Binary Digital Systems

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:

Building Blocks

The Bit

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

Groupings

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:

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

Related

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