Hex, Binary, and Decimal: A Plain-English Guide to Number Systems

Positional Notation: The Key Idea

All number systems share the same underlying idea: positional notation. The value of a digit depends on its position, and each position is worth a power of the base.

In decimal (base 10), the number 347 means: 3 × 10² + 4 × 10¹ + 7 × 10⁰ = 300 + 40 + 7 = 347. The digits available are 0 through 9 — ten symbols for base 10.

Every number system works the same way — it just uses a different base and a different set of digit symbols.

Binary: Base 2

Binary uses only two symbols: 0 and 1. Each position is worth a power of 2. The binary number 1011 means: 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 0 + 2 + 1 = 11 in decimal.

Computers use binary because transistors — the physical components of processors and memory — have two stable states: on and off, high voltage and low voltage, conducting and not conducting. Mapping these two states to 1 and 0 is natural and reliable. Building reliable hardware for ten states (to support decimal) would be vastly more complex.

A single binary digit is a bit. Eight bits form a byte, which can represent 2⁸ = 256 different values (0 through 255).

Decimal: Base 10

Decimal is the number system humans use naturally, almost certainly because we have ten fingers. It needs no further explanation — but it is worth noting that decimal is not fundamentally superior to other bases. Its dominance is purely a product of human anatomy and historical convention.

Hexadecimal: Base 16

Hexadecimal (hex) uses sixteen symbols: 0–9 and A–F (where A=10, B=11, C=12, D=13, E=14, F=15). Each position is worth a power of 16. The hex value FF means: 15 × 16¹ + 15 × 16⁰ = 240 + 15 = 255 in decimal.

Why do programmers love hex? Because one hex digit exactly represents four bits, and two hex digits exactly represent one byte. This makes hex a compact, human-readable shorthand for binary data:

• Binary: 11111111 (8 characters to express one byte)
• Hex: FF (2 characters to express one byte)
• Decimal: 255 (3 characters, and no clean relationship to bits)

Where You See Hex in the Wild

• Colors in CSS: #FF5733 — FF = 255 red, 57 = 87 green, 33 = 51 blue (in decimal)
• MAC addresses: 00:1A:2B:3C:4D:5E — six pairs of hex digits, one pair per byte
• IPv6 addresses: 2001:0db8:85a3::8a2e:0370:7334 — groups of four hex digits
• SHA-256 hashes: A 256-bit hash printed as 64 hex characters (32 bytes × 2 hex digits each)
• Memory addresses: Debuggers display addresses as 0x7fff5fbff8b0
• UUID values: 550e8400-e29b-41d4-a716-446655440000

ASCII: Where Text Meets Binary

The ASCII standard (1963) assigned a numeric value to 128 characters — letters, digits, punctuation, and control codes. The capital letter A is 65 in decimal, 41 in hex, and 01000001 in binary. All three refer to the same byte value.

Understanding this relationship explains why hex is used when examining file contents, network packets, or cryptographic keys: text is bytes, bytes are numbers, and hex is the most concise way to write those numbers.

UTF-8: Multi-Byte Characters

ASCII covers 128 characters — enough for English but not for the world's scripts. UTF-8 extends ASCII by using multiple bytes for characters beyond the 127 ASCII range. The character é is encoded as two bytes: 0xC3 0xA9. The Japanese character 日 is three bytes: 0xE6 0x97 0xA5.

UTF-8 is backward-compatible with ASCII: any valid ASCII text is also valid UTF-8. Characters in the ASCII range (0–127) are stored as a single byte with the same value. This compatibility is why UTF-8 became the dominant encoding for the web.

Bitwise Operations

Understanding binary also clarifies bitwise operations — fundamental operations in cryptography, compression, and low-level programming:

• AND: Both bits must be 1. 1010 AND 1100 = 1000
• OR: At least one bit must be 1. 1010 OR 1100 = 1110
• XOR: Bits must be different. 1010 XOR 1100 = 0110 — XOR is used extensively in encryption
• Left shift: Shift all bits left by n positions — equivalent to multiplying by 2ⁿ
• Right shift: Shift all bits right — equivalent to dividing by 2ⁿ