Digital system and binary numbers (2'nd update)

Sistem Digital
Nugroho Adi P
nugnux@gmail.com
Aravir@me.com
Thursday, September 4, 14

One characteristic of digital systems is their
ability to represent and manipulate discrete
elements of information.

Discrete Information
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Catur
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Discrete Information
26 Huruf
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Catur
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Elemen informasi diskrit diwakili di sistem
digital oleh sinyal

arus,
tegangan

Sinyal diskrit hanya mempunyai dua nilai
(Biner)

Bit bernilai 0 dan 1

Discrete elements of information are
represented with groups of bits called binary
codes

contoh kode biner
1:
2:
3:
4:
5:
0001
0010
0011
0100
0101

Kadang
(1000)2 ≠ (1000)10

tidak mungkin
(1008)2

tidak mungkin keliru
00F1

a digital system is a system that manipulates
discrete elements of information
represented internally in binary form.
Thus

Diskrit secara alami
Diambil dari proses kontinu (kuantisasi)
Informasi Diskrit

Diskrit alami

Kuantisasi

Kenapa Digital?

Dapat diprogram

Satu alat dapat digunakan untuk bermacam
keperluan

Bilangan Biner

1395

1395
seribu tiga ratus sembilan puluh lima

1395
1000 + 300 + 90 + 5

1395
1000 + 300 + 90 + 5
1 x 1000 + 3 x 100 + 9 x 10 + 5

1395
1000 + 300 + 90 + 5
1 x 1000 + 3 x 100 + 9 x 10 + 5
1 x 103 + 3 x 102 + 9 x 101 + 5 x 100

1395
1000 + 300 + 90 + 5
1 x 1000 + 3 x 100 + 9 x 10 + 5
1 x 103 + 3 x 102 + 9 x 101 + 5 x 100
4 bit

1 x 103 + 3 x 102 + 9 x 101 + 5 x 100
a3 x 103 + a2 x 102 + a1 x 101 + a0 x 100
basis 10

basis 10 secara umum
...+103a3 + 102a2 + 101a1 + 100a0 + 10-1a-1 + 10-2a-2 + 10-3a-3 + ...

secara umum hanya ditulis koeﬁsennya saja

Biner

Biner
...+23a3 + 22a2 + 21a1 + 20a0 + 2-1a-1 + 2-2a-2 + 2-3a-3 + ...

Biner
1101

Biner
1101
1 x 23 + 1 x 22 + 0 x 21 + 1 x 20

Biner
1101
1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
8 + 4 + 0 + 1 (desimal)

Biner
1101
1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
8 + 4 + 0 + 1 (desimal)
13 (desimal)

Biner
10.1

Biner
10.1
1 x 21 + 0 x 20 + 1 x 2-1

Biner
10.1
1 x 21 + 0 x 20 + 1 x 2-1
2 + 0 + 0.5 (desimal)

Biner
10.1
1 x 21 + 0 x 20 + 1 x 2-1
2 + 0 + 0.5 (desimal)
2.5 (desimal)

Basis-r

Basis-r
...+r3a3 + r2a2 + r1a1 + r0a0 + r-1a-1 + r-2a-2 + r-3a-3 + ...

Basis-r
...+r3a3 + r2a2 + r1a1 + r0a0 + r-1a-1 + r-2a-2 + r-3a-3 + ...
nilai ai antara 0 dan r-1

oktal

oktal
10.1

oktal
10.1
1 x 81 + 0 x 80 + 1 x 8-1

oktal
10.1
1 x 81 + 0 x 80 + 1 x 8-1
8 + 0 + 0.125 (desimal)

oktal
10.1
1 x 81 + 0 x 80 + 1 x 8-1
8 + 0 + 0.125 (desimal)
8.125 (desimal)

hexadesimal

hexadesimal
10.1

hexadesimal
10.1
1 x 161 + 0 x 160 + 1 x 16-1

hexadesimal
10.1
1 x 161 + 0 x 160 + 1 x 16-1
16 + 0 + 0.125 (desimal)

hexadesimal
10.1
1 x 161 + 0 x 160 + 1 x 16-1
16 + 0 + 0.125 (desimal)
16.0625 (desimal)

Biner
210 = 1024 = 1K

Biner
210 = 1024 = 1K
220 = M

Biner
210 = 1024 = 1K
220 = M
230 = G

Biner
210 = 1024 = 1K
220 = M
230 = G
240 = T

Aritmatika bilangan berbasis r mengikuti
aturan yang sama dengan bilangan desimal

Biner
11 + 0 = 11

Biner
11 + 0 = 11
1+1 = 10

Biner
11 + 0 = 11
1+1 = 10
10+11= 101

Biner
11 + 0 = 11
1+1 = 10
10+11= 101
11+11 = 110

octal

octal
1 + 1 = 2

octal
1 + 1 = 2
7 + 1 = 10

octal
1 + 1 = 2
7 + 1 = 10
17 +11 = 30

octal
1 + 1 = 2
7 + 1 = 10
17 +11 = 30
17 + 1 = 20

hexadesimal
9 + 1 = A

hexadesimal
9 + 1 = A
9 + 2 = B

hexadesimal
9 + 1 = A
9 + 2 = B
F + 2 = 11

hexadesimal
9 + 1 = A
9 + 2 = B
F + 2 = 11
FF + 1 = 100

Biner
1 - 1 = 0

Biner
1 - 1 = 0
10 - 01 = 01

Biner
1 - 1 = 0
10 - 01 = 01
1011 - 1001 =0010

Octal

Octal
10 - 1 = 7

Octal
10 - 1 = 7
17 - 10 = 7

Octal
10 - 1 = 7
17 - 10 = 7
21 - 17 = 2

Hexadesimal

Hexadesimal
10 - 1 = F

Hexadesimal
10 - 1 = F
1A - 11 = 9

Hexadesimal
10 - 1 = F
1A - 11 = 9
A0 - 1 = 9F

Konversi

Konversi
(1)2 = (1)8 = (1)16 = (1)5 = 1 (desimal)

Konversi
(1)2 = (1)8 = (1)16 = (1)5 = 1 (desimal)
(1 0000)2 = (20)8 = (10)16 = (31)5 = 16 (desimal)

Konversi
(1)2 = (1)8 = (1)16 = (1)5 = 1 (desimal)
(1 0000)2 = (20)8 = (10)16 = (31)5 = 16 (desimal)
(0.000110...)2 = 0.05 (desimal)

Konversi
(1)2 = (1)8 = (1)16 = (1)5 = 1 (desimal)
(1 0000)2 = (20)8 = (10)16 = (31)5 = 16 (desimal)
(0.000110...)2 = 0.05 (desimal)
(0.001)2 = 0.125 (desimal)

Desimal ke binary

Desimal ke binary
102

Desimal ke binary
102
102 bagi dua

Desimal ke binary
102
102 bagi dua
51 sisa 0

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0
1 sisa 1

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0
1 sisa 1
0 sisa 1

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0
1 sisa 1
0 sisa 1
tulis dari bawah

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0
1 sisa 1
0 sisa 1
tulis dari bawah
1100110

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0
1 sisa 1
0 sisa 1
tulis dari bawah
1100110
jadi

Desimal ke binary
102
102 bagi dua
51 sisa 0
25 sisa 1
12 sisa 1
6 sisa 0
3 sisa 0
1 sisa 1
0 sisa 1
tulis dari bawah
1100110
jadi
(102)10 = (1100110)2

Desimal ke octal

Desimal ke octal
102

Desimal ke octal
102
102 bagi delapan

Desimal ke octal
102
102 bagi delapan
12 sisa 6

Desimal ke octal
102
102 bagi delapan
12 sisa 6
1 sisa 4

Desimal ke octal
102
102 bagi delapan
12 sisa 6
1 sisa 4
0 sisa 1

Desimal ke octal
102
102 bagi delapan
12 sisa 6
1 sisa 4
0 sisa 1
jadi

Desimal ke octal
102
102 bagi delapan
12 sisa 6
1 sisa 4
0 sisa 1
jadi
(102)10 = (146)2

Desimal ke heksadesimal

102

102
102 bagi enambelas

102
102 bagi enambelas
6 sisa 6

102
102 bagi enambelas
6 sisa 6
0 sisa 6

102
102 bagi enambelas
6 sisa 6
0 sisa 6
jadi

102
102 bagi enambelas
6 sisa 6
0 sisa 6
jadi
(102)10 = (6)16

Pecahan

Pecahan
(0.2)10

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
.....

Pecahan
(0.2)10
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
0.4 x 2 = 0.8 + 0
0.8 x 2 = 0.6 + 1
0.6 x 2 = 0.2 + 1
0.2 x 2 = 0.4 + 0
.....
(0.22)10 = (0.0011001100...)2

Pecahan
(0.25)10

Pecahan
(0.25)10
0.25 x 2 = 0.5 + 0

Pecahan
(0.25)10
0.25 x 2 = 0.5 + 0
0.5 x 2 = 0.0 + 1

Pecahan
(0.25)10
0.25 x 2 = 0.5 + 0
0.5 x 2 = 0.0 + 1
.....

Pecahan
(0.25)10
0.25 x 2 = 0.5 + 0
0.5 x 2 = 0.0 + 1
.....
(0.25)10 = (0.01)2

Pecahan
(0.5)10

Pecahan
(0.5)10
0.5 x 2 = 0.0 + 1

Pecahan
(0.5)10
0.5 x 2 = 0.0 + 1
(0.5)10 = (0.1)2

Pecahan
(0.125)10

Pecahan
(0.125)10
0.125 x 2 = 0.25 + 0

Pecahan
(0.125)10
0.125 x 2 = 0.25 + 0
0.25 x 2 = 0.5 + 0

Pecahan
(0.125)10
0.125 x 2 = 0.25 + 0
0.25 x 2 = 0.5 + 0
0.5 x 2 = 0.0 + 1

Pecahan
(0.125)10
0.125 x 2 = 0.25 + 0
0.25 x 2 = 0.5 + 0
0.5 x 2 = 0.0 + 1
(0.5)10 = (0.001)2

Oktal dan Heksadesimal

(111111111111)2

(111111111111)2
(111 111 111 111)2 = (7 7 7 7)8

(111111111111)2
(111 111 111 111)2 = (7 7 7 7)8
(1111 1111 1111)2 = (F F F )16

Bilangan biner sulit untuk digunakan

Bilangan biner sulit untuk digunakan
Mengunakan digit yang lebih banyak dari
desimal

Bilangan desimal lebih mudah digunakan

namun komputer menggunakan biner

namun komputer menggunakan biner
konversi desimal ke biner rumit

Octal dan Hexadecimal memperkecil digit

Complements

‘negatif’-nya sebuah bilangan

menyederhanakan pengurangan

(dilakukan dengan penjumlahan)

(dilakukan dengan penjumlahan)
hemat sirkuit

Komplemen bilangan berbasis-r

komplemen r (radix)

komplemen r (radix)
komplemen r-1 (diminished radix)

jika sebuah bilangan N dengan basis r
memiliki n digit,

memiliki n digit,
maka akan memiliki diminished radix
komplemen (rn-1)-N

memiliki n digit,
komplemen (rn-1)-N
mis 7 desimal akan memiliki diminished radix
komplemen (101-1)-7 = 2

memiliki n digit,
komplemen (rn-1)-N
mis 7 desimal akan memiliki diminished radix
komplemen (101-1)-7 = 2
07 desimal akan memiliki diminished radix
komplemen (102-1)-7 = 92

diminised radix complement tuk binary

100 -> (23-1)-100

100 -> (23-1)-100
= (1000-1)-100

100 -> (23-1)-100
= (1000-1)-100
= (111)-100

100 -> (23-1)-100
= (1000-1)-100
= (111)-100
= 011

111 -> (23-1)-111

111 -> (23-1)-111
= (1000-1)-111

111 -> (23-1)-111
= (1000-1)-111
= (111)-111

111 -> (23-1)-111
= (1000-1)-111
= (111)-111
= 000

101 -> (23-1)-101

101 -> (23-1)-101
= (1000-1)-101

101 -> (23-1)-101
= (1000-1)-101
= (111)-101

101 -> (23-1)-101
= (1000-1)-101
= (111)-101
= 010

memiliki n digit,
maka akan memiliki radix komplemen (rn)-N

memiliki n digit,
mis 7 desimal akan memiliki radix
komplemen (101)-7 = 3

memiliki n digit,
mis 7 desimal akan memiliki radix
komplemen (101)-7 = 3
07 desimal akan memiliki radix komplemen
(102)-7 = 93

radix complement tuk binary

100 -> (23)-100

100 -> (23)-100
= (1000)-100

100 -> (23)-100
= (1000)-100
= (1000)-100

100 -> (23)-100
= (1000)-100
= (1000)-100
= 100

111 -> (23)-111

111 -> (23)-111
= (1000)-111

111 -> (23)-111
= (1000)-111
= 001

101 -> (23-1)-101

101 -> (23-1)-101
= (1000)-101

101 -> (23-1)-101
= (1000)-101
= 011

Jika memiliki radix point, hilangkan!

Jika memiliki radix point, hilangkan!
Kembalikan lagi setelah hitungan

Pengurangan dengan komplemen

(lebih mudah bagi hardware)

(lebih mudah bagi hardware)
(tak ada pinjaman)

pengurangan bilangan M-N berdigit n berbasis r

tambahkan M dengan radix komplemen dari N

tambahkan M dengan radix komplemen dari N
M + ( rn - N ) = M - N + rn

jika M≥N maka penjumlahan akan menghasilkan
(end carry)

(end carry)
buang end carry

(end carry)
buang end carry
yang tersisa merupakan hasilnya

Jika M < N, maka penjumlahan tidak
menghasilkan end carry

hasilnya merupakan komplemen (N-M)

agar ‘enak’ dilihat, ambil komplemen hasil
penjumlahan

agar ‘enak’ dilihat, ambil komplemen hasil
penjumlahan
beri tanda negatif

Bilangan Biner Bertanda
(Signed Binary Number)

Tanda minus tak mungkin digunakan;

Tanda minus tak mungkin digunakan;
diganti dengan bit paling kiri

bilangan positif memiliki bit terkiri nol

(0100)2 = +4

(0100)2 = +4
(0111)2 = +7

bilangan positif memiliki bit terkiri satu

(1100)2 = -4

(1100)2 = -4
(1111)2 = -7

Hati-hati!

(1100)2 = -4 = 12

(1100)2 = -4 = 12
(1111)2 = -7 = 15

Pengguna harus tahu apakah sebuah bilangan
biner yang sedang ditinjau itu bertanda atau
tidak

Biasanya ada petunjuk, informasi atau deﬁnisi
yang menyertai sebuah bilangan biner
(biasanya tidak ada masalah)

Biner Negatif

Biner Negatif
signed-magnitude representation

Biner Negatif
signed-1’s-complement representation

Biner Negatif

Signed-magnitude
Representation

Signed-magnitude
Representation
-4 = (-0100)2 =(1100)2 -> empat bit

Signed-magnitude
Representation
-4 = (-0100)2 =(1100)2 -> empat bit
-4 = (-00100)2 =(10100)2 -> enam bit

Signed-magnitude
Representation
-4 = (-0100)2 =(1100)2 -> empat bit
-4 = (-00100)2 =(10100)2 -> enam bit
-4 = (-00000100)2 = (10000100)2 -> delapan bit

Signed-1’s-Complement
Representation

Representation
-4 = (-0100)2 = (1011)2 -> empat bit

Representation
-4 = (-0100)2 = (1011)2 -> empat bit
-4 = (-000100)2 = (111011)2 -> enam bit

Representation
-4 = (-0100)2 = (1011)2 -> empat bit
-4 = (-000100)2 = (111011)2 -> enam bit
-4 = (-00000100)2 = (11111011)2 -> delapan bit

Representation

Representation
-4 = (-0100)2 = (1100)2 -> empat bit

Representation
-4 = (-0100)2 = (1100)2 -> empat bit
-4 = (-000100)2 = (111100)2 -> enam bit

Representation
-4 = (-0100)2 = (1100)2 -> empat bit
-4 = (-000100)2 = (111100)2 -> enam bit
-4 = (-00000100)2 = (11111100)2 -> delapan bit

Selesai
Dandiahidupbahagiaselama-lamanya...

Digital system and binary numbers (2'nd update)

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