- Octal System Definition: The octal system is defined as a base-8 number system using digits from 0 to 7.
- Binary System Definition: The binary system is defined as a base-2 number system using digits 0 and 1.
- Direct Octal to Binary Conversion: Replace each octal digit with its three-bit binary equivalent.
- Replace each octal digit with its three-bit binary equivalent.: Group binary digits into sets of three and convert each set to its octal equivalent.
- Binary to Octal Conversion Examples: Converting binary numbers to octal numbers directly using grouping and conversion tables.
Octal is base 8 and uses the digits 0 through 7. Binary is base 2 and uses 0 and 1. Because 8 = 2³, each octal digit corresponds to exactly three binary digits. This article explains direct binary to octal conversion and the longer decimal cross-check.
What is Octal?
An octal numeral uses positional powers of 8. A radix subscript prevents the base from being mistaken for part of the value. For example, (123)₈ expands as:
(123)₈ = (1 × 8^2) + (2 × 8^1) + (3 × 8^0)
(123)₈ = 64 + 16 + 3
(123)₈ = (83)₁₀
Octal gives a compact form for long binary numbers. Three bits cover the values 0 through 7, so each group maps to one octal digit without arithmetic through decimal. Octal remains familiar in contexts such as Unix file permissions.
What is Binary?
A binary numeral uses positional powers of 2. For example, (1011)₂ expands as:
(1011)₂ = (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0)
(1011)₂ = 8 + 0 + 2 + 1
(1011)₂ = (11)₁₀
A binary digit, or bit, has the abstract value 0 or 1. Digital circuits encode those values with defined voltage, current, charge, magnetic or optical states. The physical encoding therefore depends on the hardware and is not always a literal off or on switch.
Direct Conversion from Octal to Binary
Direct conversion works because each base-8 place equals one group of three base-2 places:
- Step 1: Separate the octal digits while keeping their order.
- Step 2: Replace every octal digit with the corresponding three-bit group:
| Octal | Binary |
|---|---|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
- Step 3: Join the three-bit groups in the original digit order.
- Step 4: For an ordinary integer value, remove optional leading zeros from the complete result.
Convert (154)₈ to binary:
- Step 1: Separate the digits:
1 5 4 - Step 2: Replace each digit with its three-bit representation:
| Octal | Binary |
|---|---|
| 1 | 001 |
| 5 | 101 |
| 4 | 100 |
- Step 3: Join the groups in order:
001 101 100 - Step 4: Remove the two optional leading zeros:
1101100
Therefore, (154)₈ = (1101100)₂.
Direct Conversion from Binary to Octal
For the reverse conversion, split the binary integer into three-bit groups:
- Step 1: Starting at the right, separate the binary digits into groups of three. Add leading zeros only to complete the leftmost group.
- Step 2: Replace each three-bit group with the corresponding octal digit:
| Binary | Octal |
|---|---|
| 000 | 0 |
| 001 | 1 |
| 010 | 2 |
| 011 | 3 |
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
- Step 3: Join the octal digits in the same order as the binary groups.
Convert (11100)₂ to octal:
- Step 1: Group from the right and pad the leftmost group:
011100 - Step 2 continues with the grouped values shown below.
Complete the conversion as follows:
- Step 2: Replace each group with its octal digit:
| Binary | Octal |
|---|---|
| 011 | 3 |
| 100 | 4 |
- Step 3: Join the octal digits in order:
34
Therefore, (11100)₂ = (34)₈.
Indirect Conversion from Octal to Binary
An indirect conversion can use decimal as an intermediate base. This method is longer than three-bit substitution, but it provides an independent arithmetic check:
- Step 1: Convert the octal numeral to decimal by multiplying each digit by its positional power of 8. For
(350)₈:
(350)₈ = (3 × 8^2) + (5 × 8^1) + (0 × 8^0)
(350)₈ = 192 + 40 + 0
(350)₈ = (232)₁₀
- Step 2: Repeatedly divide 232 by 2. Read the remainders from the final division back to the first:
| Decimal | Quotient | Remainder |
|---|---|---|
| 232 | 116 | 0 |
| 116 | 58 | 0 |
| 58 | 29 | 0 |
| 29 | 14 | 1 |
| 14 | 7 | 0 |
| 7 | 3 | 1 |
| 3 | 1 | 1 |
| 1 | 0 | 1 |
Read the remainders from bottom to top: (11101000)₂.
Therefore, (350)₈ = (11101000)₂.
Indirect Conversion from Binary to Octal
The reverse indirect method converts binary to decimal, then decimal to octal:
- Step 1: Multiply each binary digit by its positional power of 2. For
(1101100)₂:
(1101100)₂ = (1 × 2^6) + (1 × 2^5) + (0 × 2^4) + (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (0 × 2^0)
(1101100)₂ = 64 + 32 + 0 + 8 + 4 + 0 + 0
(1101100)₂ = (108)₁₀
- Step 2: Repeatedly divide 108 by 8, then read the remainders from the final division back to the first:
| Decimal | Quotient | Remainder |
|---|---|---|
| 108 | 13 | 4 |
| 13 | 1 | 5 |
| 1 | 0 | 1 |
The remainders read from bottom to top give (154)₈.
Therefore, (1101100)₂ = (154)₈.
Summary and Conclusion
Direct three-bit substitution is the shortest conversion between octal and binary. Decimal conversion is useful as a separate check. Keep these rules in view:
- Octal is base 8 and uses only the digits 0 through 7.
- Binary is base 2 and uses only 0 and 1.
- For direct octal-to-binary conversion, replace each octal digit with its three-bit equivalent.
- For a binary integer, group bits from right to left in threes, pad the leftmost group if needed and replace each group with one octal digit.
- For an indirect octal-to-binary check, evaluate the powers of 8 in decimal. Repeated division by 2 then gives the binary digits in reverse remainder order.
- For an indirect binary-to-octal check, evaluate the powers of 2 in decimal. Repeated division by 8 then gives the octal digits in reverse remainder order.
Always mark the radix clearly. In (34)₈, the subscript 8 names the base and is not a third value digit. A final decimal check confirms that (34)₈, (11100)₂ and (28)₁₀ represent the same integer.
