Number Representation with Binary

Description:

For integer:

• Signed Magnitude Representation
• Two’s Complement

For float:

• Fixed-point Binary
  • The float is divided into 2 integers, 1 before decimal and 1 after
  • Each is individually represented with their binary
  • The number of bits for each side is agreed beforehand
• Floatting-point Number
  • floats to the right of the most significant 1
  • $\pm M\times B^E$
    • Mantissa * base ^ exponent
  • There are different standard:
    • IEEE-754: demo
      • 1 sign bit (pos/neg)
      • 8 bias exponent
        • value of 8 bias exponent = real exponent +127
        • to be able to repre negative exponent, ex: $2^{-3}\to E=124$
        • meaning the most left bit is -1,
      • 24 normalized mantissa
        • 1 hidden (always 1) + 23 fraction bit (start from 1/2,1/4,…)
        • meaning the mantissa has value > 1
      • To calculate: turn a to x/y*2^n
        • where y is min
        • and x is min that larger than y
  • Can be different precision:
    • Single-precision:
      • 32 bits
      • 1 sign bit, 8 exponent bits, 23 fraction bits
      • bias = 127
    • Double-precision:
      • 64 bits
      • 1 sign bit, 11 exponent bits, 52 fraction bits
      • bias = 1023