🔢 Binary to Decimal Converter
Convert binary numbers to decimal and decimal to binary instantly. Includes step-by-step calculation explanation and powers of 2 reference table. Perfect for students learning number systems, programmers, and digital electronics enthusiasts.
Decimal Result
Enter a valid number to see step-by-step calculation.
Powers of 2 Reference
Use this table for manual binary to decimal conversion
| Power | Value | Binary | Hex |
|---|---|---|---|
| 20 | 1 | 1 | 0x1 |
| 21 | 2 | 10 | 0x2 |
| 22 | 4 | 100 | 0x4 |
| 23 | 8 | 1000 | 0x8 |
| 24 | 16 | 10000 | 0x10 |
| 25 | 32 | 100000 | 0x20 |
| 26 | 64 | 1000000 | 0x40 |
| 27 | 128 | 10000000 | 0x80 |
| 28 | 256 | 100000000 | 0x100 |
Click any row to try that binary value
Frequently Asked Questions About Binary & Decimal
How do I convert binary to decimal manually?
Follow these steps to convert binary to decimal:
- Write down the binary number (e.g., 1011)
- List the powers of 2 from right to left, starting at 20 (1,2,4,8,16...)
- Multiply each binary digit by its corresponding power of 2
- Add all the results together
Example: Binary 1011 = (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11
Use our Show Calculation Steps button to see the process for any number!
How do I convert decimal to binary manually?
Use the division-by-2 method:
- Divide the decimal number by 2
- Write down the remainder (0 or 1)
- Continue dividing the quotient by 2 until you reach 0
- Read the remainders from bottom to top
Example: Convert 13 to binary:
- 13 ÷ 2 = 6 remainder 1
- 6 ÷ 2 = 3 remainder 0
- 3 ÷ 2 = 1 remainder 1
- 1 ÷ 2 = 0 remainder 1
- Read bottom to top: 1101
Why do computers use binary?
Computers use binary because they're built with transistors that have two physical states: On (1) and Off (0). This makes binary the most efficient and reliable way to represent data electronically. Advantages include:
- Simple hardware design (transistors as switches)
- Resistant to noise and signal degradation
- Maps directly to Boolean logic (AND, OR, NOT gates)
- Error detection and correction is easier
What is the largest binary number?
There's no theoretical limit - binary numbers can be arbitrarily long. Common sizes in computing:
- 8-bit (byte): 11111111 = 255
- 16-bit (word): 1111111111111111 = 65,535
- 32-bit (double word): 11111111111111111111111111111111 = 4,294,967,295
- 64-bit (quad word): 264-1 ≈ 1.84 × 1019
What is the difference between binary, decimal, and hexadecimal?
- Binary (Base-2): Uses digits 0 and 1. Native language of computers.
- Decimal (Base-10): Uses digits 0-9. What humans use daily.
- Hexadecimal (Base-16): Uses 0-9 and A-F. Compact representation of binary (1 hex digit = 4 bits).
Example: Decimal 255 = Binary 11111111 = Hex FF
Understanding Binary Number System
What is Binary?
Binary is a base-2 number system that uses only two digits: 0 and 1. Each digit is called a bit (binary digit). 8 bits make a byte.
Place Values (Powers of 2)
From right to left, each position represents: 2⁰ (1), 2¹ (2), 2² (4), 2³ (8), 2⁴ (16), etc.
Common Binary Terms
- Bit: Single binary digit (0 or 1)
- Nibble: 4 bits
- Byte: 8 bits
- Word: 16 or 32 bits (depends on system)
Quick Conversion Tips
- • Binary 1010 = 10 (decimal)
- • Binary 1111 = 15 (decimal)
- • Binary 10000 = 16 (decimal)
| Decimal | Binary | Hex | Description |
|---|---|---|---|
| 0 | 0 | 0x0 | Zero |
| 1 | 1 | 0x1 | One |
| 2 | 10 | 0x2 | Two |
| 3 | 11 | 0x3 | Three |
| 4 | 100 | 0x4 | Four |
| 5 | 101 | 0x5 | Five |
| 6 | 110 | 0x6 | Six |
| 7 | 111 | 0x7 | Seven |
| 8 | 1000 | 0x8 | Eight |
| 9 | 1001 | 0x9 | Nine |
| 10 | 1010 | 0xA | Ten |