5.2 - Number Bases

• Number Base
  • specifies how many different digts used in constructing a numeral and by how much to multiply each digit
  • Decimal (Base 10)
    • Has 10 different digits: 0,1,2,3,4,5,6,7,8,9. Base is 10: \(10^n\) where \(n \in \Z\)
    • 
  • Binary (Base 2)
    • Has two digits: 0,1. \(2^n\) where \(n \in \Z\)
    • 
  • Hexadecimal (Base 16)
    • 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. \(16^n\) where \(n \in \Z\)
    • 
  • n (Base n)
    • some number \(a\) in base \(n\) represented as: \(a_n\)
• Decimal to Binary
  • Take the decimal number and repeatedly divide by 2 until 0 is reached, writing the remainder each time
  • Then read the remainders from the bottom to the top
  • 
• Decimal to Hexadecimal
  • Can use the succesive division method (see above), but this time, dividing by 16
  • 
• Hexadecimal to binary
  • Replace each hexadecimal digit by its binary equivalent (each should 4 bits)
  • 
  • Why
    • Hexadecimal digits (0-F) map to 0-15 in decimal, and this decimal range can be represented by exactly 4 binary digit
  • Useful result:
• Binary to Hexadecimal
  • Method
    • add leading 0s to LHS of digit such that number of bits is multiple of 4 if necessary
    • convert each block of 4 with hex equivalent
• Why hex good
  • more convenient than binary
  • takes up less space
  • less error-prone
  • less cumbersome
  • needs 4x fewer digits than binary
  • decimal needs 3.2x fewer = awkward factor to work with
  • Usages:
    • debugging (memory dumps)
    • colours