(a a + b + ) E. + (p.88) (p.89) = +, = + = 4, =, = ( ) = + = ( + ) = 4 ( + ) = = ( + )( + ) = 4 (4 ) = 5 + = ( + ) = ( + ) ( + ) = 4 4 = ( + )( + ) = = ( + )( + ) = ( + )( + ) ( + ) = = 74 6 = 7 +, = 7 () + () () (4) = 7, =, = 7 = 5 ()= ( + ) = 8 0 = 8 () = ( )( + + ) = (8 + 5) = 46 ( ) = + = ( ) + = ( ) + ( ) = ( ) + 5 = 46 () ( + ) = = ( + ) = 8 5 = 4 50 = 74 (4) ( + )( ) = = ( + )( ) + = ( + )( ) + ( ) = = = (4.00) th-note
101 4 cm5 cm7 cm. 90. (tan) (cos) (sin). B (hpotenuse) BC * B BC (opposite side) C (base) C BC B BC C B F E DEF D C D C B F E : : : DE : EF : FD D * th-note 9
102 LMMNNL PQQRRP N M Q P L R MN NL LM PR QR PQ B. (tan) C B C = 0.75 CB 0.75 C = CB B C C BC 0.75 B C B C (tan) BC B tan = = CB C * (tangent) tan C tan t t C CB tan tan Btan C 4 4 B C tan = 4 tan B = 4 = tan C = = 4 B 4 C * tan t a n tan tan sincos 94 (4.00) th-note
103 C. (cos) (sin) B B = C B = 0.75 C 0.75 B = C B = B C B = 0.75 BC 0.75 B = BC B C C BC (cos) (sin) B cos = = C B (cosine) cos C B sin = = BC B (sine) C sin cos, sin c, s tan 4.. cos sin. cos Bsin B 4 4 B. = 4 + = 5 = 5 = 4 + = 0 = 5. cos = 5 sin = 4 5. cos B = 4 5 = 5 5 sin B = 5 = 5 5 c s th-note. 95
104 5 () cos sin tan () cos Bsin Btan B () cos Csin Ctan C (4) cos Dsin Dtan D B C 0 D () = 5 cos = 5 sin = tan = 5 () 7 5 = 4 = 6 cos B = 5 7 sin B = 6 tan B = (5 ) ( ) () + 0 = 45 = 5 5 cos C = 5 = sin C = 0 5 = tan C = 0 = 5 (4) cos D = 0 5 = sin D = 5 5 = tan D = 5 0 = 4 D. (trigonometric ratio) p.4 6 p.4 0 < < 90. cos 40 sin = cos B sin 0 B. p.4 cos = 76. p.4 sin 0 0.4B (p.05) B = 70 E cos 0 sin 0 tan 0. cos 45 sin 45 tan 45. cos 60 sin 60 tan (4.00) th-note
105 . cos 0 =, sin 0 =, tan 0 = =. cos 45 = =, sin 45 = =, tan 45 = = 0. cos 60 =, sin 60 =, tan 60 = = p tan = tan = tan = t tan t {}}{ tan = cos, sin z z z c z z z s {}}{{}}{ z cos = z sin = z 8 sin = 5, cos = 4 5, tan B = 6, cos B =. s CD B 5 B C D CD = sin =. D c B c = (D cos ) cos B = D cos cos B =. : D : 5 5 =. : BC : = 4 6 th-note. 97
106 9 () D = 6 6 sin 6 cos sin B D () C = 5 CDBD B B B C () 6 D s CD 6 sin = CD 6 D c C C s B 6 cos sin B = C sin B = B () 5 C t CD CD = 5 tan 5 C s B B = 5 sin B 5 D cos = 5 D = cos B. BC 0, 60, 90 B : BC = : 0 = C 6 0 B C BC = = B BC = 6 cos = C 45 B.. BCRQPR 60 = P 4 60 R Q. : : :. BC = = RQ = 4 = PR = 4 = 6 98 (4.00) th-note
107 C..5 m 5.0 m * 4 m p.4.5 m m TH T T TH TH = H tan m m = 6.0 m.5 m m H p.4 tan D a b a b (comple fraction) *4 *5 7 = 7 = 7 7 = 7 7 = = = = 5 5 = 0 * *4 (compound fraction) *5 7 7 th-note. 99
108 m.0 m m p.4 TH T TH TH = T sin m = 8. m = 9. m m 50.0 m T H p.4 sin C B BC = 90, BC = 5, C = 40 m () BC m p.4 () C 80 m D BDC p.4 () BC = 40 m tan 5 = (m) p.4 tan 5 = () tan BDC = BC DC = 8 80 = 0.5 p.4 9 tan 9 = 0.44 tan 0 = B 5 5 () 4 () 8 () (4) a () () () = = = = = 6 = = = = = = (4.00) th-note
109 (4) a = a = a = 4a E p.46 * BC = 75, B = 60, C = 45 BC *7 DB C E BD = () BD () CBC () BEE (4) cos 5, sin 5, tan 5 (5) cos 75, sin 75, tan 75 () BD DB : B : D = : : BD = B =, D = () CD D : DC : C = : : D = C = D = 6CD = D = BC = BD + CD = + () BEC BE : EC : CB = : : BC = + BE = BC = EC = E = C + CE = = (4) EB B cos 5 = BE + 6 B = 6 + = 4 sin 5 = E 6 B = 6 = 4 ( ) ( ) = = ( ) ( ) = tan 5 = E BE = (5) EB 6 6+ B 75 E D C cos 75 = E B = sin 75 = BE B = tan 75 = BE E = = = = = ( 6 + ) ( 6 + ) ( 6 ) ( 6 + ) = + * *7 BC BC BC th-note. 0
110 .. tan = sin cos p.97 = z cos, = z sin tan = = z sin z cos = sin cos tan = sin cos B. cos + sin = + = z (z cos ) + (z sin ) = z z (cos ) + z (sin ) = z (cos ) + (sin ) = z (cos ) (sin ) (tan ) cos sin tan *8 cos + sin = 7. sin = sin cos cos. sin = 5 cos tan. sin = (sin ) = 4 9 cos = sin = 5 9 cos > 0 cos = 5 9 = 5. cos + sin = ii) ( ) cos = sin = = cos > 0 cos = 6 5 = 4 5 tan = sin cos tan = = 4 i) (p.99) *8 cos cos( ) cos (cos ) cos cos cos( ) (cos ) cos 0 (4.00) th-note
111 8 0 < < 90 () cos = sin tan () sin = cos tan ( ) () sin = cos = = 8 9 ii) sin > 0 sin = 8 9 = tan = sin cos tan = ( ) () cos = sin = = 5 5 = i) (p.99) 9 ii) cos > 0 cos = = tan = sin cos tan = = = i) (p.99) C. tan sin, cos tan + tan = 9tan cos cos + sin = + tan = cos cos cos + sin = cos + sin cos = cos + tan = cos 0 0 < < 90 tan = 7 cos sin + tan = cos cos = + tan = + 7 = 50 cos > 0 cos = 50 = 5 = tan = sin cos sin = tan cos = 7 0 = iv) i) th-note. 0
112 0 < < 90 tan = 5 cos sin + tan = cos cos = + tan = + ( ) = = 5 6 iv) cos > 0 cos = tan = sin cos sin = tan cos = = 5 6 = i) i) tan = sin sin cos tan cos ii) cos + sin = iii) + tan = cos sin cos cos tan sin tan i) ii) iii) cos iii) ii) cos tan tan = sin cos tan cos, sin tan = sin cos tan = sin cos tan = cos sin 04 (4.00) th-note
113 D. 90. cos, sin, tan. cos(90 ), sin(90 ), tan(90 ) cos = sin = 5 tan = 5. cos(90 ) = 5 sin(90 ) = tan(90 ) = cos(90 ) = z = sin sin(90 ) = z = cos z 90 z tan(90 ) = = tan () 45 ) sin 80 ) cos 46 ) tan 8 () sin 0 + sin 70 () ) sin 80 = sin(90 0 ) = cos 0 sin(90 ) = cos ) cos 46 = cos(90 44 ) = sin 44 cos(90 ) = sin ) tan 8 = tan(90 8 ) = () sin 70 = cos 0 sin 0 + sin 70 = sin 0 + cos 0 = tan 8 tan(90 ) = tan ii) sin + cos = 45 < < 90 0 < < 45 p.4 cos 89 = sin cos 88 = sin th-note. 05
114 PQ sin θ = PQ P = PQ, Q cos θ = P = Q *9 θ P Q sin θ cos θ 5. PQ P Q Q Q P cos θ, sin θ. X θ P Q 5 4 P Q 60 X. P Q PQ 5 P : P = 5 : Q = Q 5 = 4 5, Q P = QP 5 = 5 P Q PQ cos θ = Q = 4 cos θ = Q = 4 Q P, sin θ = = P 5, cos θ = 5 5 QP P = 5.,, X, = X 60 *9 θ φ p.i 06 (4.00) th-note
115 B. (unit circle) PQ P(cos θ, sin θ) cos θ = Q =P sin θ = QP =P θ Q tan θ = QP Q = P P = P 6. P. (p.4) P 60 Q. P 0. P 50. PQ,, P,. P. P,. P cos 50 sin 50 (p.4) P (0.648, 0.766) θ P P θ Q P = θ Q P = θ Q P θ Q P θ Q θ Q P (angular point) * 0 P θ P θ P P (radial vector) X (initial line) *0 th-note θ X th-note. 07
116 7 () B () 45 B (),, 45 () ( ) + = = = = 5 5 = 5 = 45 B 8 () () P P 45 () () P,, P P, C. P 0 80 P X PX = θ (0 θ 80 ) cos θ = P sin θ = P P tan θ = P = * P P(, ) P 0 θ = 90 tan θ cos θ sin θ θ X * P = P P 08 (4.00) th-note
117 9 I P II III 0 80 Q X S X X. I P II S. cos 0, sin 0, tan 0, cos 80, sin 80, tan 80. X = 5 III 4. sin 5 cos 5 tan 5. PQ = 60 PQ,, P, II S(, 0). cos 0 =, sin 0 =, tan 0 = = cos 80 =, sin 80 = 0, tan 80 = 0 = 0,. B P, 4. cos 5 =, sin 5 =, tan 5 =, = B 5 X D. p.4 0 cos θ = θ m P P = m : = P Q θ X PQ = θ θ = : : PQ : θ = PX = 80 PQ = 0 PQ = 60 th-note. 09
118 I II III X X X. PX = 0 P I cos 0, sin 0, tan 0. QX = 50 Q II cos 50, sin 50, tan 50. RX = 90 R III cos 90, sin 90. P ( ),, P, cos 0 =, sin 0 =, tan 0 = =. P ( ),, P, cos 50 =, sin 50 =, tan 50 = =. P P (0, ) cos 90 = 0, sin 90 = P 50 P 0 P 90 sin θ = θ = m : = PP m P Q θ P θ X Q PQ P Q sin θ = θ =, : : PQ P Q, PQ = P Q = 0, : PX = 0 : P X = 80 P Q = 50 0 (4.00) th-note
119 θ 0 θ 80 () cos θ = () sin θ = () tan θ = (4) sin θ = () = P PQ : : PQ = 0 θ = 80 0 = 50 () = PP PQ P Q PQ = 45, P Q = 45 P P Q Q θ θ θ Q P X X θ = 45 θ = = 5 P () P P PQ : : PQ = 60 θ = = 0 = Q θ = X (4) = P θ = 90 θ X 4 θ 0 θ 80 cos θ sin θ > tan θ > 50 θ 80 > P P 50 X P 45 < θ < 5 > P 5 45 X 0 θ < 90 0 < θ 80 0 X = th-note.
120 θ sin θ 0 0 cos θ 0 tan θ p θ (p.6) 80 θ (p.5).. 0 θ 80 θ 0 θ 80 i), iii) 0 i) tan θ = sin θ cos θ ii) cos θ + sin θ = iii) + tan θ = cos θ sin θcos θtan θ sin θ cos θ cos θ tan θ sin θ tan θ i) ii) iii) cos θ cos θ =, sin θ = tan θ = = sin θ cos θ P(, ) sin θ tan * + = θ sin θ + cos θ = cos θ iii), iv) * p.08 (4.00) th-note
121 6 0 α 80. sin α = 5 cos αtan α. cos α = sin αtan α ii)iii)iv) sin costan. cos α + sin α = ii) ( ) cos α = sin α = = cos α = ± 6 5 = ± 4 5 tan α = sin α cos α cos α = 4 5 tan α = 5 = cos α = 4 5 tan α = 5 = i) cos α = ± 4 5, tan α = ± 4. cos α + sin α = ii) ( ) sin α = cos α = = 8 9 sin α 0 sin α = 8 9 = tan α = sin α cos α tan α = = i) 0 α 80 sin α 0 7tan θ cos θ + sin θ = + tan θ = cos θ, + tan θ = sin θ cos θ + sin θ = cos θ + sin θ cos θ = cos θ + tan θ = cos θ cos θ + sin θ = sin θ cos θ sin θ + = sin θ tan θ + = sin θ th-note.
122 8 0 α 80 7 () cos α = sin αtan α 4 () sin α = cos αtan α () tan α = 7 cos αsin α () cos α + sin α = ii) ( ) sin α = cos 7 α = = sin α 0 sin α = 9 6 = 4 0 α 80 sin α 0 tan α = sin α cos α = = 7 = 7 7 () cos α + sin α = ii) ( ) cos α = sin α = = 7 9 cos α = ± = ± tan α = sin α cos α i) 7 4 cos α = tan α = = = cos α = ±, tan α = ± 7 4 cos α = 7 tan α = 7 = 7 () tan α + = cos α = cos α + tan α = cos α = ± 50 = ± tan α = sin α cos α 7 = sin α ± 0 sin α = = 50 0 ( ± 0 ) = ± 7 0 iv) i) 0 sin α sin α = 7 0 tan cos cos α = 0, sin α = (4.00) th-note
基礎数学I
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More information5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i
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More informationNo δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
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More information.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +
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