Representing numbers

• Easiest class is just natural numbers
  • Just convert to binary
  • We usually have a fixed size (8, 16, 32, 64 bits)
  • unsigned int
• How about negative numbers
  • Sign magnitued
    • Just what we do in regular life
    • -4, -9, +8 (or 8)
    • Use the first bit to represent the sign
    • fixed bit size again, choose 8
    • 00001001 is 9
    • 10001001 is -9
    • Format (-1)^s*magnitued
    • SMMM MMMM
    • What is 0000 0000, how about 1000 0000
    • How do we add sign-magnitude numbers?
  • Two's compliment.
    • With two decimal digits we can represen 0 - 99
    • (With two binray digits we can represent 0-3)
    • With sign magnitude we can shift this range, but math becomes hard
    • There are other equally vialble systems where the math is easier.
    • REMEMBER WE ARE WORKING ON REPRESENTATIONS HERE
      • We need to have an encoding that is efficient
      • Easy to convert from/to
      • Easy to perform math in
      • represents as many numbers as possible
    • One way to move the range is to represent everything in two's compliment
      • Positive integers just give n bit binary representation
      • Negative integers, give n bit positive representation, flip the bits and add one.
      • in 6 bits, 13 = 001101
      • -13 = 001101+1 = 110011
      • Notice 001101 + 110011 = 1000000
      • Try -6, 4 and math on these two.
      • Look at the posiblities in 3 bits.
    • Overflow (in three bits signed, add 2+3)
  • There are other methods (excess, one's compliment) but we will let these for another class
• Real numbers
  • Scientific notation
  • 4.243 x 10 ⁴
  • 3.61 x 10 ^-2
  • Can represent very large or very small numbers
  • exponent, sign, mantissa
  • adding in scientific notation.
  • in 5 decimal digits, add the two above numbers
  • floating point roundoff error, representation error
  • Computer representation [+|-] d.d^*[[+|-]ed⁺