Hexadecimal Number System

Every microcontroller datasheet, memory map, and hardware register uses hexadecimal notation. Understanding hex is non-negotiable for reading CPU instruction encodings, peripheral base addresses, and embedded firmware.

Core Concept

The hexadecimal system uses base 16. It has sixteen symbols: 0 through 9, then A through F representing decimal 10 through 15. Each single hex digit maps exactly to a 4-bit nibble.

This 1:4 mapping is the key advantage. A 32-bit processor address like 0xBFFF0000 is far more readable than its 32-bit binary equivalent. CPU datasheets, UART baud-rate registers, and I2C device addresses all use hex notation for this reason.

In embedded C and assembly, hex constants are written with the prefix 0x (e.g., 0x1A3C). In GATE problems hex values may also appear as (2AF3)₁₆. The radix subscript confirms the base.

Boolean Expression

The positional value formula is: N = dₙ × 16ⁿ + ... + d₁ × 16¹ + d₀ × 16⁰. Each digit dᵢ is in the range 0–15. For letter digits, substitute A=10, B=11, C=12, D=13, E=14, F=15 before multiplying.

Given:
  Convert (3B7)₁₆ to decimal

Formula / Rule:
  N = d₂×16² + d₁×16¹ + d₀×16⁰

Step by step:
  d₂ = 3, d₁ = B = 11, d₀ = 7
  3 × 256 = 768
  11 × 16  = 176
  7 × 1    = 7
  768 + 176 + 7 = 951

Final Answer:
  (3B7)₁₆ = (951)₁₀

Exam Tip: The most common GATE trap is forgetting that A=10 not 1 and 0. Students also confuse (1A)₁₆ = 26₁₀ with 1A meaning 1 and A separately. Another trap: when converting decimal to hex, divide repeatedly by 16 and read remainders bottom-up. Reading remainders top-down is the classic error.

Key Properties

• Base 16 system; 16 unique symbols per digit position
• One hex digit represents exactly 4 binary bits (one nibble)
• Two hex digits represent one byte (8 bits), range 00 to FF
• Widely used in memory addressing: x86 segment:offset and ARM Cortex-M peripheral base addresses
• Hex to binary conversion: replace each hex digit with its 4-bit group, no calculation needed
• Binary to hex: group binary digits in sets of 4 from LSB side, convert each group
• Hex arithmetic follows standard carry rules but carry occurs at 16, not 10

Quick Revision

• Base 16; digits 0–9 and A–F
• A=10, B=11, C=12, D=13, E=14, F=15
• 1 hex digit = 4 bits; 2 hex digits = 1 byte
• Hex to decimal: multiply each digit by 16 raised to its position
• Decimal to hex: divide by 16, collect remainders from bottom to top
• Hex to binary: each digit directly expands to a 4-bit group
• Used in UART, SPI, I2C register maps and CPU memory-mapped I/O
• Exam trap: reading the repeated-division remainders in the wrong order gives the reversed hex digit sequence