Binary Conversion Formula – Positional Notation & Bit-Grouping Reference
The binary-to-decimal positional notation formula and the binary-to-octal/hex bit-grouping shortcuts, with full variable definitions, step-by-step procedure, and three worked examples.
Formula
To convert a binary number to its decimal value, each bit dᵢ is multiplied by 2 raised to its positional index i (counting from 0 at the rightmost bit) and all products are summed. For octal or hexadecimal output, bit-grouping shortcuts bypass the decimal step entirely — group bits into 3s for octal, or 4s for hex, then map each group directly to a digit.
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| N₁₀ | Decimal result | The decimal (base-10) value of the binary number — the sum of all bit contributions | — |
| dᵢ | Bit at position i | The binary digit (0 or 1) at position i. A bit of 1 contributes 2^i to the total; a bit of 0 contributes nothing | — |
| i | Bit position | Integer index of the bit, starting at 0 for the rightmost bit and increasing by 1 for each step to the left | — |
| n | Number of bits | The total length of the binary number — determines the highest positional power used (2^(n-1) for the leftmost bit) | bits |
How to Use
- Write out the binary number and label each bit's position from 0 (rightmost) to n−1 (leftmost).
- For each bit at position i, compute its contribution: if dᵢ = 1, note the value 2^i; if dᵢ = 0, skip it.
- Sum all non-zero contributions — the total is the decimal value N₁₀.
- For binary → octal: pad the binary number on the left with zeros to make its length a multiple of 3, then group into 3-bit chunks from right, and replace each chunk with the octal digit it represents (000=0 through 111=7).
- For binary → hex: pad to a multiple of 4 bits, group into 4-bit chunks from right, and replace each chunk with its hex digit (0000=0 through 1111=F).
Examples
1. Convert (1011)₂ to decimal using positional notation
| Bit | 1 | 0 | 1 | 1 |
|---|---|---|---|---|
| Position | 3 | 2 | 1 | 0 |
| Value 2^i | 8 | 4 | 2 | 1 |
| Contribution | 8 | 0 | 2 | 1 |
(1011)₂ = 11₁₀
2. Convert (101101)₂ to octal using 3-bit grouping
Six bits — divide evenly into two groups of 3 from right to left.
| 3-bit group | 101 | 101 |
|---|---|---|
| Octal digit | 5 | 5 |
(101101)₂ = (55)₈. Verify: 5×8 + 5 = 45₁₀. Binary check: 32+8+4+1 = 45₁₀ ✓
3. Convert (11101001)₂ to hexadecimal using 4-bit grouping
Eight bits — split into two groups of 4 from right to left.
| 4-bit group | 1110 | 1001 |
|---|---|---|
| Decimal value | 14 | 9 |
| Hex digit | E | 9 |
(11101001)₂ = (E9)₁₆. Verify: 14×16 + 9 = 224+9 = 233₁₀. Positional sum: 128+64+32+8+1 = 233₁₀ ✓
Related pages
- Use the Calculator — Interactive calculator for this formula
- Read the Notes — Step-by-step explanation with worked examples