Binary

Binary (Base-2) is the fundamental language of computers. Every piece of data (code, images, integers), is ultimately stored as a sequence of 0s (off) and 1s (on).

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Units of Binary Data

• Bit (b): A single 0 or 1.
• Byte (B): 8 bits. Range: $0$ to $255$ ($2^8-1$).
  • This is the smallest addressable unit of memory in C (char).
• Word: The natural size of the CPU architecture (usually 64-bit).

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Hexadecimal (Base-16)

We use Hex because it maps perfectly to binary. One Hex digit represents exactly 4 bits

• 0b1111 = 15 (Decimal) = 0xF (Hex).
• 0b1010 = 10 (Decimal) = 0xA (Hex).

Example: A 32-bit memory address like 11000000101010000000000100000001 is unreadable. In Hex: 0xC0A80101.

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Integer Overflow & Modular Arithmetic

Computer integers are finite. When a variable exceeds its maximum value (e.g., a standard uint32_t exceeds $2^{32}-1$), it wraps around to 0.

• This behavior is mathematically identical to Modular Arithmetic.
• A 32-bit unsigned integer behaves like the ring of integers modulo $2^{32}$.

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