Conversion from Binary Number to Octal/Hexadecimal Number and Vice-Versa

Converting between binary and octal/hexadecimal never needs division or positional sums — it is pure grouping and substitution. A binary number is converted to an octal or hexadecimal number by making groups of 3 bits (octal) or 4 bits (hexadecimal) and replacing each group with its equivalent digit; the reverse direction expands each digit back into its bit group.

(A) Binary number to octal number

Group the bits 3 at a time, from right to left, and replace each 3-bit group with the corresponding octal digit (Table 2.5). If the number of bits is not a multiple of 3, add the required number of 0s at the most significant (left) end.

Example 2.9 — Convert (10101100)2 to octal.

Binary        :  10101100          (8 bits — pad to 9 with a leading 0)
3-bit groups  :  010 | 101 | 100   (grouped right to left)
Octal digits  :   2  |  5  |  4

(10101100)2 = (254)8

(B) Octal number to binary number

Each octal digit is an encoding of a 3-digit binary number, so simply replace every octal digit by its 3-bit binary group.

Example 2.10 — Convert (705)8 to binary.

Octal digits  :   7  |  0  |  5
3-bit groups  :  111 | 000 | 101

(705)8 = (111000101)2

Why are 4 bits grouped together to get a hexadecimal number?

The base of hexadecimal is 16, and 16 = 2^4 — so four binary digits are exactly sufficient to represent all 16 hexadecimal symbols.

(C) Binary number to hexadecimal number

Group the bits 4 at a time, from right to left, and substitute each 4-bit group with its hexadecimal symbol (Table 2.6). If needed, add 0 bits at the most significant position to make the bit count a multiple of 4.

Example 2.11 — Convert (0110101100)2 to hexadecimal.

Binary        :  0110101100                (10 bits — pad to 12)
4-bit groups  :  0001 | 1010 | 1100        (grouped right to left)
Hex symbols   :   1   |  A   |  C

(0110101100)2 = (1AC)16

(D) Hexadecimal number to binary number

Each hexadecimal symbol encodes a 4-digit binary number, so substitute the 4-bit binary equivalent of each hexadecimal digit and join the groups together.

Example 2.12 — Convert (23D)16 to binary.

Hex digits    :   2   |  3   |  D
4-bit groups  : 0010  | 0011 | 1101

(23D)16 = (001000111101)2
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