Number Systems Review

Introduction

Everything inside a computer is represented as numbers — and not just the decimal numbers you're used to. Computers work with binary (base-2) because their circuits have two states: on and off, high voltage and low voltage, 1 and 0. But binary numbers get long and hard to read, so computer scientists also use octal (base-8) and hexadecimal (base-16) as convenient shorthand. Mastering these number systems and converting between them is the foundation of everything in computer organization.

The Concept of a Base (Radix)

Every number system has a base (also called radix) — the number of unique digits it uses.

In any base-r system, the value of a number is calculated using positional notation:

d₃ × r³ + d₂ × r² + d₁ × r¹ + d₀ × r⁰

Where d is each digit and r is the radix.

Binary Number System (Base-2)

Why Binary?

Electronic circuits are naturally bistable — they rest in one of two states. This maps perfectly to binary: 0 = low voltage (typically 0V), 1 = high voltage (typically 3.3V or 5V). Building circuits that reliably distinguish between two states is much easier than distinguishing between ten states.

Counting in Binary

Positional Values

Each bit position represents a power of 2:

Example: 10110₂ = 1×16 + 0×8 + 1×4 + 1×2 + 0×1 = 22₁₀

Binary Terminology

• Bit: A single binary digit (0 or 1)
• Nibble: 4 bits (e.g., 1010)
• Byte: 8 bits (e.g., 10110011)
• Word: Processor-dependent (16, 32, or 64 bits)

Octal Number System (Base-8)

Why Octal?

Octal provides a compact representation of binary. Since 8 = 2³, each octal digit represents exactly 3 binary digits. This was especially popular in older systems with 12-bit, 24-bit, or 36-bit words (all multiples of 3).

Converting Binary to Octal

Group binary digits into sets of 3 (from right to left), then convert each group:

Converting Octal to Binary

Replace each octal digit with its 3-bit binary equivalent:

Hexadecimal Number System (Base-16)

Why Hexadecimal?

Hexadecimal is the most popular shorthand for binary in modern computing. Since 16 = 2⁴, each hex digit represents exactly 4 binary digits (one nibble). This makes it perfect for byte-oriented architectures — one byte is always exactly 2 hex digits.

Hex Digits

Converting Binary to Hexadecimal

Group binary digits into sets of 4 (from right to left):

Where You'll See Hex

• Memory addresses: 0x7FFE0000
• Color codes: #FF5733 (RGB colors)
• MAC addresses: 00:1A:2B:3C:4D:5E
• Machine code: 8B 45 FC

Conversion Methods

Decimal to Binary (Repeated Division)

Divide by 2 repeatedly, collect remainders from bottom to top:

So 25₁₀ = 11001₂

Decimal to Any Base (Repeated Division by Base)

Same method works for any base. For 25 to octal:

So 25₁₀ = 31₈

Fractional Conversion (Repeated Multiplication)

For decimal fractions, multiply by the target base and collect integer parts:

So 0.625₁₀ = 0.101₂

Any Base to Decimal (Positional Expansion)

Multiply each digit by its positional value:

Practical Applications

Memory Addresses

A 32-bit memory address like 11111111111111100000000000000000 is unreadable. In hex: FFFE0000 — much cleaner.

Debugging

When examining memory contents or machine code, hexadecimal makes patterns visible that would be invisible in binary.

Color Representation

Web colors use hex: #FF0000 = Red (255,0,0), #00FF00 = Green (0,255,0)

Common Conversions Reference

Key Takeaways

1. Computers use binary because circuits naturally have two states
2. Hexadecimal is the preferred shorthand (1 hex digit = 4 bits = 1 nibble)
3. Octal groups binary into sets of 3; hex groups into sets of 4
4. Decimal-to-binary conversion uses repeated division by 2
5. Fractional conversion uses repeated multiplication by the target base
6. Mastering these conversions is essential — they appear everywhere in computer science