Understanding Number Systems: Binary, Decimal, and Hexadecimal
Introduction
When working with EVM and smart contracts, you'll frequently encounter different number systems. Understanding how to convert between binary, decimal, and hexadecimal is crucial for working with bytecode, memory addresses, and data representation.
Why Different Number Systems?
Each number system serves a specific purpose:
- Binary (base-2): Used by computers for internal processing
- Decimal (base-10): Natural for human counting and calculations
- Hexadecimal (base-16): Compact representation of binary data
Decimal (Base-10)
The decimal system is what we use in everyday life. It uses 10 digits (0-9) and each position represents a power of 10.
Example: 425₁₀
4 × 10² + 2 × 10¹ + 5 × 10⁰
4 × 100 + 2 × 10 + 5 × 1
400 + 20 + 5 = 425
Binary (Base-2)
Binary uses only two digits (0 and 1). Each position represents a power of 2. In EVM, data is processed at the binary level.
Example: 1101₂
1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
1 × 8 + 1 × 4 + 0 × 2 + 1 × 1
8 + 4 + 0 + 1 = 13₁₀
Hexadecimal (Base-16)
Hexadecimal uses 16 digits (0-9 and A-F). It's commonly used in EVM bytecode and memory addresses because it's a more compact way to represent binary data.
Example: 2A₁₆
2 × 16¹ + 10 × 16⁰
2 × 16 + 10 × 1
32 + 10 = 42₁₀
Quick Conversion Chart
| Decimal | Binary | Hexadecimal |
|---|---|---|
| 0 | 0000 | 0 |
| 1 | 0001 | 1 |
| 10 | 1010 | A |
| 15 | 1111 | F |
| 16 | 10000 | 10 |
Converting Between Systems
Decimal to Binary
To convert decimal to binary:
- Divide the number by 2
- Keep track of remainders
- Read remainders from bottom to top
Example: 13₁₀ to Binary
13 ÷ 2 = 6 remainder 1 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1 Result: 1101₂
Binary to Hexadecimal
To convert binary to hexadecimal:
- Group binary digits into sets of 4
- Convert each group to its hex equivalent
Example: 1101 1110₂ to Hex
1101 = D 1110 = E Result: DE₁₆
Common EVM Examples
In EVM, you'll often see numbers represented in hexadecimal with a "0x" prefix. Here are some common examples:
Memory Addresses
0x123f = 4671₁₀
Used for storage slots and memory locations
Bytecode Operations
0x60 = PUSH1
Opcodes are represented in hex
Practice Problems
Try converting these numbers:
1. Convert 42₁₀ to binary and hexadecimal
2. Convert 0xFF to decimal
3. Convert 1100 1010₂ to hexadecimal
Video Resources
Check out this comprehensive video playlist on number systems and conversions:
Tools and Resources
Here are some helpful tools for working with different number systems:
- RapidTables Converter- Online conversion between different number systems
- Hex Calculator- Perform arithmetic operations in hexadecimal
Conclusion
Understanding number systems is fundamental to working with EVM and smart contracts. Practice converting between different bases, and you'll find it becomes more intuitive over time. In our next posts, we'll use this knowledge to better understand EVM bytecode and memory operations.