Octal and Hexadecimal Numeration: A Practical Guide for Engineers

Binary representation is the native language of digital circuits, but its dense strings of 0s and 1s can make debugging and design laborious. To streamline communication among engineers, technicians, and programmers, we use base‑8 (octal) and base‑16 (hexadecimal) systems, both of which convert effortlessly to and from binary.

Octal is a place‑weighted system with eight symbols—0 through 7—each digit representing a power of eight. Hexadecimal expands the alphabet to include A‑F, providing sixteen distinct symbols (0–9, A–F) and a power‑of‑sixteen weight for each position.

Below is a side‑by‑side comparison of the first twenty decimal numbers expressed in four common numeral systems:

Number       Decimal      Binary        Octal      Hexadecimal
------       -------      -------       -----      -----------
Zero            0           0             0             0
One             1           1             1             1
Two             2           10            2             2
Three           3           11            3             3
Four            4           100           4             4
Five            5           101           5             5
Six             6           110           6             6
Seven           7           111           7             7
Eight           8           1000          10            8
Nine            9           1001          11            9
Ten             10          1010          12            A
Eleven          11          1011          13            B
Twelve          12          1100          14            C
Thirteen        13          1101          15            D
Fourteen        14          1110          16            E
Fifteen         15          1111          17            F
Sixteen         16          10000         20            10
Seventeen       17          10001         21            11
Eighteen        18          10010         22            12
Nineteen        19          10011         23            13
Twenty          20          10100         24            14

Octal and hexadecimal are not merely academic curiosities; they provide a concise shorthand for binary data. Because 8 = 2³ and 16 = 2⁴, groups of three or four binary bits can be mapped directly to a single octal or hexadecimal digit, respectively.

Binary to Octal Conversion

BINARY TO OCTAL CONVERSION
Convert 10110111.12 to octal:

Step 1: Pad with zeros to form complete 3‑bit groups.

      010 110 111 100

Step 2: Translate each group.

      2    6    7    4

Result: 10110111.12 = 267.48

Grouping starts at the binary point and extends outward, adding implied zeros where necessary.

Binary to Hexadecimal Conversion

BINARY TO HEXADECIMAL CONVERSION
Convert 10110111.12 to hexadecimal:

Step 1: Pad with zeros to form complete 4‑bit groups.

      1011 0111 1000

Step 2: Translate each group.

      B    7    8

Result: 10110111.12 = B7.816

Conversion in the reverse direction follows the same principle: each octal or hexadecimal digit expands into its 3‑ or 4‑bit binary counterpart.

Hexadecimal enjoys broader adoption because common digital data widths—8, 16, 32, 64, 128 bits—are multiples of four, aligning neatly with 4‑bit hexadecimal groups. Octal, requiring groups of three, rarely matches these standard word sizes.

RELATED WORKSHEETS:

• Numeration Systems Worksheet