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Binary Digital Systems

Systems that represent and process information using two discrete voltage levels mapped to 0 and 1.

PLANTED · 12 OCTOBER 2025 · ZETTEL NOTE

A binary digital system represents and stores all data using exactly two discrete values — conventionally 0 and 1. Every piece of information in a modern computer — text, images, programs, network packets — is ultimately a sequence of these bits.

Why Binary?

Physical circuits are far more reliable at distinguishing two states (voltage high / voltage low) than at representing analog magnitudes precisely. A transistor is either on or off. Noise and manufacturing variation can shift an analog signal significantly without flipping a well-designed binary signal.

The two voltage levels are:

Symbol	Meaning	Typical voltage
`0` (LOW)	Logic false	0 V
`1` (HIGH)	Logic true	3.3 V or 5 V

Building Blocks

The Bit

The bit (binary digit) is the elementary unit of information — a single 0 or 1.

Groupings

Group	Size	Range
Nibble	4 bits	0–15
Byte	8 bits	0–255
Word	16 / 32 / 64 bits	Architecture-dependent

Encoding Numbers

Unsigned binary: positional notation in base 2.

$1011_2 = 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 11_{10}$

Two’s complement: the standard for signed integers.

To negate a number: invert all bits, add 1.

$5 = 0101_2 \xrightarrow{\text{invert}} 1010_2 \xrightarrow{+1} 1011_2 = -5$

The MSB is the sign bit. This encoding makes arithmetic circuits identical for signed and unsigned addition.

Logic Gates

All computation reduces to combinations of three fundamental gates:

Gate	Symbol	Behavior
AND	`A · B`	Output 1 only if both inputs are 1
OR	`A + B`	Output 1 if at least one input is 1
NOT	`¬A`	Output 1 if input is 0

From these three, all other gates (NAND, NOR, XOR, XNOR) and all arithmetic circuits (adders, multipliers) can be constructed. NAND and NOR are individually functionally complete — any logic function can be built from only NANDs or only NORs.

Encoding Text

Raw binary numbers need a mapping to represent human-readable characters.

ASCII — 7-bit encoding, covers English letters, digits, and control codes (128 characters)
UTF-8 — variable-width encoding (1–4 bytes) that covers all of Unicode while staying ASCII-compatible

'A' = 65 = 0100 0001
'a' = 97 = 0110 0001
'0' = 48 = 0011 0000

Algorithm — processes sequences of bits
Data Structure — organizes bits into meaningful structures