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Base Conversion

Base Conversion. Number bases. Digital computers usually store integers in base 2 (binary), base 8 (octal), or base 16 (hexadecimal) 266 10 = 100001010 2 = 412 8 = 10A 16. Arbitrary Bases. In base 2, we only need 2 digits, 0,1. In base 16, we need 16 digits

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Base Conversion

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  1. Base Conversion

  2. Number bases • Digital computers usually store integers in base 2 (binary), base 8 (octal), or base 16 (hexadecimal) • 26610 = 1000010102 = 4128 = 10A16

  3. Arbitrary Bases • In base 2, we only need 2 digits, 0,1. • In base 16, we need 16 digits • Use the letters "ABCDEF" to represent the digits from 10 to 15. • With enough digits, integers can be expressed in any base!

  4. Other interesting bases • Base 36. 0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ • Base 51. 0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+-*/=()!@#$%^&_

  5. 4011 (10->16) 57005 (10->16) 1767707668033969 (10->36) 7194 (10->51) 31071116227 (10->51) GQE8RABO5H0JZQGTD709CK (36 -> 51) FAB DEAD HELLOWORLD 2/3 1/2=2/4 This_is_not_a_number Some interesting numbers

  6. Convert one base to another • Convert base 16 to base 10. • Convert any base to base 10. • Convert base 10 to base 16 • Convert base 10 to any base. • Combine Steps 2 and 4.

  7. 1. Base 16 to base 10 • Represent Base 16 numbers as strings. • Digits in Base 16: digits = '0123456789ABCDEF' • digits.find('F') = 15 (15 is base 10 equivalent of digit 'F')

  8. Example • In Base 16: 20A = 20*10 + A • In Base 10: 522 = 32*16 + 10 • If we had a subroutine to convert hex to decimal, this equation would read: • decimal(20A) = decimal(20)*16 + decimal(A)

  9. Code for base 16 to 10 def deciamal(s): d = '0123456789ABCDEF' if len(s) == 1: return d.find(s) else: return decimal(s[:-1])*16 + d.find(s[-1])

  10. 2: Any Base to 10 def decimal(s,d): b = len(d) if len(s) == 1: return d.find(s) else: return decimal(s[:-1],d)*b + d.find(s[-1])

  11. 3. Base 10 to 16 • Think of 522 = a2162 + a116 + a0 • Then a2a1a0 is the hexadecimal representation of 522. • Clearly 522%16 = a0 • Also 522//16 = a216 + a1 = (a2a1) 16

  12. Code decimal to hex def hex(n): d = '0123456789ABCDEF' if n < 16: return d[n] else: return hex(n//16) + d[n%16]

  13. 4. Code decimal to any base def anyBase(n,d): b = len(d) if n < b: return d[n] else: return anyBase(n//b,d) + d[n%b]

  14. 5. Any base to any base def changeBase(s,digits0,digits1): n = decimal(s,digits0) t = anyBase(n,digits1) return t

  15. More usable: def defaultDigits(n): d = '01234567890ABCDEFGHI' + \ 'JKLMNOPQRSTUVWXYZ' + \ 'abcdefghijklmnopqrstuvwxyz'+\ '+-*/!@#$%^&()_' return(d[:n])

  16. 5'. Base to Base def base2base(s,a,b): d0 = defaultDigits(a) d1 = defaultDigits(b) return anyBase(decimal(s,d0),d1)

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