Binary to Decimal Converter
Convert binary numbers (base-2) to decimal numbers (base-10) instantly. Binary uses only two digits (0 and 1), whilst decimal is the standard numbering system we use every day with digits 0-9.
Quick Conversions
Binary to Decimal Conversion Table
Here’s a reference table showing common binary numbers and their decimal equivalents. This is particularly helpful when you’re learning the pattern of binary counting.
| Binary | Decimal | Binary | Decimal |
|---|---|---|---|
| 0 | 0 | 10000 | 16 |
| 1 | 1 | 10001 | 17 |
| 10 | 2 | 10010 | 18 |
| 11 | 3 | 10011 | 19 |
| 100 | 4 | 10100 | 20 |
| 101 | 5 | 11000 | 24 |
| 110 | 6 | 11001 | 25 |
| 111 | 7 | 11111 | 31 |
| 1000 | 8 | 100000 | 32 |
| 1001 | 9 | 1000000 | 64 |
| 1010 | 10 | 10000000 | 128 |
| 1011 | 11 | 11111111 | 255 |
| 1100 | 12 | 100000000 | 256 |
| 1101 | 13 | 1111111111 | 1023 |
| 1110 | 14 | 10000000000 | 1024 |
| 1111 | 15 | 11111111111111111111111111111111 | 4294967295 |
Conversion Formula & Method
Converting binary to decimal involves multiplying each binary digit by powers of 2, based on its position. The rightmost digit represents 2⁰, the next is 2¹, then 2², and so on.
Decimal = (d₀ × 2⁰) + (d₁ × 2¹) + (d₂ × 2²) + … + (dₙ × 2ⁿ)
Where d represents each binary digit (0 or 1) from right to left.
Step-by-Step Method
Step 1: Write down the binary number and assign powers of 2 from right to left
- Position 0 (rightmost): 1 × 2⁰ = 1 × 1 = 1
- Position 1: 1 × 2¹ = 1 × 2 = 2
- Position 2: 0 × 2² = 0 × 4 = 0
- Position 3 (leftmost): 1 × 2³ = 1 × 8 = 8
8 + 0 + 2 + 1 = 11
Result: 1011₂ = 11₁₀
This method is simpler for mental arithmetic. Start from the left, double the previous total, and add the current digit.
Example: Convert 1011₂
- Start with 0, read first digit (1): (0 × 2) + 1 = 1
- Read second digit (0): (1 × 2) + 0 = 2
- Read third digit (1): (2 × 2) + 1 = 5
- Read fourth digit (1): (5 × 2) + 1 = 11
Powers of 2 Reference
Knowing powers of 2 makes binary conversion much faster. Here are the most commonly used values.
| Power | Value | Power | Value |
|---|---|---|---|
| 2⁰ | 1 | 2⁸ | 256 |
| 2¹ | 2 | 2⁹ | 512 |
| 2² | 4 | 2¹⁰ | 1,024 |
| 2³ | 8 | 2¹¹ | 2,048 |
| 2⁴ | 16 | 2¹² | 4,096 |
| 2⁵ | 32 | 2¹⁶ | 65,536 |
| 2⁶ | 64 | 2²⁰ | 1,048,576 |
| 2⁷ | 128 | 2³² | 4,294,967,296 |
Everyday Examples
Binary isn’t just for computers. Here are some places you encounter it in daily life.
- Computer Memory: Your laptop’s RAM is measured in gigabytes. 1 GB equals 1,073,741,824 bytes (2³⁰), a power of 2 because computers process information in binary.
- IP Addresses: Every device on a network has an IP address. IPv4 addresses like 192.168.1.1 are actually four 8-bit binary numbers (octets) written in decimal.
- File Permissions: On Unix/Linux systems, file permissions like 755 or 644 represent binary values. 7 is 111₂ (read, write, execute), 5 is 101₂ (read, execute).
- Digital Images: Each pixel’s colour is stored as binary. An 8-bit greyscale image has 256 shades (00000000₂ to 11111111₂ in decimal 0-255).
- QR Codes: Those black and white squares? They’re binary data encoded visually. Your phone reads them and converts the pattern back to text or URLs.
Number System Conversions
Binary is part of a larger family of number systems used in computing and mathematics.
| System | Base | Digits Used | Example |
|---|---|---|---|
| Binary | 2 | 0, 1 | 1010₂ |
| Octal | 8 | 0-7 | 12₈ |
| Decimal | 10 | 0-9 | 10₁₀ |
| Hexadecimal | 16 | 0-9, A-F | A₁₆ |
Conversion Between Systems
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 8 | 1000 | 10 | 8 |
| 10 | 1010 | 12 | A |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 255 | 11111111 | 377 | FF |
| 256 | 100000000 | 400 | 100 |
FAQs
What is binary code?
Binary code is a numbering system that uses only two digits: 0 and 1. It’s the fundamental language of computers because electronic circuits have two states—on (1) and off (0). Every piece of data in a computer, from text to images to videos, is ultimately stored as sequences of binary digits.
How do you convert binary to decimal quickly?
The fastest method for mental arithmetic is the double dabble technique. Start from the leftmost digit with 0, then repeatedly double your running total and add the next digit. For example, for 101: start with 0, first digit is 1 so (0×2)+1=1, next digit is 0 so (1×2)+0=2, last digit is 1 so (2×2)+1=5.
Why do computers use binary?
Computers use binary because it’s the simplest and most reliable way to represent data electronically. Electronic circuits can easily detect two states (voltage on or off), making binary perfect for digital electronics. It’s also resistant to errors—distinguishing between two states is more reliable than trying to detect multiple voltage levels.
What’s the largest number you can represent with 8 bits?
With 8 bits, you can represent numbers from 0 to 255. That’s 2⁸ = 256 different values. The largest is 11111111₂ which equals 255₁₀. This is why you often see 255 as a maximum value in computing (like RGB colour values ranging from 0-255).
Can binary represent negative numbers?
Yes, binary can represent negative numbers using several methods. The most common is two’s complement, where the leftmost bit indicates the sign (0 for positive, 1 for negative). This allows computers to perform both addition and subtraction using the same circuitry.
What’s the difference between bits and bytes?
A bit is a single binary digit (0 or 1), whilst a byte is a group of 8 bits. Bytes are the standard unit for measuring computer storage and memory. For example, 1 kilobyte (KB) is 1,024 bytes, and 1 megabyte (MB) is 1,024 kilobytes.
How do you write binary numbers?
Binary numbers are typically written with a subscript 2 (like 1010₂) or with a ‘0b’ prefix (like 0b1010) to distinguish them from decimal numbers. When writing them out, you group them in sets of 4 or 8 for readability—for example, 1010 1100 instead of 10101100.
What happens if I enter an invalid binary number?
A valid binary number contains only 0s and 1s. If you enter any other digits (2-9) or letters, the converter will display an error message. Spaces are typically ignored, so you can write 1010 1100 or 10101100—both are valid.