Binary in Computing
How Computers Think in 1s and 0s
Understand BinaryEvery photo, video, program, and website ultimately exists as patterns of 1s and 0s. Binary isn't just another number system—it's the foundation of all digital technology. Understanding binary reveals how computers work at their most fundamental level.
Bits and Bytes
The Bit
- Smallest unit of data
- Single binary digit: 0 or 1
- "Binary digit" shortened to "bit"
- Can represent two states (yes/no, on/off, true/false)
The Byte
- 8 bits grouped together
- Can represent 2⁸ = 256 different values (0-255)
- Standard unit for character storage
- Foundation for larger units (KB, MB, GB)
Larger Units
| Unit | Size | Values |
|---|---|---|
| Byte | 8 bits | 256 |
| Word (16-bit) | 2 bytes | 65,536 |
| Double word (32-bit) | 4 bytes | ~4.3 billion |
| Quad word (64-bit) | 8 bytes | ~18.4 quintillion |
How Data Is Represented
Text (Characters)
- ASCII: 7 bits, 128 characters
- Extended ASCII: 8 bits, 256 characters
- Unicode (UTF-8): Variable length, millions of characters
Example: 'A' = 01000001 (65 in decimal)
Numbers
- Integers: Direct binary representation
- Negative numbers: Two's complement
- Decimals: Floating-point (IEEE 754)
Images
- Pixels represented as numbers
- RGB: 3 bytes per pixel (8 bits each for Red, Green, Blue)
- 1920×1080 image ≈ 6.2 million bytes uncompressed
Audio
- Sound waves sampled as numbers
- CD quality: 16-bit samples, 44,100 times per second
Binary Arithmetic
Addition
Same as decimal, but carry at 2:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (0, carry 1)
- 1 + 1 + 1 = 11 (1, carry 1)
Example: 1011 + 1101
1011 + 1101 ------ 11000
= 11 + 13 = 24 ✓
Logic Gates
Hardware implements binary operations through logic gates:
Basic Gates
| Gate | Function | Truth |
|---|---|---|
| AND | Both inputs must be 1 | 1 AND 1 = 1 |
| OR | At least one input is 1 | 1 OR 0 = 1 |
| NOT | Inverts input | NOT 1 = 0 |
| XOR | Exactly one input is 1 | 1 XOR 1 = 0 |
| NAND | NOT AND | 1 NAND 1 = 0 |
Complex operations (addition, comparison) are built from combinations of these simple gates.
Signed Numbers: Two's Complement
How computers represent negative numbers:
Method
- Invert all bits
- Add 1
Example: -5 in 8-bit
- 5 = 00000101
- Invert: 11111010
- Add 1: 11111011
- -5 = 11111011
Why Two's Complement?
- Addition works naturally (no special cases)
- One representation for zero
- Easy to implement in hardware
Bitwise Operations in Programming
Programming languages provide operators for bit manipulation:
Common Operations
- AND (&): Mask certain bits
- OR (|): Set certain bits
- XOR (^): Toggle bits, encryption
- NOT (~): Invert all bits
- Left shift (<<): Multiply by 2ⁿ
- Right shift (>>): Divide by 2ⁿ
Example: Check if number is even
n & 1 == 0 means n is even
(Last bit determines odd/even)
Binary in Modern Computing
Memory Addresses
- 32-bit: Can address 2³² = 4 GB
- 64-bit: Can address 2⁶⁴ = 16 exabytes
Network Addresses
- IPv4: 32 bits (e.g., 192.168.1.1)
- IPv6: 128 bits
File Sizes
- All files are sequences of bytes
- File type determined by content/structure
Encryption
- AES uses 128, 192, or 256-bit keys
- SHA-256 produces 256-bit hashes
Conclusion
Binary is the language of all digital systems because electronic components naturally have two states. Every piece of digital data—text, images, audio, video, programs—is ultimately represented as patterns of bits. Understanding binary illuminates how computers work: from the logic gates performing operations to the bytes storing characters to the bits flying across networks. While we rarely interact with binary directly, it underlies everything in the digital world.