Number Systems: Binary, Octal, and Hexadecimal

A Room Full of Light Switches

Imagine a room with one billion light switches. Each one is either ON or OFF. Nothing in between — no dimmer, no partial state, just two positions.

That's your computer's memory.

Every piece of data — this sentence, a YouTube video, a bank transaction, a 3D game world — exists inside your machine as a colossal arrangement of switches, each either ON (1) or OFF (0). The reason is physics: transistors are the world's best switches, and switches have two stable states. Building a reliable three-state transistor at nanometer scale is extraordinarily difficult. Binary is not a human preference. It's the language of electrons in silicon.

Why Binary? The Physics Explanation

A modern CPU transistor operates at either ~0V (logic 0) or ~1.0–3.3V (logic 1), depending on the process node. There is a forbidden zone in between — any voltage there represents an error.

This design gives us noise immunity: even if a wire picks up interference and the voltage wavers from 1.0V to 1.15V, the transistor still reliably reads it as "1". With three or more states, those margins shrink dramatically and errors multiply.

The result: Every circuit in your CPU, RAM, SSD, and GPU operates in binary. Choosing base-2 is not arbitrary — it is thermodynamically optimal for silicon-based electronics.

Binary: Base-2

In the decimal system (base-10), each digit position represents a power of 10:

365 = 3×10² + 6×10¹ + 5×10⁰

In binary (base-2), each digit represents a power of 2:

1011 = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11

Each binary digit is called a bit (binary digit). Eight bits form a byte.

Decimal to Binary Conversion (Repeated Division)

To convert 42 to binary, repeatedly divide by 2 and record remainders:

42 ÷ 2 = 21  remainder 0  ← least significant bit
21 ÷ 2 = 10  remainder 1
10 ÷ 2 =  5  remainder 0
 5 ÷ 2 =  2  remainder 1
 2 ÷ 2 =  1  remainder 0
 1 ÷ 2 =  0  remainder 1  ← most significant bit

Read remainders bottom-up: 101010

42 in decimal = 101010 in binary ✓

Binary to Decimal Conversion

101010 = 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1
       = 32 + 8 + 2
       = 42 ✓

Octal: Base-8

Octal uses digits 0–7, where each position represents a power of 8. It was widely used in early computing because 3 binary digits map perfectly to 1 octal digit.

Primary use today: Unix/Linux file permissions.

When you type chmod 755 in a terminal, those three digits are octal:

• 7 = 111 in binary = read + write + execute (owner)
• 5 = 101 in binary = read + execute (group)
• 5 = 101 in binary = read + execute (others)

Octal 377 = 3×64 + 7×8 + 7×1 = 192 + 56 + 7 = 255 (decimal)
Binary: 011 111 111 = 11111111

Hexadecimal: Base-16

Hexadecimal (hex) uses 16 symbols: 0–9 then A–F (where A=10, B=11, C=12, D=13, E=14, F=15).

The power of hex: 4 binary digits = 1 hex digit exactly. This makes hex a compact, human-readable representation of raw binary data.

Decimal to Hex Conversion

Converting 255 to hex:

255 ÷ 16 = 15  remainder 15 → F
 15 ÷ 16 =  0  remainder 15 → F

Read bottom-up: FF

255 = 0xFF = 0b11111111 = 0o377

Why Programmers Use Hex

• Memory addresses: 0x7fff5b8a0000 (stack address on x86-64 macOS)
• RGB colors: #FF5733 = Red 255, Green 87, Blue 51
• CPU opcodes: 0x90 is the NOP (no operation) instruction on x86
• Debugging: Memory dumps print as hex because it's 1/4 the length of binary while still showing exact bit patterns

The Master Conversion: 255 in Every System

This number is significant: 255 is the maximum value of an 8-bit unsigned integer — the largest number a single byte can store.

Two's Complement: Negative Numbers in Binary

How does a computer store -5? It can't use a minus sign — those don't exist in binary.

Sign-magnitude (naive approach): use the leftmost bit as a sign. Problem: you get +0 and -0, and addition hardware needs special cases.

Two's complement (the universal standard): To negate a number, flip all bits, then add 1.

Start with +5:   00000101
Flip all bits:   11111010
Add 1:           11111011  ← this is -5 in two's complement

Verify: +5 + (-5) should equal 0.

  00000101  (+5)
+ 11111011  (-5)
──────────
 100000000  → the 9th bit overflows and is discarded → 00000000 ✓

The genius of two's complement: subtraction is just addition with a negated operand. The CPU needs only an adder, not a separate subtractor.

Overflow: When Numbers Exceed Capacity

An 8-bit unsigned integer can hold values 0–255. What happens when you add 200 + 100?

200 = 11001000
100 = 01100100
Sum = 100101100  ← 9 bits! The carry bit is lost.
      00101100  = 44 (decimal)

200 + 100 = 44 in an 8-bit system. This is integer overflow — a real bug source. The classic example: the Ariane 5 rocket explosion in 1996 was caused by a 64-bit float converted to a 16-bit integer overflowing, triggering a self-destruct command.

Number System Conversion Flowchart

Complete Number Systems Reference Table

Key Takeaways

• Binary is binary because transistors have two stable voltage states — physics, not preference
• Each binary digit is a bit; 8 bits = 1 byte
• Decimal → Binary: divide repeatedly by 2, read remainders from bottom up
• Octal (base-8) groups 3 bits; historically used in Unix permissions (chmod 755)
• Hexadecimal (base-16) groups 4 bits; used for memory addresses, colors, and opcodes because it's compact
• 255 = 0xFF = 0b11111111 = 0o377 — the maximum value of 1 byte
• Two's complement is how all modern CPUs represent negative integers — flip bits, add 1
• Overflow occurs when a result exceeds the bit width — a real-world source of security vulnerabilities and bugs