Tushen kowane tsarin lambobi

Tushen yana ƙayyade yawan lambobi na musamman da ake amfani da su da kuma yadda darajar wuri ke ƙaruwa. Tushe 10 yana amfani da lambobi 0-9. Tushe 2 (binary) yana amfani da 0-1. Tushe 16 (hexadecimal) yana amfani da 0-9 da A-F.

A cikin tushe 8 (octal): 157₈ = 1×64 + 5×8 + 7×1 = 111₁₀

Alamomin da ke wakiltar darajoji a cikin tsarin lambobi

Kowane tushe yana buƙatar alamomi na musamman don darajoji daga 0 zuwa (tushe-1). Binary yana amfani da {0,1}. Decimal yana amfani da {0-9}. Hexadecimal ya faɗaɗa zuwa {0-9, A-F} inda A=10...F=15.

2F3₁₆ a hexadecimal = 2×256 + 15×16 + 3 = 755₁₀

Fassara lambobi tsakanin tsarin daban-daban

Canza ya ƙunshi faɗaɗawa zuwa decimal ta amfani da darajar matsayi, sannan a canza zuwa tushen da ake so. Daga kowane tushe zuwa decimal: jimlar lamba×tushe^matsayi.

1011₂ → decimal: 8 + 0 + 2 + 1 = 11₁₀

Harshen kwamfutoci - 0 da 1 kawai

Binary shine tushen dukkan tsarin dijital. Kowane aikin kwamfuta yana raguwa zuwa binary. Kowane lamba (bit) yana wakiltar yanayin kunnawa/kashewa.

• Lambobi: {0, 1} - mafi ƙarancin saitin alamomi
• Byte ɗaya = bit 8, yana wakiltar 0-255 a cikin decimal
• Ƙarfin 2 lambobi ne masu zagaye: 1024₁₀ = 10000000000₂
• Sauƙaƙan tarawa: 0+0=0, 0+1=1, 1+1=10
• Ana amfani da shi a cikin: CPUs, ƙwaƙwalwar ajiya, hanyoyin sadarwa, ma'anar dijital

Wakilcin binary mai ƙanƙanta ta amfani da lambobi 0-7

Octal yana haɗa lambobin binary a cikin rukuni na uku (2³=8). Kowane lamba na octal = daidai bit 3 na binary.

• Lambobi: {0-7} - babu 8 ko 9
• Kowane lamba na octal = bit 3 na binary: 7₈ = 111₂
• Izinin Unix: 755 = rwxr-xr-x
• Na tarihi: farkon ƙananan kwamfutoci
• Ba a cika amfani da shi a yau ba: hexadecimal ya maye gurbin octal

Tsarin lambobi na ɗan adam na duniya

Decimal shine ma'aunin sadarwa na ɗan adam a duk duniya. Tsarin tushe-10 ɗin sa ya samo asali ne daga ƙidaya akan yatsu.

• Lambobi: {0-9} - alamomi goma
• Na halitta ga mutane: yatsu 10
• Rubutun kimiyya yana amfani da decimal: 6.022×10²³
• Kudi, ma'auni, kalandarku
• Kwamfutoci suna canzawa zuwa binary a ciki

Gajeren rubutu na mai shirye-shirye don binary

Hexadecimal shine ma'aunin zamani don wakiltar binary a taƙaice. Lamba ɗaya na hexadecimal = daidai bit 4 (2⁴=16).

• Lambobi: {0-9, A-F} inda A=10...F=15
• Kowane lamba na hexadecimal = bit 4: F₁₆ = 1111₂
• Byte ɗaya = lambobi 2 na hexadecimal: FF₁₆ = 255₁₀
• Launukan RGB: #FF5733 = ja(255) kore(87) shuɗi(51)
• Adireshin ƙwaƙwalwar ajiya: 0x7FFF8A2C

Tushe mafi inganci a fannin lissafi

Ternary yana amfani da lambobi {0,1,2}. Tushe mafi inganci don wakiltar lambobi (mafi kusa da e=2.718).

• Ingancin lissafi mafi kyau
• Daidaitaccen ternary: {-,0,+} daidaito
• Ma'anar ternary a cikin tsarin da ba a sani ba
• An ba da shawarar don kwamfutar quantum (qutrits)

Madadin decimal mai amfani

Tushe 12 yana da masu rarraba da yawa (2,3,4,6) fiye da 10 (2,5), wanda ke sauƙaƙa ɓangarori. Ana amfani da shi a cikin lokaci, dozin, inci/ƙafa.

• Lokaci: agogo na sa'o'i 12, mintuna 60 (5×12)
• Imperial: inci 12 = ƙafa 1
• Sauƙaƙan ɓangarori: 1/3 = 0.4₁₂
• Kungiyar Duodecimal tana goyon bayan amfani da shi

Ƙidaya da ashirin

Tsarin tushe 20 ya samo asali ne daga ƙidaya akan yatsu da yatsun ƙafa. Misalan Mayan, Aztec, Celtic, da Basque.

• Tsarin kalandar Mayan
• Faransanci: quatre-vingts (80)
• Turanci: 'score' = 20
• Ƙidaya na gargajiya na Inuit

Mafi girman tushen haruffa da lambobi

Yana amfani da dukkan lambobin decimal (0-9) da dukkan haruffa (A-Z). Mai ƙanƙanta kuma mai sauƙin karantawa ga mutane.

• Masu gajartar da URL: hanyoyin haɗi masu ƙanƙanta
• Makullan lasisi: kunna software
• ID na bayanai: alamun da za a iya bugawa
• Lambobin bibiya: fakiti, umarni

Tsohon Roma (500 BC - 1500 AD)

Ya mamaye Turai tsawon shekaru 2000. Kowane alama yana da daraja tabbatacciya: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

• Har yanzu ana amfani da su: agogo, Super Bowl, zane-zane
• Babu sifili: matsalolin lissafi
• Dokokin cirewa: IV=4, IX=9, XL=40
• Iyakantacce: ma'auni ya kai 3999
• An maye gurbinsu da lambobin Hindu-Arabic

Tsohon Babila (3000 BC)

Tsarin da ya fi dadewa da ya rage. 60 yana da masu rarraba 12, wanda ke sauƙaƙa ɓangarori. Ana amfani da shi don lokaci da kusurwoyi.

• Lokaci: daƙiƙa 60/minti, mintuna 60/awa
• Kusurwoyi: da'ira 360°, mintuna 60 na baka
• Ikon rarraba: 1/2, 1/3, 1/4, 1/5, 1/6 mai tsabta
• Lissafin taurari na Babila

Kowane lamba na decimal an yi masa kodin a matsayin bit 4

BCD yana wakiltar kowane lamba na decimal (0-9) a matsayin binary na bit 4. 392 ya zama 0011 1001 0010. Yana kauce wa kurakuran lissafi na floating-point.

• Tsarin kuɗi: decimal daidai
• Agogon dijital da kalkuleta
• Manyan kwamfutocin IBM: sashin decimal
• Layukan maganadisu na katin kiredit

Darajoji masu kusa da juna sun bambanta da bit ɗaya

Gray code yana tabbatar da cewa bit ɗaya kawai ke canzawa tsakanin lambobi masu zuwa. Yana da matuƙar muhimmanci ga canza daga analog zuwa dijital.

• Na'urorin kodin masu juyawa: na'urorin gano wuri
• Canza daga analog zuwa dijital
• Taswirar Karnaugh: sauƙaƙe ma'ana
• Lambobin gyara kuskure

Masu shirye-shirye suna aiki da tushe da yawa a kullum:

• Adireshin ƙwaƙwalwar ajiya: 0x7FFEE4B2A000 (hexadecimal)
• Tutar bit: 0b10110101 (binary)
• Lambobin launi: #FF5733 (hexadecimal RGB)
• Izinin fayil: chmod 755 (octal)
• Gyara kurakurai: hexdump, binciken ƙwaƙwalwar ajiya

Dokokin hanyar sadarwa suna amfani da hexadecimal da binary:

• Adireshin MAC: 00:1A:2B:3C:4D:5E (hexadecimal)
• IPv4: 192.168.1.1 = rubutun binary
• IPv6: 2001:0db8:85a3:: (hexadecimal)
• Mask na ƙananan hanyar sadarwa: 255.255.255.0 = /24
• Binciken fakiti: Wireshark hexadecimal

Tsarin kayan aiki a matakin binary:

• Ƙofofin ma'ana: AND, OR, NOT binary
• Rijistar CPU: 64-bit = lambobi 16 na hexadecimal
• Harshen taro: lambobin aiki a cikin hexadecimal
• Shirye-shiryen FPGA: rafukan binary
• Gyara kurakuran kayan aiki: masu nazarin ma'ana

Ka'idar lambobi tana bincika kaddarori:

• Lissafin modular: tushe daban-daban
• Rubutun sirri: RSA, lanƙwasa na elliptic
• Samar da fractal: saitin Cantor na ternary
• Tsarin lambobi na farko
• Combinatorics: tsarin ƙidaya

Faɗaɗa ta amfani da darajar matsayi:

• Gano tushe da lambobi
• Sanya matsayi daga dama zuwa hagu (0, 1, 2...)
• Canza lambobi zuwa darajar decimal
• Ninka: lamba × tushe^matsayi
• Hada dukkan kalmomi

Raba da tushen da ake so akai-akai:

• Raba lamba da tushen da ake so
• Yi rikodin ragowar (lamba ta dama)
• Raba sakamakon da tushen kuma
• Maimaita har sai sakamakon ya zama 0
• Karanta ragowar daga ƙasa zuwa sama

Hada bit na binary:

• Binary → Hexadecimal: haɗa da bit 4
• Binary → Octal: haɗa da bit 3
• Hexadecimal → Binary: faɗaɗa kowane lamba zuwa bit 4
• Octal → Binary: faɗaɗa zuwa bit 3 ga kowane lamba
• Tsallake canza zuwa decimal gaba ɗaya!

Hanyoyi don canza abubuwa da aka saba:

• Ƙarfin 2: haddace 2¹⁰=1024, 2¹⁶=65536
• Hexadecimal: F=15, FF=255, FFF=4095
• Octal 777 = binary 111111111
• Ninkawa/rabi: canza binary
• Yi amfani da yanayin mai shirye-shirye na kalkuleta

Duk lokacin da ka duba agogo, kana amfani da tsarin tushe-60 na Babila mai shekaru 5000. Sun zaɓi 60 saboda yana da masu rarraba 12, wanda ke sauƙaƙa ɓangarori.

A cikin 1999, na'urar NASA ta Mars mai darajar dala miliyan 125 ta lalace saboda kurakuran canza raka'a - wata ƙungiya ta yi amfani da na'ura na imperial, wata kuma ta yi amfani da na'ura na metric. Darasi mai tsada a cikin daidaito.

Lambobin Roman ba su da sifili kuma ba su da lambobi marasa kyau. Wannan ya sa lissafi mai zurfi ya kusan yiwuwa har sai da lambobin Hindu-Arabic (0-9) suka kawo sauyi a lissafi.

Kwamfutar Jagorar Apollo ta nuna komai a cikin octal (tushe 8). 'Yan sama jannati sun haddace lambobin octal don shirye-shiryen da suka saukar da mutane a duniyar wata.

Lambobin launi na RGB suna amfani da hexadecimal: #RRGGBB inda kowane ɗaya yake 00-FF (0-255). Wannan yana ba da launuka 256³ = 16,777,216 da za a iya samu a cikin launi na gaskiya na 24-bit.

Masu bincike na Soviet sun gina kwamfutoci na ternary (tushe-3) a cikin shekarun 1950-70. Kwamfutar Setun ta yi amfani da ma'anar -1, 0, +1 maimakon binary. Kayan aikin binary ya yi nasara.

Binary ya yi daidai da da'irorin lantarki: kunnawa/kashewa, babban/ƙaramin ƙarfin lantarki. Tsarin yanayi biyu abin dogaro ne, mai sauri, kuma mai sauƙin ƙerawa. Decimal zai buƙaci matakan ƙarfin lantarki daban-daban guda 10, wanda zai sa da'irori su zama masu rikitarwa da sauƙin samun kuskure.

Haddace taswirar hexadecimal-zuwa-binary guda 16 (0=0000...F=1111). Canza kowane lamba na hexadecimal daban: A5₁₆ = 1010|0101₂. Hada binary da 4 daga dama don juyawa: 110101₂ = 35₁₆. Ba a buƙatar decimal!

Yana da muhimmanci ga shirye-shirye (adireshin ƙwaƙwalwar ajiya, ayyukan bit), hanyar sadarwa (adireshin IP, adireshin MAC), gyara kurakurai (juji na ƙwaƙwalwar ajiya), lantarki na dijital (tsarin ma'ana), da tsaro (rubutun sirri, hashing).

Hexadecimal yana daidaita da iyakokin byte (bit 8 = lambobi 2 na hexadecimal), yayin da octal ba haka ba (bit 8 = lambobi 2.67 na octal). Kwamfutoci na zamani suna da alaƙa da byte, wanda ya sa hexadecimal ya fi dacewa. Izinin fayil na Unix kawai ke sa octal ya kasance da muhimmanci.

Babu hanya mai sauƙi kai tsaye. Octal yana haɗa binary da 3, hexadecimal da 4. Dole ne a canza ta hanyar binary: octal→binary (bit 3)→hexadecimal (bit 4). Misali: 52₈ = 101010₂ = 2A₁₆. Ko kuma yi amfani da decimal a matsayin matsakaici.

Al'ada da kyau. Ana amfani da su don al'amuran yau da kullum (Super Bowl, fina-finai), bambanci (zane-zane), rashin lokaci (babu rudanin ƙarni), da kuma kyawun zane. Ba su da amfani don lissafi amma suna dawwama a al'ada.

Kowane tushe yana da dokoki masu tsauri. Tushe 8 ba zai iya ƙunsar 8 ko 9 ba. Idan ka rubuta 189₈, ba shi da inganci. Masu canzawa za su ƙi shi. Harsunan shirye-shirye suna tilasta wannan: '09' yana haifar da kurakurai a cikin mahallin octal.

Tushe 1 (unary) yana amfani da alama ɗaya (alamun ƙidaya). Ba ainihin matsayi ba ne: 5 = '11111' (alamomi biyar). Ana amfani da shi don ƙidaya na farko amma ba mai amfani ba. Wasa: unary shine tushe mafi sauƙi - ci gaba da ƙidaya kawai!