zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {

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Download "zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {"

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1 04 zz + iz z) + 5 = 0 + i z + i = z i z z z y zz + i z z) + 5 = 0 z i) z + i) = 9 5 = 4 z i = i) zz i z z) + = a {zz + i z z) + 4} a ) zz + a + ) z z) + 4a = 0 4a a = 5 a = x i) i) : c Darumafactory -- UltimateMath

2 z > 5 4 z w = z z w T { T = w w = z z z 5 } 4 T w w w 005 z z = w α, β α + β =, αβ = w z 5 4 α 5 4, β 5 4 β 4 i 4 i α 5 4, α 5 4 α = 5 4, α = 5 4 α = ± 4 i w = αβ = α α 5 6 α = ± 4 i w w = ± 4 ) i ± 4 ) i = 9 6 = 5 6 c Darumafactory -- UltimateMath

3 a),b) z a) z, z b) z 999 z + z = z + z, z + z = z + z 0 z + z z + z + = z + = 0, ±, ± z i ) z + = 0 z + z = 0 z z z ii ) z + z = z + z = zz ; z = z z + z c Darumafactory -- UltimateMath

4 z = ± i, ± i, iii) z + = z + z = zz ; z z = 4 z z + z z = z + =, z z + z 0 z ) z = ± ) ± i z = ± ± i) z = ±) z = ±) c Darumafactory -4- UltimateMath

5 4 α + i i β ) α + β + π ) αβ π ) α ) z = α + β β = β z α = α + i i 4 + π ) z = αβ, β = β z = z = α α α z z π c Darumafactory -5- UltimateMath

6 ) y z = α = + ti) = t + ti t ) z = x + yi x = t, y = t, t z 4 y ) y + ) dy = ) = 8 5 t = ) t = ) x = y 4 x ABCD 4 A,B,C,D α, β, γ, δ α = i, γ = 0 + 5i, β > δ β = + i, δ = + i β, δ = γ α) {cos 45 ± i sin 45} + α β γ = 0 + 5i = 0 + 4i) ± i) + i = 7 + 8i, 7 + 8i δ 7 + 8i = 7, 7 + 8i = 4 α = i β = 7 + 8i, δ = 7 + 8i c Darumafactory -6- UltimateMath

7 6 n α n = cos π n + i sin π p, q n p + qα n A n A =, A 4 =, A 5 = n = A p + qα n = p + qα n ) p + qα n ) = p + q + pq α n + α n ) = p + q + pq cos π n = ) n = p + q pq = q pq + p = 0 q p 4 p ) 0 p 4 0 p = 0, p = ± p, q) = 0, ±),, 0),, ),, 0),, ) A = 6 ) n = 4 p + q = c Darumafactory -7- UltimateMath

8 p, q) =, ),, ),, ),, ) A 4 = 4 ) n = 5 p + q + pq cos π 5 = cos π 5 pq = 0 p, q) =, 0),, 0), 0, ), 0, ) A 5 = 4 n = + ) + i = + i + 0 ) + i = 0 + ) + i = + i 0 ) + i = i ) + i = i + 0 ) + i = 4 6 = c Darumafactory -8- UltimateMath

9 7 6 A,7 + 7i B 4t ) i t P i)t 7 ) APB ) P t 00 P ) B 7 + 7i 4t ) i)t 7 6 4t ) i)t 7 = 7 + 7i) t 7 it) 4 t ) 6 t 7 it) 4 t ) = 7 + i) t = 7 t APB = 45 cos 45 + i sin 45 ) A ) ) P AB i 6) cos 45 + i sin 45) + 6 = + 7i) + i) + 6 = + 4i + 4i P P 6 + 8i 6 + 8i = t = 8 4 t ) i) t 7 t > 0 c Darumafactory -9- UltimateMath

10 8 a, a,, a n, { a =, a = i a n+ = a n+ + a n n =,, ) b n = a n+ a n n =,, ) i ) b, b, b C ) b n n =,, ) C 00 ) a = a + a = i + b a 4 = a + a = i + b = a a = i b = a a = i b = a 4 a = i + i + = + i b b b b = i b b i = i) + i) 0 = i b b = b b, b b b = 90 b 5 c Darumafactory -0- UltimateMath

11 ) a n+ = a n+ + a n a n+ = + a n a n+ a n+ b n+ = + b n babababababababababababababababab w = + z w z z z 5 = w w z b n = b n+ b n 5 = b n+ 5 = b n+ b n+ = 5 b n+ = 5 b n+ b n+ b n+ + b n+ ) + 9 = 5 { b n+ b n+ + b n+ ) + } 4 b n+ ) b n+ + b n+ + 4 = 0 b n+ 5 = b n C c Darumafactory -- UltimateMath

12 RadicalMath 9 Pα),Qβ) l, l l Rw) γ C Cγ) ) w = αβ Pα),Qβ) A 000 RP = RQ = 90 w w α = α, w β = β w αw + αw) = 0 w ) βw + βw = 0 w = αβ c Darumafactory -- UltimateMath

13 αβ ) ααβ + ααβ = 0 αβ ) βαβ + βαβ = 0 { α β ) } β + β = 0 { β α } α + α) = 0 β ) β + β = 0 α α + α) = 0 α = α = β = 0 a, b x + ax + bx + = 0 α, α, α i, j α i α j = a, b 00 i, j α i α j = α c Darumafactory -- UltimateMath

14 α α c α α c α α c R ) α = c +, α = c + i, α = c i α α α = c + ) c ) + i c ) i { c + ) c ) } + = 4 c + ) c c + ) = c + = c = a = c + + c + i + c a = b = c + ) c ) + c c + ) = c = 4 ) i α = c, α = c + + i, α = c + + i = c Darumafactory -4- UltimateMath

15 α α α = c ) c + ) + i c + ) i { c ) c + ) } + = 4 c + ) c + c + ) = c = c = 0 a = c + c + + i + c + a = 0 b = 0 i z, z, z, z 4 w = z z )z z 4 ) z z 4 )z z ) = ) z z = z ) a, z, z w z ) z = zz = z = z ) z i = z i i =,, ) w R w = w c Darumafactory -5- UltimateMath

16 z z ) z z 4 ) z z 4 ) z z ) = z z ) z z 4 ) z z 4 ) z z ) ) ) z z ) z z 4 ) z z 4 ) z z ) = z z z z 4 ) ) z z 4 z z z z ) z z 4 ) z z 4 ) z z ) = z z ) z z 4 ) z z 4 ) z z ) z z 4 = z z 4 z z 4 z z 4 z z 4 ) z z 4 ) = z z 4 ) z z 4 ) z z 4 z z z 4 + z z 4 = z z 4 z z z 4 + z z 4 z z ) z 4 ) = 0 z 4 = α, β 0 α β + α β = 0,α, β 005 α β + α β = k k R) [ ] α β = + ki 0, α, β α k > kiβ β α, β, γ α + β + γ = α + β + γ = 0 α, β,, γ 005 c Darumafactory -6- UltimateMath

17 α, β, γ 0 γ α γ + β γ + = 0 ) ) α β + + = 0 γ γ ) ) α β = 0 γ γ ) ) α β, γ γ t + t + = 0 α γ, β ) = cos 0 ± i sin 0, cos 0 i sin 0 ) γ α, β,, γ 4 ABC α, β, γ α, β, γ i ) ABC ii ) α + β + γ = iii) αβγ ) z = α β γ z ) α, β, γ 0 arg α arg β arg γ < c Darumafactory -7- UltimateMath

18 ) α + β + γ = α + β + γ = ) α ±0 β, γ β, γ = α ) cos 0 ± sin 0 ) β = + z ) ± i γ = + z ) i ) ω β = + zω, γ = + zω αβγ = z + ) + zω) + zω ) = z + ) z + z ) = + z z = z = z z z ) z = cos 0 + i sin 0 z = cos 40 + i sin 40 + z z = cos 0 + i sin 0 z = cos θ + i sin θ z = cos θ + i sin θ θ = n θ = n θ = 40, 60, 80 c Darumafactory -8- UltimateMath

19 z = cos θ + i sin θ, w = cos φ + i sin φ z + w θ φ 80 arg z + w) = θ + φ φ θ φ > 80 arg z + w) = 80 + θ + φ φ y θ + ϕ y θ + ϕ 80 + θ + ϕ θ ϕ x θ ϕ x + z = + cos θ + i sin θ θ 80 + θ θ = 40, 60, 80 + z 0, 80, 0 arg α = 0, arg β = 80, arg γ = 0 c Darumafactory -9- UltimateMath

20 5 0 z, z, z ) arg z = arg z + 0 ) z z, z 0 ) z z z ) α = cos 60 + i sin 60 αz = pz + qz, αz = sz + tz p, q, s, t z, z ) z = az + bz z, z 997 ),z ), z ), z ), αz ), αz ) z z αz αz z,z ), z ) αz = z z z c Darumafactory -0- UltimateMath

21 p = 0, q = k k αz ) z )z ) αz = z z z + z s = z z, t = ), z ), z ), z ) z z z + z z z = z z z z + z = z z = z z αz = z + z z = z + z z z z z + z ) z + z + z z z a = z + z, b = z + z z z,z, z w w = az + bz z i = k i i =, ) α = cos 60 +i sin 60 α = arg z = arg z +0 z = α z k k αz = pz + qz α α z k k = pz + qz z k k = pz + qz p = 0, q = k k c Darumafactory -- UltimateMath

22 αz = sz + tz α z = sαz + tαz k z = s z ) z + tαz k z, αz k z = s k ) z + t sz + tz ) k k k = ts, s k ) + t = 0 k k t =, s = k k z z = α z z ) z = z + αz αz = z + k z + z + k z k k = + k ) z + + k ) z k k 6 x, y) x + 4y = = u + iv x + iy u, v) x, y, u, v 965 z = x + iy, w = u + vi w = z z = w z x + 4y = z + z + 4 z z i w = c Darumafactory -- UltimateMath

23 z w w + w w w + 4 = i w + w) 4i w w) = w ) ) w i w + i w = 0 )) )) w + i w i = + i ) w + i = 5 5π ) + i 5 7 ) z 0 z 7 z + = 0 z ) n z n + z n = z + z z C z C:0 a a az + z) = z 974 c Darumafactory -- UltimateMath

24 [ ] a a z + z) = z 0 a i ) z + z = z = 0 z, z = ± i ii ) z + z \= a = z z + z 0 a 0 z z + z 0 z z + z z < 0 z + z z w = 9 iz z ) w z ) a w a z 004 ) w = iz z w c Darumafactory -4- UltimateMath

25 iz z = iz z z z ) = z z ) zz z z = 0 z = z \= ) w a iz z a z a z z a z a z a z + a i ) a < ii ) a = a ) z + a z + a z a z + a a a a z + z iii) a > a ) z a z a z a z a a a a a a a a c Darumafactory -5- UltimateMath

26 0 z w = z z w 005 w = z z z = cos θ + i sin θ 0 θ < π) w = cos θ + i sin θ) cos θ + i sin θ) cos θ cos θ = 0 cos θ cos θ = 0 cos θ = θ { x = cos θ cos θ y = sin θ sin θ r = + cos θ y y x x c Darumafactory -6- UltimateMath

27 A,P,Q,R,S,R,S AP= a,pq= b,qr=qr = c, RS=RS = d, APQ= SRQ= S R Q =α0 α π) RQ= PQR =β0 β π) AS AS sin α, sin β a, b, c, d A a R P α b S β β α d c Q c α R d S Q QP z = cos π α) + i sin π α) w = cos β + i sin β A : b + az) S : cw + dwz) S : cw + dwz) S d R α c β Q c β α R d b S A a α P AS = b + az cw dwz = b + az + cw + dwz b + az) cw + dwz) b + az) cw + dwz) AS = b + az cw dwz = b + az + cw + dwz b + az) cw + dwz) b + az) cw + dwz) c Darumafactory -7- UltimateMath

28 AS AS = b + az) cw + dwz) + b + az) cw + dwz) b + az) cw + dwz) b + az) cw + dwz) = b + az) c + dz) w + b + az) c + dz) w b + az) c + dz) w b + az) c + dz) w = bc + acz + bdz + ad) w + bc + acz + bdz + ad) w bc + acz + bdz + ad) w bc + acz + bdz + ad) w = ac bd) z z) w ac bd) z z) w = ac bd) z z) w w) = ac bd) i sin π α) i sin β = 4 ac bd) sin α sin β c Darumafactory -8- UltimateMath

mobius1

mobius1 H + : ω = ( a, b, c, d, ad bc > 0) 3.. ( c 0 )... ( 5z + 2 : ω = L (*) z + 4 5z + 2 z = z =, 2. (*) z + 4 5z+ 2 6( z+) ω + = + = z+ 4 z+ 4 5z+ 2 3( z 2) ω 2 = 2= z+ 4 z+ 4 ω + z + = 2 ω 2 z 2 x + T ( x)