d 001472541
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Transcript of d 001472541
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Pertemuan 25-26
BILANGAN KOMPLEKS 1. Bilangan kompleks Bilangan yang terjadi dari bilangan real dan bilangan imaginer. a) Bilangan imaginer
( )
( )( )
3i 9 . 1- 9-
5i 5 . 1- 5-
i i . 1 i . (-1) i . i i . i i
1 (-1) i i
i- i . (-1) i .i i1- 1- i
i 1-
50502100100
2224
23
22
==
==
=====
===
===
==
=
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b) Bilangan kompleks
Z = x + y i dimana : x dan y = bagian real
i = imaginer Z = x – y i disebut sekawan Z = -x – yi disebut berlawanan c) Operasi Hitung Bil. Kompleks
29i - 31 29i- 21 10 21(-1)- 29i - 10
21i - 6i 35i - 10 7i)-3i)(2(5 Perkalian Operasi c.
10i 3 7i 2 - 3i 5 7i)-(2 - 3i) (5
nPenguranga Operasi b.
4i - 7 7i)-(2 3i) (5 nPenjumlaha Operasi a.
2
=+=
=+=+
+=++=+
=++
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d)
i 5341
5311-
5341i 11-
49(-1) - 441i 21- 10
i 49 - 421(-1) 41i 10
(7i)- )2(i 21 6i 35i 10
7i)7i)(2-(27i)3i)(2(5
7i 27i 2 .
7i - 23i 5
7 - 23i 5
Pembagian Operasi
2
22
2
+=
+=
+=
++=
+++=
+++
=
+++
=+
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Sifat sifat operasi aljabar bilangan kompleks 1. 2. 3. 4. 5. 6. 7. 8.
1221 z z z +=+z
1221 z . z z . =z
321321 z )z (z )z (z ++=++z
3121321 z z z )z (z zz +=+
)z ( z z - 1121 −+=z
(z) Re 2 z =+z
iy x z dimana , y x z . 22 +=+=z
2iz - z )Im( =z
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9. 10. 11. 12. 13 Z = x + yi (bentuk caertesius) Bentuk polar bilangan kompleks Z = r (cos θ + i sin θ) Dimana :
2121 z z z z +=+
2121 z . z z . z =
2
1
2
1
zz
zz
=
2112 z ke zjarak z - →z
| z | |z | z 2121 +≤+z
22 y x +=r
xy tgarc
x y =→= θθtg
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Jika : Maka Dan BILANGAN KOMPLEKS DALAM BENTUK EKSPONENSIAL
(Rumus Euler)
)sin i cos(r zdan )sin i (cosr z 22221111 θθθθ +=+=
)] (sin i ) [cos(rr z z 21212121 θθθθ +++=
)] -(sin i ) - [cos(rr 2121
2
1
2
1 θθθθ +=zz
θθθ
θ
sin icose dimana
e .r Z
i
i
+=
=
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Sehingga :
Teorema De Moivre
θθθθθ nsin i n cos n cos )sin i (cos n ++=+
) - i(
2
1
2
1
) i(21 21
21
21
e . rr
ZZdan
e r Z
θθ
θθ
=
= +rZ
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AKAR AKAR BILANGAN KOMPLEKS
1-n 0,1,2,..., k untuk
nk 2 sin i
nk 2 cos r
)]2 .k (sin i )2 .k ( [cos r
)]sin i (cos[r Z
)sin i r(cos Z
n
n1
n1
n1
k
n
=
+
+
+
=
+++=
+=
+=
πθπθ
πθπθ
θθ
θθ
k
k
Z
Z