Hex numbers explained

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Hex Numbers Explained

What Are Hexadecimal Numbers?

Hexadecimal (or "hex") is a base-16 number system used to represent numbers in a compact, human-readable way, especially in computing. Unlike the decimal system (base-10), which uses digits 0–9, hexadecimal uses 16 symbols: 0–9 for values 0–9 and A–F (or a–f) for values 10–15.

Hex Symbol Values:

  • 0–9 represent 0–9
  • A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Hex numbers are widely used in programming, digital electronics, and color codes (e.g., #FF0000 for red in web design) because they align with binary (base-2), the language of computers, where one hex digit represents four binary digits (bits).

How Hex Numbers Work

Hex numbers are read in blocks of digits, where each digit’s position represents a power of 16. To convert a hex number to decimal (base-10), multiply each digit by 16 raised to the power of its position (starting from 0 on the right) and sum the results.

For a two-digit hex number (e.g., XY):

  • X (left digit) is multiplied (x16) by 16¹ (16).
  • Y (right digit) is multiplied (x1) by 16⁰ (1).
  • Add the results to get the decimal value.

Examples:

  • 01: (0 × 16) + (1 × 1) = 0 + 1 = 1
  • A4: (10 × 16) + (4 × 1) = 160 + 4 = 164
  • E4: (14 × 16) + (4 × 1) = 224 + 4 = 228
  • FF: (15 × 16) + (15 × 1) = 240 + 15 = 255


Why Use Hexadecimal?

Hexadecimal is compact and efficient for representing large binary numbers. For example:

  • Binary 11111111 = Hex FF = Decimal 255.
  • One hex digit represents four bits, so two hex digits can represent a byte (8 bits), making it easier to read and write than long binary strings.

Hex is common in:

  • Programming: Memory addresses, machine code.
  • Color Codes: RGB values (e.g., #FFFFFF = white).
  • Digital Systems: Representing binary data in hardware design.

A Brief History of Hexadecimal

The hexadecimal system was popularized in the 1950s and 1960s with the rise of digital computing. Early computers used binary, but it was cumbersome for humans. Hexadecimal emerged as a practical solution because it maps directly to binary while being more concise than decimal. IBM and other computer manufacturers adopted hex in their documentation and programming tools. The use of A–F for 10–15 was standardized to avoid confusion with decimal digits.

Converting Between Hex and Other Systems

  • Hex to Binary: Each hex digit equals four bits. E.g., A = 1010, F = 1111. So, A4 = 10100100.
  • Decimal to Hex: Divide the decimal number by 16, note the remainder (0–15, using A–F for 10–15), and repeat for the quotient until it’s 0. Read remainders backward.
    • Example: 164 ÷ 16 = 10 remainder 4 (4); 10 ÷ 16 = 0 remainder 10 (A). Result: A4.
  • Tools: Most programming languages (e.g., Python’s hex() function) and calculators handle conversions automatically.

Fun Fact

Hexadecimal is often prefixed with 0x (e.g., 0xFF) in programming to distinguish it from decimal numbers. This convention started with early programming languages like C.

Try It Yourself

  • Convert 7B to decimal: (7 × 16) + (11 × 1) = 112 + 11 = 123.
  • Convert 200 to hex: 200 ÷ 16 = 12 (C) remainder 8. Result: C8.
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