φ + 5 2 φ : φ [ ] a [ ] a : b a b b(a + b) b a 2 a 2 b(a + b). b 2 ( a b ) 2 a b + a/b X 2 X 0 a/b > 0 2 a b + 5 2 φ φ :
2 5 5 [ ] [ ] x x x : x : x x : x x : x x 2 x 2 x 0 x ± 5 2 x x φ : φ 2
: φ ( ) r ae bθ φ φ 2 φ 0 φ + φ φ φ + + φ 3
φ + + + φ φ + + + + + + ( ) 0 2 3 a n+2 a n+ + a n, a a 2 {a n } 202 (Liber Abaci) [ ] 2 2 3 8 5,, 2, 3, 5, 8, 3,... 4
a n+ /a n a n+ lim n a n φ [ ] φ 2 φ + a n+2 a n+ + a n 2 a n+2 φa n+ ( φ)(a n+ φa n ), a n+2 ( φ)a n+ φ(a n+ ( φ)a n ) {a n+ φa n }, {a n+ ( φ)a n } a 2 φa φ, a 2 ( φ)a φ, φ, φ a n+ φa n ( φ) n, a n+ ( φ)a n φ n a n+ a n φn ( φ) n 2φ φn ( φ) n 5 a n+ a n φn+ ( φ) n+ φ n ( φ) n φ ( + φ φ )n φ 2 φ φ > φ φ < φ 2 φ ( φ φ )n ( ) n φ 0, φ ( ) n φ (n ) φ a n+ lim n a n φ 5
2 x x x x 3.76543 x 3 x 23.2 x 24 x x 0 x x < x > 0 x x x a 0 x x a 0 x x 0 x (x a 0 ) 0 < x a 0 < x > x a 0 + (x a 0 ) a 0 + x a 0 a 0 + x a x x x a 0 x 2 (x a ) x a + /x 2 x a 0 + a + x 2 0 < x a < x 2 > 0 x k a k x k x k a k 0 x k+ (x k a k ) x k 0 x a 0 + a + a 2 + a 3 +... x x [a 0 ; a, a 2, a 3,... ] a 0 a a k 6
2 < 2 < 2 a 0 2 2 a 0 2 x x 2 2 + ( 2 )( 2 + ) 2 + 2 2 2 + 2 < 2 + < 3 x a 2 + 2, ( 2 + ) 2 2 2 x k 2 +, a k 2 2 2 + 2 + 2 + 2 +... [; 2, 2, 2,... ] φ φ + /φ φ + + + +... [;,,,... ] φ a n+2 a n+ + a n a n+ b n a n+ /a n b b n+ + b n {b n } φ, + 2, + + 3 2, + + + 5 3, + + + 8 5,... + 7
lim n a n+ /a n φ x x [ ] x p, q x p/q q > 0 p q a 0, r p a 0 q + r. x p q a 0 + r q, 0 r < q x a 0 x a 0 r /q < x q/r q r a, r 2 q a r + r 2 x q/r a + r 2 /r x a, x a r 2 /r < x i+ r i /r i+ r i r i+ a i+, r i+2 r i a i+ r i+ + r i+2 x i+ r i r i+ a i+ + r i+2 r i+, (0 r i+2 < r i+ ) 0 < r i+ < r i < < r < q N r N+ 0 x N a N ax + b 0 2 2 n n n 8
2, 3 φ X 2 2 0, X 2 3 0, X 2 X 0 2 3 2 3 2 2 2 3 < 3 < 2 a 0 3, x 3 3 + ( 3 )( 3 + ) 3 + 2 a x x 2 3 + 2 3 3 + 2 a 2 x 2 2 x 3 ( 3 + ) 2 3 3 + 2 x k x k x, a k k x k x 2, a k 2 3 + [;, 2,, 2,... ] + 2 + + 2 +... 3,2 2 2 φ 7,,, 4 9
( ) x 2 x 2 2 A B D M C 3 E φ ABCD BC M M D BC E AB BE MC /2, CD MD (/2) 2 + 2 5/2 BE BM + MD /2 + 5/2 φ. AB : BE : φ n n n n π ( ) e ( ) π e π [3; 7, 5,, 292,,,, 2,, 3,, 4, 2,,, 2,... ] e [2;, 2,,, 4,,, 6,,, 8,,, 0,... ] 0
3 x x a 0 + a + a 2 + a 3 + a 4 +... a 0 x, x (x a 0 ), a x x 2 (x a ) x k a k x k, x k+ (x k a k ) x x k a k 0 x k x k+ x k a k + x k+ a kx k+ + x k+ () ( ) xk ( ak 0 ) ( ) xk+ (2) 2 (2) x k / (2) a k x k+ +, 2 x k+ x k x k+ () 2 2 x k x k+2 x k a kx k+ + x k+ a k a k+ x k+2 + x k+2 + a k+ x k+2 + x k+2 ( ) ( ) ( ) ( ) ( ) ( ) xk ak xk+ ak ak+ xk+2 0 0 0 (a ka k+ + )x k+2 + a k a k+ x k+2 + ( ak a k+ + a k a k+ ) ( ) xk+2
(2) ( ( ) ( ) ( ) ( ) ( ) ( ) x a0 x a0 a ak xk+ ) 0 0 0 0 ( ) ( ) ( ) pk r k a0 a q k s k 0 0 ( ) ak 0 ( ) ( ) ( ) ( ) pk+ r k+ pk r k ak+ pk a k+ + r k p k q k+ s k+ q k s k 0 q k a k+ + s k q k r k+ p k, s k+ q k {p k }, {q k } { p0 a 0, p a 0 a +, p k+ a k+ p k + p k, { q0, q a, q k+ a k+ q k + q k (3) a 0, a,... a 0 a (3) p k q k r k+ p k, s k+ q k ( ) ( ) ( ) ( ) ( ) pk p k a0 a ak ak q k q k 0 0 0 0 a j 0 k + p k q k ( ) k+ p k q k (4) p k q k p k q k ( ) k+ (k, 2, 3,... ) (5) p n lim x n q n 2
[ ] p k, q k (3) p 0 q 0 a 0, p q a 0 + a, p 2 q 2 a 0 + a + a 2,..., p k q k a 0 + a +... a k + a k ( ( ) ( ) x pn p n xn+ ) q n q n ( ) ( ) ( ) ( ) ( ) xn+ pn p n x ( ) q n q n n+ qn p n x q n p n x n+ q n x p n q n x + p n x n+ > 0 q n x p n q n x + p n > 0 ( q n x + p n ) 2 x 2 (q n x p n )(q n x p n ) < 0 q n q n x p n /q n p n /q n p n q n p n q n ( ) n+ q n q n p n q n p n q n ( )n+ q n q n p n x q n < p n p n q n q n q n q n q k q k+ a k+ q k + q k lim n q n lim n p n /q n x x φ a 0, a,... {q k } k {p k } k. 3
2 x 2 x 2 x 2 2 [ ] x 2 ax 2 + bx + c 0, a X 2 + b X + c 0 X x ax 2 + bx + c 0, a x 2 + b x + c 0 a a x 2 (a b ab )x + (a c ac ) 0 x a b ab 0 a c ac 0 a : b : c a : b : c a /a 2 2 x 2 ax 2 + bx + c 0 a > 0 a, b, c x 2 0 x 2 x x x x b ± b 2 4ac 2a x b b 2 4ac 2a ( ) ( x b 2 4ac 0 x x.) 4
3 α, β, γ, δ αδ βγ ± x, y ( ( ( ) y α β x ) γ δ) x 2 y 2 x 2 x 2 y 2 x, y ( ) ( ) ( ) y α β x γ δ [ ] y 2 2 ax 2 + bx + c 0 a 0 ay 2 + by + c 0 x y ( ) 2 ( ) αx + β αx + β a + b + c 0 γx + δ γx + δ Ax 2 + Bx + C 0 A aα 2 + bαγ + cγ 2, B 2aαβ + b(αδ + βγ) + 2cγδ, C aβ 2 + bβδ + cδ 2 x AX 2 + BX + C 0 2 X 2 A 0 aα 2 + bαγ + cγ 2 0 γ 0 aα 2 0 a 0 α 0 αδ βγ ± γ 0 γ 2 ( ) 2 α a + b α γ γ + c 0 y 2 ax 2 + bx + c 0 α/γ A 0 y 2 x y x x y B 2 4AC (αδ βγ) 2 (b 2 4ac) b 2 4ac 5
ax 2 + bx + c 0 AX 2 + BX + C 0 y y a(y ) 2 + by + c 0 αδ βγ y (αz + β)/(γz + δ) z Az 2 + Bz + C 0 z x z x y (αx + β)/(γx + δ) x x k x x k ± x 2 x k x i 2 x j 2 4 d 2 ax 2 + bx + c 0 d ac < 0 [ ] b 2 4ac d ac < 0 0 > 4ac b 2 d b 2 < d b b 4ac b 2 d (a, c) 2 x 2 x > 0 x < 0 x x > 0 3 x 2 x x x x x > 0 x 2 2 ax 2 + bx + c 0 ( ) ( ) ( ) x pk p k 2 xk q k q k 2 6
3 x k 2 A k X 2 + B k X + C k 0 (k, 2, 3,... ) B 2 k 4A kc k k b 2 4ac 3 x, x k ( ) ( ) ( ) x pk p k 2 x k q k 2 q k ( ) ( ) x ( ) ( ) ( ) k pk p k 2 x ( ) q k q k 2 k qk 2 p k 2 x q k p k x k q k 2x p k 2 q k x + p k q k 2 q k x p k 2 q k 2 x p k q k q k, q k 2 lim n p n /q n x k x x x k < 0 ( ) xk+ ( ak 0 ) ( ) xk ( ) ( ) 0 xk a k 3 x k, x k+ x k+ x k a k x k < 0 < x k+ < 0 x k < 0 m < x m < 0 x k x k > m x m x m < 0 x m x m 2 2 A m X 2 + B m X + C m 0 A m C m < 0 B 2 m 4A m C m m b 2 4ac 4 2 7
x N < 0 N x N+i (i, 2,...) 2 A N+i X 2 + B N+i X + C N+i 0 n n + l 2 x n x n+l 2 x n x n+l x i x i+l (i n, n +,...) a j x j a i a i+l (i n, n +,...) x a n, a n+,..., a n+l 2 x l n n i a i a l+i x i x l+i x i x l+i (i n, n +,... ) x n x n+l ( ) ( ) ( ) ( ) ( ) xn+l xn an an+l xn+l 0 0 ( ) α β γ δ ( ) an 0 ( ) an+l 0 γ a n+ a n+l 0 (l γ ) ( ) ( ) ( ) xn+l α β xn+l γ δ x n+l αx n+l + β γx n+l + δ γx2 n+l + (δ α)x n+l β 0 x n+l 2 3 x 2 2 x () a : x [a; a, a,... ] (2) a, b : x [a; b, a, b,... ] 2 x x N > < x N < 0 N x x 0 8
[ ] x x 0, x, x 2,... 2 x N x N > < x N < 0 N m x m > < x m < 0 x m > < x m < 0 n x n > < x n < 0 n N n > N n x n x n+l l ( ) ( ) ( ) xn an xn, 0 ( xn+l ) x n x n+l a n a n+l ( an+l 0 ) ( ) xn+l ( an+l 0 ) ( ) xn n > N n N, n + l N x n x n+l x n, x n+l 2 d 2 r, r 2, s, s 2 x n s + d, x n+l s 2 + d r r 2 x n x n+l s s ( 2 + ) d r r 2 r r 2 d x n x n+l r r 2 r r r 2 x n x n+l s r s 2 r x n x n+l a n a n+l x n x n+l x n x n+l n > N n N, n+l N < x n < 0 < x n+l < 0 x n x n+l < a n a n+l < a k a n a n+l x n x n+l x n x n+l x n+2l x n+3l x n n N n 9
4 n n a, b, c,..., c, b, a [ ] x n + n x 2 X 2 2 n X + ( n ) 2 n 0 2 x x [a 0 ; a, a 2, a 3,... ] x x n n x >, < x < 0 x a 0, a, a 2,..., a l a l a 0, a l+ a, a l+2 a 2,... l x a 0 + a +... + a l + x a 0 x 2 n x x a 0 ( n + n) 2 n n n n n n [ n ; a, a 2,..., a l, a 0 ] 20
a,..., a l, a 0 a,..., a l, a 0 /x x /x 2 X 2 +2 n X+( n ) 2 n 0 2 x x, x 2,... x k a kx k+ + x k+ x k x k+ a k x k+ + /x k+ + a k ( ) ( ) ( ) /xk 0 /xk+ a k x l+ x ( ) ( ) ( ) ( ) /x 0 0 /x a a l /x y 3 ( ( ) ( ) ( ) y 0 0 y ) a a l ( 0 a k ) ( ) ak 0 ( ( ) y al ) 0 ( ( ) a y 0) y a l + a l +... + a + y ( ) y ( n n ) n n x x y x a l + a l +... + a + x 2
x a k a l k (k, 2,..., l ) [ n n ; a, a 2, a 3,..., a 3, a 2, a, 2 ] n a, a 2,..., a 2, a 9 [4; 2,, 3,, 2, 8] 24 [; 7, 2,,,, 3,, 4,, 3,,,, 2, 7, 22] 39 [;, 3,, 3, 7,,, 2,, 2,,, 7, 3,, 3,, 22] 240 [46; 3,, 5, 2, 2,,, 9,, 2, 3,,, 22,,, 3, 2,, 9,,, 2, 2, 5,, 3, 92] 22