Math 441

Numbers in Computers

Numbers in Computers

• Finite memory
  • no irrationals!
  • limited number of decimals
  • limited size of denominator
  • limited precision!
  • numbers cannot be arbitrarily small
  • numbers cannot be arbitrarily large
• Speed concerns
  • operations
    • hardware: very fast but rigid
    • software: slow but flexible

“Hardware” numbers

• Binary
• Fixed number of “bits”
• Several different “types” of numbers:
  • Integers:
  • unsigned
  • signed
  • Floating point numbers

Unsigned Integers

• Non-negative
• Each bit is one binary digit
• \(2^{\text{number of bits}}\) different values
• Smallest is 0
• Largest is \(2^{\text{number of bits}} - 1\)
• What happens when a result of a calculation is larger than that?

Signed Integers

• Different designs
• The highest bit represents the sign
• Two’s complement:
  • Positive numbers are regular binary numbers using all except the highest bit
  • To represent a negative number, we represent the corresponding positive number, flip all the digits (including the sign bit), and add 1.
  • What is the largest (most positive) number?
  • What is the smallest (most negative) number?