Conversion among binary, octal, and hexadecimal systems

1. The Key Relationship

Octal and hexadecimal are not independent of binary — they are shorthand for it, because their bases are powers of 2:

8 = 2^{3}, 16 = 2^{4}

System	Bits per digit	Grouping rule
Binary ↔ Octal	3	Group the bits in 3s
Binary ↔ Hex	4	Group the bits in 4s

Because the grouping is exact, binary ↔ octal/hex conversion needs no division — you simply re-group the bits, which is why programmers use octal/hex to write machine code and addresses compactly.

2. Binary to Octal

Group from the LSB towards the MSB in threes; pad the most significant group with zeros if it is short. For fractions, group from the binary point to the right, padding the last group.

Example: $(1011101)_2$

Grouping	001	011	101
Octal digit	1	3	5

$(1011101)_2 = (135)_8$

With a fraction: $(1\ 011\ 101\ .\ 110\ 1)_2$ → integer 1|011|101 = 135, fraction 110|100 (padded) = .64 → 135.64₈

3. Octal to Binary

Replace each octal digit by its 3-bit equivalent (write leading zeros as needed).

Example: $(745)_8 → 111\ 100\ 101 = (111100101)_2$

4. Binary to Hexadecimal

Group from the LSB in fours and replace each group by its hex symbol (A-F for 10-15).

Example: $(101110101101)_2$

Grouping	1011	1010	1101
Hex	B	A	D

$(101110101101)_2 = (BAD)_{16}$

5. Hexadecimal to Binary

Expand each hex digit into 4 bits.

Example: $(2F5)_{16} = 0010\ 1111\ 0101 = (1011110101)_2$
$(A3)_{16} = 1010\ 0011$ — note that each hex digit always produces 4 bits, even if the leading bit is 0.

6. Octal ↔ Hexadecimal (through binary)

There is no power relationship between 8 and 16, so convert via binary (or via decimal):

(72)_8 → 111\ 010 → 0011\ 1010 → (3A)_{16}

(3A)_{16} → 0011\ 1010 → 111\ 010 → (72)_8

7. Decimal to Octal / Hexadecimal — Repeated Division

Divide by the base and read the remainders upwards.

Example: $(431)_{10} = (?)_{8}$

Division	Quotient	Remainder
431 ÷ 8	53	7
53 ÷ 8	6	5
6 ÷ 8	0	6

431₁₀ = 657₈

Example: $(431)_{10} = (?)_{16}$

Division	Quotient	Remainder
431 ÷ 16	26	15 = F
26 ÷ 16	1	10 = A
1 ÷ 16	0	1

431₁₀ = 1AF₁₆

8. Quick Reference Table

Decimal	Binary (4-bit)	Octal	Hex
8	1000	10	8
10	1010	12	A
15	1111	17	F
16	1 0000	20	10
255	1111 1111	377	FF
256	1 0000 0000	400	100

9. Common Mistakes to Avoid

Grouping from the left (MSB) instead of the right — the LSB group must be complete (pad on the left).
Using 4-bit groups for octal or 3-bit groups for hex.
Forgetting to pad the fractional group at the right end.
Writing a valid hex digit sequence but forgetting that A-F are single digits (10-15).

10. Exam Points

Convert binary to octal and hex and back, showing the grouping — 5 marks, almost certain.
Convert a decimal number to octal and hexadecimal by repeated division — 5 marks.
Why is hex a shorthand for binary but octal is not a shorthand of hex? — 3 marks.
Convert an octal number to hexadecimal via binary — 3-5 marks.
Given a number in one base, express it in all four bases — 5 marks.

Conversion among binary, octal, and hexadecimal systems