Octal and Hexadecimal

• Binary numbers can become quite large.
• It is helpful to represent these numbers in other bases for shorthand.
• Base 10 is too much work.
• However, because of a math trick, base 8 (octal) and base 16 (hexadecimal, or hex) are quite easy .
• Octal
  • Octal or base 8 uses the digits 0 through 7.
  • A conversion table from binary to octal:

  • Binary	Octal
    000	0
    001	1
    010	2
    011	3
    100	4
    101	5
    110	6
    111	7

  • To convert from binary to octal
    • From the right, group the binary digits in sets of three.
    • Pad on the left with 0 if required.
    • Replace the groups of three with the corresponding octal digit.
    • Example:

      Convert $10010110111101_2$ to  octal
      Group by groups of three from the right
                 10 010 110 111 101
      Pad with a 0
                 010 010 110 111 101
      Replace the groups with the corresponding octal digits
                 010 010 110 111 101
                  2   2   6   7   5
      Answer
          $22675_8$
                   

    • Example

      Convert $1010000100000_2$ to  octal
           1 010 000 100 000
         001 010 000 100 000
           1  2   0   4  0
         $12040_8$
                   

  • To convert from octal to binary
    • replace each octal digit with the corresponding binary triplet
  • Example:

    Convert $7235_8$ to binary
          7   2   3   5
         111 010 011 101
          $111010011101_2$
             

• The same is true for hex but
  • Hex needs extra digits:
    • 1 - 9 stay the same.

      Hex	Decimal
      A	10
      B	11
      C	12
      D	13
      E	14
      F	15

  • Groups of four.

    Binary	Hex
    0000	0
    0001	1
    0010	2
    0011	3
    0100	4
    0101	5
    0110	6
    0111	7
    1000	8
    1001	9
    1010	A
    1011	B
    1100	C
    1101	D
    1110	E
    1111	F

  • Example:

    Convert $10110110101011110001_2$ to hex
        1011 0110 1010 1111 0001
         B    6    A    F     1
        $B6AF1_{16}$
             

  • Example

    Convert $F2C4_{16}$ to binary
        F   2   C   4
      1111 0010 1100 0100
      $1111001011000100_2$
             

• It is generally a good skill to have to convert between binary and hex and octal.
• You will make fewer copying mistakes.