Sistem Nombor. Sistem Nombor. Sistem Nombor. Sistem Nombor. Sistem Nombor. Elektronik Digit. Sistem Nombor 1. SEE Elektronik Digit
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1 SEE - Elektronik Digit Bab Kebanyakan sistem komputer (sistem Digital) melakukan operasi pengiraan nombor dalam kuantiti yang banyak! Maka, Sistem Pernomboran yang digunakan oleh Sistem Digital perlu diketahui dari segi:- Bagaimana pernyataan nombor tersebut! Bagaimana operasi arithmetik dilakukan! Jenis-jenis :- Decimal (asas ) Binary (asas ) Octal (asas ) Hexadecimal (asas ) Nombor Decimal Terdiri daripada angka iaitu,,,,,,,,, Ia merupakan nombor Asas Salah satu contoh dalam sistem nombor Decimal adalah atau Kedudukan setiap digit menunjukkan magnitud bagi setiap digit tersebut iaitu:- Nilai - - = x + x + x + x Nombor Binary Terdiri daripada angka iaitu, Ia merupakan nombor Asas Salah satu contoh dalam sistem nombor Binary adalah atau Kedudukan setiap digit menunjukkan magnitud bagi setiap digit tersebut iaitu:- Nilai = x + x + x + x Dalam no Binary, bilangan digit dipanggil bit - - Nombor Octal Terdiri daripada angka iaitu,,,,,,, Ia merupakan nombor Asas Salah satu contoh dalam sistem nombor Octal adalah atau Kedudukan setiap digit menunjukkan magnitud bagi setiap digit tersebut iaitu:- Nilai = x + x + x + x - -
2 Nombor Hexadecimal Terdiri daripada angka iaitu,,,,,,,,,, A, B, C, D, E, F Ia merupakan nombor Asas Salah satu contoh dalam sistem nombor hexadecimal adalah BF atau BF Kedudukan setiap digit menunjukkan magnitud bagi setiap digit tersebut iaitu:- Nilai BF = x + B x + x + F x B F - - Rumusan: setiap sistem nombor di atas dinyatakan dalam bentuk Tatanda Nilai Kedudukan Berpemberat (Weighted Positional Value Notation) Secara umumnya, suatu nombor N = a n- a boleh dinyatakan dalam asas b seperti berikut:- (N) b = a n- b n- + a n- b n- + +a b + a b Most Significant Bit (MSB) Least Significant Bit (LSB) Decimal Binary Octal Hexadecimal A B C D E F Penukaran Binary Decimal = x + x + x + x = = Penukaran N X Decimal Binary = =? = = Berhenti apabila N = = = = Baki (N-X) LSB MSB Latihan Tukar no Decimal kepada no Binary, (a) (b) (c) Penukaran Octal Decimal = x + x + x + x = Tukar no Binary kepada no Decimal, (a) (b) (c) Tukar no pecahan kepada no Binary, (a) (b) = Penukaran N X Decimal Octal = = =? Berhenti apabila N = = = = Baki (N-X) LSB MSB
3 Penukaran Octal Binary dan Binary Octal Ada dua kaedah penukaran iaitu secara terus (direct conversion) atau melalui decimal (octal decimal binary) Untuk melaksanakan penukaran Octal Binary melalui kaedah direct conversion, hubungan antara no octal digit dan no binary digit perlu diketahui Kaedah kedua, iaitu melalui decimal telah dipelajari! Tukarkan no berikut:- (a)? (b)? Penukaran Hexadecimal Decimal E = x + E x + x = + X + = Penukaran Decimal Hex N X =? = = Berhenti apabila N = = E = Baki (N-X) =E LSB MSB E Penukaran Hex Binary dan Binary Hex Ada dua kaedah penukaran iaitu secara terus (direct conversion) atau melalui decimal (hex decimal binary) Untuk melaksanakan penukaran melalui kaedah direct conversion, hubungan antara no hex digit dan no binary digit perlu diketahui Kaedah kedua, iaitu melalui decimal telah dipelajari! Tukarkan no berikut:- (a)f? (b)? Rumusan bit N N bit (c)? (d) A? N N Kenapa perlu ada berbagai sistem nombor? Peralatan/peranti terdahulu melakukan pengiraan komputer dengan menggunakan suis dan geganti Suis/geganti hanya ada dua keadaan sahaja iaitu tutup dan buka Maka operasi komputer tersebut dapat dinyatakan dalam digit nombor Binary, yang terdiri daripada dua keadaan sahaja Jika sistem Decimal digunakan, maka ia memerlukan suatu peralatan/peranti yang boleh menghasilkan keadaan! Hasilnya adalah suatu peranti yang kompleks, tak reliable, tidak cekap dan mahal! Ok, no Binary sesuai digunakan dalam sistem komputer, manakala no Decimal tidak cekap Bagaimana pula dengan nombor Octal &Hexadecimal? Kenapa perlu wujudkan sistem nombor Octal dan Hex? Rujuk jadual! Bagi suatu nombor yang kecil,(cthnya ), hanya digit no Decimal shj diperlukan, tetapi digit Binary diperlukan Bagi no Hex, hanya digit diperlukan! In terms of numbers representation/expression, decimal is better, but Hex is the best! Bagi digit no Decimal, digit Binary diperlukan, digit no Octal diperlukan dan hanya digit Hex diperlukan!
4 Maka didapati, pernyataan nombor dapat dilakukan dengan lebih berkesan, dengan mengkodkan dan memadatkan maklumat dalam sistem nombor tersebut Secara praktikal, ia banyak digunakan! Contohnya dalam penyataan alamat dalam memory Kod BCD Decimal Binary BCD Kod BCD (Binary Coded Decimal) Mengungkapkan setiap digit Decimal kepada digit Binary Salah satu kod yang popular adalah kod BCD, dimana pemberat bagi digit binary hanya terhad kepada,,, Tukarkan Decimal BCD dan Binary (a) (b) Tukarkan BCD Decimal (a) (b) Kod Excess - Decimal Binary BCD Excess - Kod Excess - Kod ini terbit dengan menambah kepada kod BCD iaitu N BCD + = N BCD + Kod ini tidak berpemberat, dan ia merupakan salah satu kod BCD Tukarkan Decimal BCD dan Binary (a) (b) Tukarkan BCD Decimal (a) (b)
5 Kod Gray Dlm kod ini, perubahan satu nombor ke nombor selepasnya hanya bit sahaja yang berubah! Ctrhnya =, nombor seterusnya, =, bukan Kod ini tidak berpemberat, dan ia juga merupakan salah satu kod BCD Binary ke kod Gray = = = Binary Gray Kod Gray ke Binary + = + = + = Gray Binary Kod ASCII ASCII merupakan singkatan kepada American Standard Code of Information Interchange Merupakan kod Alphanumeric iaitu kod yang dinyatakan dalam nombor dan abjad Terdiri daripada nombor bit, utk mewakilkan aksara, iaitu n Huruf A cthnya diwakilkan oleh kod ASCII, ESC diwakilkan oleh Dapatkan maklumat yang dinyatakan oleh kod ASCII berikut: Saiz Nombor Julat sesuatu nombor ditentukan oleh saiz atau bilangan atau jumlah bit yang digunakan! Ditentukan oleh persamaan n dimana n adalah bilangan bit! Contohnya, Suatu nombor bit boleh mewakili = nombor! (-) Suatu nombor bit boleh mewakili = nombor! (-) Saiz bit dan gelarannya bit nibble bit byte bit word bit long word Tambah Aritmetic Binary Penambahan adalah berasaskan kombinasi di bawah: + = + = + = + = bawa/carry Contohnya, () () +() +() () () Tolak Aritmetic Binary Penolakan adalah berasaskan kombinasi di bawah: - = - = pinjam - = - = Contohnya, () () - () - () () ()
6 Dalam sistem no Decimal, nombor negatif ditandakan dengan - Dalam sistem no Binary, no positif ditandakan dengan bit yang berada paling kiri! Bit menandakan no positif, bit menandakan no negatif! Ada jenis nombor bertanda iaitu: Perwakilan magnitud bertanda (sign-magnitude) Perwakilan pelengkap ( s complement) Perwakilan pelengkap ( s complement) Sistem Magnitud Bertanda (sign-magnitude) Pada bit paling kiri, bit menandakan no positif, bit menandakan no negatif! Contohnya, nombor +, ditukarkan kepada nombor magnitud bertanda adalah Tanda +ve Bit Magnitud Tetapi nombor pula adalah Tanda -ve Bit Magnitud Didapati magnitud masih sama tetapi tanda sahja yang berubah Sistem Pelengkap ( s complement) Nombor pelengkap (pelengkap (r-)) bagi sesuatu nombor N dalam dasar r yang mengandungi n digit adalah (r n ) N Suatu no Binary ditukarkan kepada pelengkap (r-) r = ; n = ; N = ; Maka, pelengkap bagi () = ( -) = Pelengkap hanya digunakan untuk menandakan nombor negatif! Nombor negatif untuk pelengkap diperolehi dengan menggunakan persamaan di atas atau tukarkan bit kepada dan bit kepada Sistem Pelengkap ( s complement) Nombor pelengkap (pelengkap r) bagi sesuatu nombor N dalam dasar r yang mengandungi n digit adalah [(r n ) N]+ Suatu no Binary ditukarkan kepada pelengkap (r) r = ; n = ; N = ; Maka, pelengkap bagi () = ( -) + = Pelengkap juga hanya digunakan untuk menandakan nombor negatif! Nombor negatif untuk pelengkap diperolehi dengan menggunakan persamaan di atas atau tukarkan Sistem Pelengkap ( s complement) Nombor negatif untuk pelengkap diperolehi dengan menggunakan persamaan di atas, atau menambahkan kepada no pelengkap, atau mengekalkan bit yang paling kanan, dan semua bit di sebelah kanannya, manakala bit di sebelah kiri ditukarkan kepada dan kepada Contoh + = - = (magnitud tanda) = (pelengkap ) = (pelengkap ) Julat nombor Julat nombor tak bertanda ; n -, dimana n adalah bilangan bit bagi nombor bit, julatnya adalah; -() Julat nombor bertanda; - n- n- -, n adalah bilangan bit cth; bagi nombor bit, julatnya adalah; - - (-) - -()
7 Arithmetic nombor bertanda Dalam no pelengkap, semua operasi dilakukan dengan penambahan sahaja () + () () (-) + (-) (-) abaikan () +(-) () (-) + () () abaikan
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