Number Systems: Base Ten Conversions

• Let's look at some code involving the discussion of base 10.
• Programming problem
  • Given a string of digits representing a positive integer, provide the decimal equivalent. Do not use any string to number libraries, write the code yourself.
  • There are two ways to do this, a brute force method and a more elegant method.
  • Brute Force is to just apply the above formulas. The index is off, but that is easy to fix.
    • I will use a fact from the ASCII table
      • The code for '1' is one higher than the code for '0'
      • The ascii code for '1' is 48
      • The ascii code for '0' is 49
      • so '1' - '0' = 49 - 48 = 1
      • This is true for all of the digits.
      • So '7' - '0' = 7
    • General idea:

      power = 0
      value = 0
      Loop though all of the letters of the string starting at the last position
         digit = numberString[i] - '0'
         value = value + digit * pow(10, power) 

    • Take a look at StringToInt1.cpp
    • This code has a few problems
      • It is awkward, but we can live with that.
      • It computes powers of 10 over and over again.
      • We could fix that
      • Take a look at StringToInt2.cpp
  • There is another way to evaluate polynomials
    • This is called Horner's Method
    • A polynomial in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ can be rewritten as $a_0 + x(a_1 + x(a_2 + x(... + x(a_{n-1} + xa_n$.

    • $a_nx^n + a_{n-1}x^{n-1} + ... + a_0$
      $ = a_0 + a_1x^1 + a_2x^2 + ... + a_{n-2}x^{n-2} + a_{n-1}x^{n-1} + a_nx^n$
      $ = a_0 + a_1x^1 + a_2x^2 + ... + a_{n-2}x^{n-2} + x^{n-1}(a_{n-1} + a_nx)$
      $ = a_0 + a_1x^1 + a_2x^2 + ... + x^{n-2}(a_{n-2}+ x(a_{n-1} + a_nx))$
      $ = a_0 + x(a_1 + x(a_2 + x(... + x(a_{n-1} + xa_n)...)$

  • In our number system, $x = 10$ and $a_i$ are the digits of the number.
  • But why would we want to do this?
    • Notice that we are only multiplying by $x$ and adding in each coefficient.
    • We do not have to compute $x^n$, and we don't have to store a value of $x^n$.
  • This leads to StringToInt3.cpp.
• Problem : Given a positive integer, find the string representation, do not use any conversion functions or libraries.
  • This is really the opposite problem from before.
  • But the solution is straight forward if you remember divide and mod.
  • What is a % b or a modulus b?
    • It is the remainder when simple division is applied.
    • 7/4 = 1 R 3
    • So 7 % 4 = 3.
    • Some interesting facts to remember
      • if a % b = c then 0 ≤ c < b
  • So our solution is fairly simple
    • Consider 824
    • 824 % 10 = 4
    • 824 / 10 = 82
    • We have found the LSD, 4
    • If we repeat this, we can find
    • 82 % 10 = 2
    • 82 / 10 = 8
    • And repeat until the number we have left is less than 10.
  • This leads to IntToString.cpp.