The foundation of any number system

The base determines how many unique digits are used and how place values increase. Base 10 uses digits 0-9. Base 2 (binary) uses 0-1. Base 16 (hexadecimal) uses 0-9 plus A-F.

In base 8 (octal): 157₈ = 1×64 + 5×8 + 7×1 = 111₁₀

Symbols representing values in a number system

Each base requires unique symbols for values 0 through (base-1). Binary uses {0,1}. Decimal uses {0-9}. Hexadecimal extends to {0-9, A-F} where A=10...F=15.

2F3₁₆ in hex = 2×256 + 15×16 + 3 = 755₁₀

Translating numbers between different systems

Converting involves expanding to decimal using positional values, then converting to target base. From any base to decimal: sum digit×base^position.

1011₂ → decimal: 8 + 0 + 2 + 1 = 11₁₀

The language of computers - only 0s and 1s

Binary is the foundation of all digital systems. Every computer operation reduces to binary. Each digit (bit) represents on/off states.

• Digits: {0, 1} - minimal symbol set
• One byte = 8 bits, represents 0-255 in decimal
• Powers of 2 are round numbers: 1024₁₀ = 10000000000₂
• Addition simple: 0+0=0, 0+1=1, 1+1=10
• Used in: CPUs, memory, networks, digital logic

Compact binary representation using digits 0-7

Octal groups binary digits in sets of three (2³=8). Each octal digit = exactly 3 binary bits.

• Digits: {0-7} - no 8 or 9 exists
• Each octal digit = 3 binary bits: 7₈ = 111₂
• Unix permissions: 755 = rwxr-xr-x
• Historical: early minicomputers
• Less common today: hex has replaced octal

The universal human number system

Decimal is standard for human communication worldwide. Its base-10 structure evolved from counting on fingers.

• Digits: {0-9} - ten symbols
• Natural for humans: 10 fingers
• Scientific notation uses decimal: 6.022×10²³
• Currency, measurements, calendars
• Computers convert to binary internally

Programmer's shorthand for binary

Hexadecimal is the modern standard for representing binary compactly. One hex digit = exactly 4 bits (2⁴=16).

• Digits: {0-9, A-F} where A=10...F=15
• Each hex digit = 4 bits: F₁₆ = 1111₂
• One byte = 2 hex digits: FF₁₆ = 255₁₀
• RGB colors: #FF5733 = red(255) green(87) blue(51)
• Memory addresses: 0x7FFF8A2C

Most efficient base mathematically

Ternary uses digits {0,1,2}. Most efficient radix for representing numbers (closest to e=2.718).

• Mathematical efficiency optimal
• Balanced ternary: {-,0,+} symmetric
• Ternary logic in fuzzy systems
• Proposed for quantum computing (qutrits)

The practical alternative to decimal

Base 12 has more divisors (2,3,4,6) than 10 (2,5), simplifying fractions. Used in time, dozens, inches/feet.

• Time: 12-hour clock, 60 minutes (5×12)
• Imperial: 12 inches = 1 foot
• Fractions easier: 1/3 = 0.4₁₂
• Dozenal Society advocates adoption

Counting by twenties

Base 20 systems evolved from counting fingers + toes. Mayan, Aztec, Celtic, and Basque examples.

• Mayan calendar system
• French: quatre-vingts (80)
• English: 'score' = 20
• Inuit traditional counting

Maximum alphanumeric base

Uses all decimal digits (0-9) plus all letters (A-Z). Compact and human-readable.

• URL shorteners: compact links
• License keys: software activation
• Database IDs: typeable identifiers
• Tracking codes: packages, orders

Ancient Rome (500 BC - 1500 AD)

Dominated Europe for 2000 years. Each symbol has fixed value: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

• Still used: clocks, Super Bowl, outlines
• No zero: calculation difficulties
• Subtractive rules: IV=4, IX=9, XL=40
• Limited: standard goes to 3999
• Replaced by Hindu-Arabic numerals

Ancient Babylon (3000 BC)

Oldest surviving system. 60 has 12 divisors, making fractions easier. Used for time and angles.

• Time: 60 seconds/minute, 60 minutes/hour
• Angles: 360° circle, 60 arcminutes
• Divisibility: 1/2, 1/3, 1/4, 1/5, 1/6 clean
• Babylonian astronomical calculations

Each decimal digit encoded as 4 bits

BCD represents each decimal digit (0-9) as 4-bit binary. 392 becomes 0011 1001 0010. Avoids floating-point errors.

• Financial systems: exact decimal
• Digital clocks and calculators
• IBM mainframes: decimal unit
• Credit card magnetic stripes

Adjacent values differ by one bit

Gray code ensures only one bit changes between consecutive numbers. Critical for analog-to-digital conversion.

• Rotary encoders: position sensors
• Analog-to-digital conversion
• Karnaugh maps: logic simplification
• Error correction codes

Programmers work with multiple bases daily:

• Memory addresses: 0x7FFEE4B2A000 (hex)
• Bit flags: 0b10110101 (binary)
• Color codes: #FF5733 (hex RGB)
• File permissions: chmod 755 (octal)
• Debugging: hexdump, memory inspection

Network protocols use hex and binary:

• MAC addresses: 00:1A:2B:3C:4D:5E (hex)
• IPv4: 192.168.1.1 = binary notation
• IPv6: 2001:0db8:85a3:: (hex)
• Subnet masks: 255.255.255.0 = /24
• Packet inspection: Wireshark hex

Hardware design at binary level:

• Logic gates: AND, OR, NOT binary
• CPU registers: 64-bit = 16 hex digits
• Assembly language: opcodes in hex
• FPGA programming: binary streams
• Hardware debugging: logic analyzers

Number theory explores properties:

• Modular arithmetic: various bases
• Cryptography: RSA, elliptic curves
• Fractal generation: Cantor set ternary
• Prime number patterns
• Combinatorics: counting patterns

Expand using positional values:

• Identify base and digits
• Assign positions right-to-left (0, 1, 2...)
• Convert digits to decimal values
• Multiply: digit × base^position
• Sum all terms

Repeatedly divide by target base:

• Divide number by target base
• Record remainder (rightmost digit)
• Divide quotient by base again
• Repeat until quotient is 0
• Read remainders bottom-to-top

Group binary bits:

• Binary → Hex: group by 4 bits
• Binary → Octal: group by 3 bits
• Hex → Binary: expand each digit to 4 bits
• Octal → Binary: expand to 3 bits per digit
• Skip decimal conversion entirely!

Tricks for common conversions:

• Powers of 2: memorize 2¹⁰=1024, 2¹⁶=65536
• Hex: F=15, FF=255, FFF=4095
• Octal 777 = binary 111111111
• Doubling/halving: shift binary
• Use calculator programmer mode

Every time you check the clock, you're using a 5000-year-old Babylonian base-60 system. They chose 60 because it has 12 divisors, making fractions easier.

In 1999, NASA's $125 million Mars orbiter was destroyed due to unit conversion errors - one team used imperial, another metric. A costly lesson in precision.

Roman numerals have no zero and no negatives. This made advanced mathematics nearly impossible until Hindu-Arabic numerals (0-9) revolutionized math.

The Apollo Guidance Computer displayed everything in octal (base 8). Astronauts memorized octal codes for programs that landed humans on the Moon.

RGB color codes use hex: #RRGGBB where each is 00-FF (0-255). This gives 256³ = 16,777,216 possible colors in 24-bit true color.

Soviet researchers built ternary (base-3) computers in the 1950s-70s. The Setun computer used -1, 0, +1 logic instead of binary. Binary infrastructure won.

Binary maps perfectly to electronic circuits: on/off, high/low voltage. Two-state systems are reliable, fast, and easy to manufacture. Decimal would require 10 distinct voltage levels, making circuits complex and error-prone.

Memorize the 16 hex-to-binary mappings (0=0000...F=1111). Convert each hex digit independently: A5₁₆ = 1010|0101₂. Group binary by 4 from right to reverse: 110101₂ = 35₁₆. No decimal needed!

Essential for programming (memory addresses, bit operations), networking (IP addresses, MAC addresses), debugging (memory dumps), digital electronics (logic design), and security (cryptography, hashing).

Hex aligns with byte boundaries (8 bits = 2 hex digits), while octal doesn't (8 bits = 2.67 octal digits). Modern computers are byte-oriented, making hex more convenient. Only Unix file permissions keep octal relevant.

No easy direct method. Octal groups binary by 3, hex by 4. Must convert via binary: octal→binary (3 bits)→hex (4 bits). Example: 52₈ = 101010₂ = 2A₁₆. Or use decimal as intermediate.

Tradition and aesthetics. Used for formality (Super Bowl, movies), distinction (outlines), timelessness (no century ambiguity), and design elegance. Not practical for calculation but culturally persistent.

Each base has strict rules. Base 8 cannot contain 8 or 9. If you write 189₈, it's invalid. Converters reject it. Programming languages enforce this: '09' causes errors in octal contexts.

Base 1 (unary) uses one symbol (tally marks). Not truly positional: 5 = '11111' (five marks). Used for primitive counting but impractical. Joke: unary is the easiest base - just keep counting!