Μιγαδικοί Αριθμοί - Επανάληψη

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    27-Jun-2015
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Μια γρήγορη επανάληψη-φρεσκάρισμα των Μιγαδικών Αριθμών των Μαθηματικών Κατεύθυνσης Γ Λυκείου πριν τις εξετάσεις.

Transcript of Μιγαδικοί Αριθμοί - Επανάληψη

  • 1. , , . 1 ! .86: ( C ) .87: ( , ) .88-90: C. 2 .89 .90: ( ) .90: ( i) .91: ( ) .91: : 1 + 2 = 1 + 2 .92: ( z2 + z + = 0) .93: .97: ( ) .97: .98: .98: : ( |z1 z2| = |z1| |z2| ) .99: : |z z0| = , > 0 |z z1| = |z z2| .124-5: :

2. 2 .93-94 2 . 99-100 8/95, 3/96, 4/96, 7/96 - Vieta 14A/96 z: z : 11/96, 6/96, 8/96 9/101, 1/101, 7/102, 10/102 12/96, 9/97, 4/101, 5/101, 6/101, 8/101, 2/101, 3/101, 4/102, 5/102, 6/102, 9/102, 1/123, 6/103 7/101, 8/102, 3/123 : 3. i * 4 + 4 4 , , = 4 + 0 < 4 1 , = 0 i , = 1 i = i = i i = i i = i = - 1 , = 2 - i , = 3 2015 =i 2 3 1996 ... =i i i i : i 1 : 4. : 1 + 2 1 2 + 2 2 3 + 3 + 3 3 1 20 2 3 + 2 3 2i 2i 2 2 i 2 2 i 3 8a i 3 8a 10 2 64i i, i , i : 5. z + , z - z+z=2Re(z) z-z=2Im(z)i : z w + z w z w - z w z z w w z z w w 2Re(z w) 2Im(z w) i 2Re z w 2Im i z w : 6. z=z1+z2i z1 , z2 C, : = 1 2 : = 1 2 , z . . ( z1 < z2 z1 , z2 C ) ! : 7. < 0 : , = , , ! , : z1+ z2= 2Re(z1) z1z2=|z1|2 Vieta: + = = : 1. 14 96 . 2. : z2 i z + 2 = 0 , z C. ! : z2 + z + = 0 , , R 0 : 8. z , , . z , z = x + y i x , y. : 9. z : z = + i : = 0 : = . z : z = + i : = 0 : = - . : z , w |z|=|w|= 3 , z1= 3+ . : 10. : z1 2 + z2 2 = 0 z1 0 z2 0 . z1 2 + z2 2 = 0, : 1 2 + 2 2 = 0 1 2 - i2 2 2 = 0 (z1 + iz2) (z1 iz2) = 0 z1 = -iz2 z1 = iz2 : iz2= 2 ! : 11. z = + i , , R z : w = i ( w = - + i ). : i = -i( + i) w = -i z + i = i( i) z = i w - + i = i( + i) -w = i z : z 4+2 + w 4+2 = ( i w )4+2 + w 4+2 = - w 4+2 + w 4+2 = 0 : : (3-i)2010 + (1+3i)2010 : 12. : z1 , z2 0 1 2 + 2 1 = 1. : . 1 3= -2 3 . 1 2 2010 + 2 1 2010 = 2 : 13. z = + i , : |z|= + |z|= |-z|=|| |z|2 = z |z1 z2|=|z1||z2| = , z20 : |iz| = |i| |z| = |z| : z1 , z2 , z3 : z1 = 1, z2 = 3, z3 = 5 :|z1 + z2 + z3|= 1 15 |z2z3+9z1z3+25z1z2| : 14. ||z1|-|z2|| |z1 + z2| |z1|+|z2| . |z|=2 z C w = 3 4i , : 3 |z+w| 7 : 15. z1 , z2 1 , 2 , |z1 z2| = (M1M2) , . : ||z1|-|z2|| |z1 - z2| |z1|+|z2| z1 , z2 , z3 : |z1|=|z2|=|z3|=1 z1+z2+z3=1, : ) + + = 1 ) |z1 2z2|2 9 ) Re(1 2) -1 : 16. : |z z0| = , > 0 (x0,y0) z0 . : |z z1|=|z z2| 12 , 1 , 2 z1 , z2 . z w, : |2z+3-2i|=2 |w-2+i|=|w+2i| z w z = w. : 17. : |z z1| + |z z2| = 2 , > 0 1 , 2 1 , 2 z1 , z2 : 2=(12)=|z1 z2| 0 1 , 2 1 , 2 z1 , z2 : 2=(12)=|z1 z2|>2 : 18. |z z0| = , > 0 (x0,y0) |z z0| , > 0 (x0,y0) |z z0|< , > 0 (x0,y0) |z z0|> , > 0 (x0,y0) |z z0| , > 0 (x0,y0) : 19. z :|z z0| = , > 0 z |z| , : max|z| = (KO) + min|z| = (KO) z0 . , : 20. z (), z . , () . min|z|= d(O,) : 2 .99 , 7 .101 , 8 .102 , 3 . 123 : 21. z (,) w ,: max |z w|=(NB)=(N)+ min |z w|=(NA)=|(NK)-| w ( , ) : max |z w|=(NB)=2 min |z w|=0 O K y x N B A : 22. z () w,: max |z w| min |z w|= d(N,) O y x N : 23. 1(x1,y1) :x+By+ = 0 : d(M1 , ) = + + + : 24. O(0 , 0) C: x2 + y2 = 2 (x0 , y0) C: (x x0)2 + (y y0)2 = 2 , C: x2 + y2 + Ax + By + = 0 2 + 2 4 > 0 : = + : 25. z , w (,), z w, : max |z w|= = 2. min |z w| O K y x : 26. z (,) w () , max |z w| min |z w|= |d(K , ) | O K y x : 27. z ( , ) w (,R) ( 1), max |z w|= () + + R min |z w|= |() - R| ( 2) min|z w|= 0. O K y x 1 2 y O K x x K O y : 28. z , w z w , max |z w|= 2 , . : 2 .99 , 7 .101, 8 .102 3 .123 O y x : 29. , . z1 + z2 = z3 (0,0) 2, : z2z3 + z1z3 = z1z2 ( , ) = 0 ( , ) = 0 : 30. . z1 , z2 , z3 (0,0) 2, : 1 + 22 33 = 1 2 2 3 + 21 3 31 2 | , | = |( , )| : 31. |z|2 = : 1. z , w C z = w |z| = |w| |z|= | | 2. z , w C z = w |z| = |w| |z| = |w| |z|2 = |w|2 = 3. [f(z)] = [g(z)] | f(z)| = |g(z)| |f(z)|=|g(z)| |f(z)|2 = |g(z)|2 ()() ( )() , ( ) : 32. 1. 6 123 2. (1 + iz) = (1 iz) z R. , ,, z , w , u, : . = = |z-w|=|u-w|=|z-u| . = |z-w|=|z-u| . =900 2 + 2 = 2 |z-w|2 + |z-u|2 = |w-u|2 : z1 , z2 , z3 : z1 + z2 + z3 = 0 |z1|=|z2|=|z3|=1 , z1 , z2 , z3 1 . : 33. z w 1. : |z-z0|=, >0 |z-z1|=|z-z2| z. z w |z|=2 w=(- +i)iz, w. 2. 1. z = x+yi w= +i , . , x y . z w |z|=2 w=2z+ , w. : 34. !! . . ! ! ! ! ! :