Binary To Decimal Converter
Enter the binary value to convert to decimal or Decimal to Binary.
Binary:
A binary number system has two digits "0" and "1" and is known as the Base-2 number system. It is used in digital electronics and computing, where 0 represents OFF and 1 represents ON.
Decimal:
A decimal number system is also called Base-10 (Hindu-Arabic) number system. It uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
How to Convert Binary to Decimal — Formula:
Decimal = Σ(dᵢ × 2ⁱ) where dᵢ is each binary digit and i is its position from right (starting at 0).
Example: Binary 1010 = (1×2³) + (0×2²) + (1×2¹) + (0×2⁰) = 8 + 0 + 2 + 0 = 10 decimal.
Technical Details:
Binary is the native language of all digital computers. A single binary digit (bit) can represent 2 states. 8 bits = 1 byte (0-255). 32-bit integers can represent values up to 4,294,967,295. JavaScript uses 64-bit floating point (IEEE 754) for all numbers.
Binary To Decimal Converter:
Our free online binary to decimal converter instantly converts binary numbers to decimal. It supports integer and fractional binary numbers with step-by-step calculation.
Binary to Decimal: How Each Bit Has a Value
128 + 32 + 16 + 4 + 2 = 182
How to Convert Binary to Decimal
- Write down the binary number
- Multiply each digit by 2 raised to its position power (starting from 0 on the right)
- Add all the values together
Formula
Example
Binary: 11010110
= 1×2⁷ + 1×2⁶ + 0×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰
= 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0
= 214
Binary to Decimal Conversion Table
| Binary | Decimal | Binary | Decimal |
|---|---|---|---|
| 0000 | 0 | 1000 | 8 |
| 0001 | 1 | 1001 | 9 |
| 0010 | 2 | 1010 | 10 |
| 0011 | 3 | 1011 | 11 |
| 0100 | 4 | 1100 | 12 |
| 0101 | 5 | 1101 | 13 |
| 0110 | 6 | 1110 | 14 |
| 0111 | 7 | 1111 | 15 |
Frequently Asked Questions
How do I convert Binary to Decimal?
Decimal = Σ(dᵢ × 2ⁱ) where dᵢ is each binary digit and i is its position from right (starting at 0).
What is the Binary number system?
A binary number system has two digits "0" and "1" and is known as the Base-2 number system. It is used in digital electronics and computing, where 0 represents OFF and 1 represents ON.
What is the Decimal number system?
A decimal number system is also called Base-10 (Hindu-Arabic) number system. It uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Where is Binary to Decimal conversion used?
Binary is the native language of all digital computers. A single binary digit (bit) can represent 2 states. 8 bits = 1 byte (0-255). 32-bit integers can represent values up to 4,294,967,295. JavaScript uses 64-bit floating point (IEEE 754) for all numbers.
Can I convert large binary numbers?
Yes. This converter handles numbers of any practical size. For very large numbers, the conversion is performed using arbitrary-precision arithmetic to ensure accuracy.
How to Convert Binary to Decimal (Base-2 to Base-10)
Binary (base-2) is the language of computers — every processor instruction, memory address, and data byte is ultimately binary. Converting binary to decimal helps programmers interpret raw data, debug memory dumps, and understand how computers represent numbers internally.
- Write down the binary number with position indices starting from 0 on the right.
- For each digit that is 1, calculate 2 raised to that position's power.
- Sum all the powers of 2 where the digit is 1.
- Example: 1101₂ → 2³ + 2² + 0 + 2⁰ = 8 + 4 + 0 + 1 = 13₁₀.
Binary to Decimal: Key Values
Important binary numbers every programmer should recognize:
| Input | Output |
|---|---|
| 0000 0001 | 1 |
| 0000 1010 | 10 |
| 0010 0000 | 32 |
| 0100 0001 | 65 |
| 0111 1111 | 127 |
| 1000 0000 | 128 |
| 1111 1111 | 255 |
| 1111 1111 1111 1111 | 65,535 |
Solved Examples: Binary to Decimal
Question 1: Convert the binary number 10110110 to decimal.
Solution:
Position values (right to left): 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, 2⁰
Binary digits: 1, 0, 1, 1, 0, 1, 1, 0
Sum where digit = 1: 128 + 32 + 16 + 4 + 2 = 182
Answer: 10110110₂ = 182₁₀ — this is 0xB6 in hex, within the extended ASCII range.
Question 2: A Unix file permission is 111 101 101 in binary. What is the decimal value?
Solution:
Split into three octal groups: 111 = 7, 101 = 5, 101 = 5
Or as a full binary: 111101101₂
= 256 + 128 + 64 + 32 + 0 + 8 + 4 + 0 + 1 = 493
Answer: 111101101₂ = 493₁₀ (octal 755) — the classic rwxr-xr-x permission set.
Question 3: Convert 11111111 11111111 (16-bit all ones) to decimal.
Solution:
This is 2¹⁶ - 1 (all bits set)
= 65,536 - 1
= 65,535
Answer: 1111111111111111₂ = 65,535₁₀ — the maximum value of an unsigned 16-bit integer.
Practice: Binary to Decimal
Try solving these on your own to test your understanding:
- Convert 1010 to decimal. (Answer: 10)
- Convert 11001100 to decimal. (Answer: 204)
- Convert 10000000 to decimal. (Answer: 128)
- Convert 01010101 to decimal. (Answer: 85)
- Convert 11111110 to decimal. (Answer: 254)
Why Computers Use Binary
Transistors have two reliable states: on (1) and off (0). Building circuits with only two states makes them resistant to electrical noise — a signal above a threshold is 1, below is 0. Early computers experimented with ternary (base-3) and decimal systems, but binary won because it requires the simplest, most reliable, and cheapest circuits. Every decimal number you type is converted to binary for processing, then converted back to decimal for display.
Signed vs. Unsigned Binary: Two's Complement
In unsigned binary, 11111111 = 255. In signed (two's complement), the same bits mean -1. The MSB (most significant bit) indicates sign: 0 = positive, 1 = negative. To negate a two's complement number: flip all bits and add 1. Example: 5 = 00000101, flip = 11111010, add 1 = 11111011 = -5. This is why signed 8-bit ranges from -128 to +127, while unsigned ranges from 0 to 255.
Key Takeaways
- Multiply each binary digit by its positional power of 2, then sum.
- Memorize key values: 128, 64, 32, 16, 8, 4, 2, 1 (one byte).
- 8 bits = 0-255 unsigned, -128 to +127 signed (two's complement).
- File permissions (chmod 755) come from interpreting binary as octal groups.
- All data in computers is binary — decimal display is just for human convenience.