Hexadecimal Number System — Digital Electronics

The hexadecimal (hex) system is the most widely used compact notation for binary data in modern computing. Every byte can be represented by exactly two hexadecimal digits.

Fundamentals

Hex Digit Mapping

Positional Value

Hexadecimal to Decimal Conversion

Example: Convert 1A3F₁₆ to decimal

Decimal to Hexadecimal Conversion

Example: Convert 750₁₀ to hex

Hexadecimal to Binary (Direct Conversion)

Each hex digit = exactly 4 binary bits:

Example: Convert A7F2₁₆ to binary

Binary to Hexadecimal

Group binary digits in sets of 4 (from decimal point outward):

Example: Convert 110101011110₂ to hex

Hexadecimal to Octal Conversion

Convert through binary as intermediate:

Example: Convert 2F₁₆ to octal

Hex Arithmetic

Hex Addition

Hex Subtraction

Common Hex Values in Computing

Hex Notation Conventions

Why Hexadecimal Dominates Computing

Interview Questions

Q1: Why is hexadecimal preferred for representing memory addresses? Each hex digit maps to exactly 4 bits, and two hex digits represent exactly one byte. A 32-bit address is exactly 8 hex digits, a 64-bit address is 16 hex digits. This clean mapping makes it easy to identify individual bytes and bit patterns visually, which is crucial for debugging and memory analysis.

Q2: Convert 0xDEADBEEF to decimal. D×16⁷ + E×16⁶ + A×16⁵ + D×16⁴ + B×16³ + E×16² + E×16¹ + F×16⁰ = 13×268435456 + 14×16777216 + 10×1048576 + 13×65536 + 11×4096 + 14×256 + 14×16 + 15 = 3,735,928,559₁₀

Q3: How do you quickly identify if a hex number is a power of 2? A power of 2 in hex will have exactly one bit set, meaning it will be one of: 1, 2, 4, 8 followed by all zeros. Examples: 0x100 = 256 = 2⁸, 0x8000 = 32768 = 2¹⁵. If the leading digit is 1, 2, 4, or 8 and all other digits are 0, it's a power of 2.

Q4: Why does hexadecimal use A-F instead of other characters? The letters A through F are the first six letters of the alphabet, providing a natural sequential extension after 0-9. This was standardized early in computing history. It keeps hex values compact (single characters) and maintains a clear ordering: 0<1<...<9<A<B<C<D<E<F.

Q5: What is 0xFF in signed vs unsigned interpretation for 8 bits? As unsigned 8-bit: 0xFF = 255₁₀ (all bits set). As signed 8-bit (two's complement): 0xFF = -1₁₀ (the two's complement representation of -1, since inverting gives 0x00 and adding 1 gives 0x01 = 1, so the original is -1).