How to Convert Binary to Decimal
A complete guide on binary to decimal conversion methods, positional notation formulas, signed binary arithmetic, and step-by-step examples.
Try it: Binary to Decimal mini-calculator
Understanding positional notation.
The binary system is a base-2 number system representing values using only two digits: 0 and 1. The decimal system is a base-10 system using ten digits (0-9). If you want to perform the opposite calculation, you can use our dedicated decimal to binary converter.
To convert from binary to decimal, we use the positional notation method. Each digit in a binary number represents a power of 2, starting with 20 on the far right (least significant bit) and increasing by a power of one as you move left. For negative values, see how to convert signed binary to decimal, or learn how to group bits to convert binary to hexadecimal directly.
The conversion formula.
For any binary number with n digits, the conversion formula is:
Where d0 is the rightmost digit, d1 is the second digit from the right, and so on.
Step-by-step conversion examples.
Example 1: Convert 1011 to Decimal
- Write down the binary digits and their weights: 1 (23), 0 (22), 1 (21), 1 (20)
- Calculate the values: (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)
- Sum the results: 8 + 0 + 2 + 1 = 11
- Result: 11
Example 2: Convert 10101 to Decimal
- Weights: 1 (24), 0 (23), 1 (22), 0 (21), 1 (20)
- Calculate: (1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
- Sum: 16 + 0 + 4 + 0 + 1 = 21
- Result: 21
Frequently Asked Questions
How do you convert binary to decimal? ▼
To convert binary to decimal, multiply each binary digit (0 or 1) by 2 raised to the power of its position index (starting at 0 on the far right). Finally, sum all these products to get the decimal equivalent.
What is the decimal equivalent of binary 1111? ▼
The binary number 1111 is equivalent to 15 in decimal. Calculation: (1 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 4 + 2 + 1 = 15.
How does signed binary to decimal conversion work? ▼
For signed numbers (usually using 8-bit Two's Complement), the MSB (most significant bit on the left) serves as the sign bit. If the MSB is 1, the number is negative. To find its value, invert all bits, add 1, convert to decimal, and add a negative sign.
Binary to Decimal Chart
| Binary | Decimal |
|---|---|
| 101 | 5 |
| 1010 | 10 |
| 1011 | 11 |
| 1111 | 15 |
| 10101 | 21 |
| 11111 | 31 |