Binary to Decimal and Decimal To Binary

• We need to be able to convert between binary and decimal.
• Fortunately we already know how to do this.
• Binary to decimal.
  • Just expand the value
  • $10110101_2$
    Brute force is hard,
    • Number the digits, write out the expansion

    • Digits Number: $76543210$
      Digits:        $10110101_2$ 
      

    • $10110101_2 = 1\times2^7 + 0\times2^6 + 1\times2^5 + 1\times2^4 + 0\times2^3 + 1\times2^2 + 0\times2^1 + 1\times2^0$
    • $10110101_2 = 1\times 128 + 0\times 64 + 1\times 32 + 1\times 16 + 0\times 8 + 1\times 4 + 0\times 2 + 1\times 1$
    • $10110101_2 = 128 + 32 + 16 + 4 + 1$
    • $10110101_2 = 181$
    • I would cheat and do 1 + no 2 + 4 + no 8 + 16 + 32 + no 64 + 128
    • AND THEN I WOULD CHECK IT WITH A COMPUTATIONAL TOOL
    • If I were computing this with a program, I would definitely use Horner's method.
  • Decimal to Binary
    • Used the div/mod method from before, but div and mod by 2.
    • Remember we had to "reverse" the digits as we went.
    • Convert 181 to to binary

    •    181 / 2 = 90 R 1
          90 / 2 = 45 R 0
          45 / 2 = 22 R 1
          22 / 2 = 11 R 0
          11 / 2 =  5 R 1
           5 / 2 =  2 R 1
           2 / 2 =  1 R 0
           1 / 2 =  0 R 1

    • So $181 = 10110101_2$
    • I use a shorthand for this (drop / 2)

    •      2|181 R 1
           2|90  R 0
           2|45  R 1
           2|22  R 0
           2|11  R 1
           2|5   R 1
           2|2   R 0
           2|1   R 1

  • Both of these techniques are illustrated in DecBin.cpp"