CSCI 235 — Elementary Arithmetic

Numbers used in finance, carpentry, science and programming

• Mixed numbers   with fractions
  • 3 5/16 is 3.3125
  • The North Carolina Standard Course of Study K-8 Mathematics requires that:
    • 7th graders be able to “Convert a fraction to a decimal using long division.”
• Scientific notation
  • float literals in FORTRAN, C and Java resemble scientific notation.
  • The North Carolina Standard Course of Study K-8 Mathematics requires that:
    • 8th graders be able to “Perform multiplication and division with numbers expressed in scientific notation to solve real-world problems, including problems where both decimal and scientific notation are used.”

SI Prefixes

• IEC prefixes
• IEC (IEEE) binary prefixes

Using exponents

• Laws of Exponents
• Exponents, radicals, and scientific notation from Kahn Academy

Algorithms for elementary arithemetic

Refresh your knowledge of the four basic operations.

• Addition
  • The elementary school algorithm, called ripple carry by computer engineers, requires time O(n) and space O(n).
  • Computers use carry lookahead adders that require time O(log n) and space O(n).
• Subtraction
  • The American elementary school algorithm can be a tad confusing, especially which many borrows are required, as in 2000-13.
  • The elementary school algorithm also requires that you first compare the sign. For example, 128-2018 is computed as -(2018-128).
  • Computers calculate a-b as a+(-b). This is similar to using nine’s complement.
  • Computers do not use “sign magnitude” numbers. You’ll learn about two’s-complement in Chapter 2.
• Multiplication
  • Start by memorizing your multiplication tables!
  • The Penrose (NC) Elementary School algorithm requires time O(n²) and paper space O(n²).
  • Today North Carolina schools often use lattice multiplication. It’s first known use was about 700 years ago by Ibn al‐Bannāʾ al‐Marrākushī al-Azdi.
  • Peasant Multiplication is an ancient binary-based algorithm that some say is 4000 years old.
  • Older (and cheaper) computers use shift-and-add multipliers but the fast ones implement lattice muliplication using array multipliers which requires O(n) time but O(n²) space.
• Division
  • Division is hard and often slow.
  • The Pentium FDIV bug cost Intel about $475,000,000.

Douglas Jones has a page, Beyond Integer Addition and Subtraction, showing how multiplication and division can be implemented with C loops without using the * and /. This is useful because some really cheap computers do not have ”machine-level“ implementations of these operators.