Number Systems: Binary, Octal, and Hexadecimal Arithmetic

Why These Number Systems Matter

Computers operate in binary - everything is ultimately 1s and 0s. However, reading long strings of binary is impractical for humans. Octal and hexadecimal provide compact representations that map cleanly to binary:

Understanding Place Values

Decimal (Base-10) Review

In decimal, each position is a power of 10:

Binary (Base-2)

Each position is a power of 2:

Powers of 2 Reference

Hexadecimal (Base-16)

Hex Digit Values

Hex to Decimal Conversion

Hex to Binary (Direct Mapping)

Each hex digit converts directly to 4 binary bits:

Common Hex Notations

Octal (Base-8)

Octal Digit Values

Octal to Decimal Conversion

Octal to Binary (Direct Mapping)

Each octal digit converts directly to 3 binary bits:

Binary Arithmetic

Binary Addition

Rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 carry 1)

Binary Subtraction

Rules: 0-0=0, 1-0=1, 1-1=0, 0-1=1 (borrow 1)

Binary Multiplication

Same as decimal: multiply and shift

Two's Complement (Negative Numbers)

Computers represent negative numbers using two's complement:

1. Invert all bits (one's complement)
2. Add 1

Hexadecimal Arithmetic

Hex Addition

Add digits, carry 16 (remember: A=10, B=11, C=12, D=13, E=14, F=15)

Hex Subtraction

Bitwise Operations

These operations work on individual bits and are fundamental to system programming:

AND (&)

Result is 1 only if both bits are 1

OR (|)

Result is 1 if either bit is 1

XOR (^)

Result is 1 if bits are different

NOT (~)

Inverts all bits

Left Shift (<<)

Shifts bits left, fills with zeros (multiply by 2 per shift)

Right Shift (>>)

Shifts bits right (divide by 2 per shift)

Linux/Unix Tools for Number Conversion

printf Command

# Decimal to hex
printf "%x\n" 255
# Output: ff

# Decimal to octal
printf "%o\n" 255
# Output: 377

# Hex to decimal
printf "%d\n" 0xff
# Output: 255

# Octal to decimal
printf "%d\n" 0377
# Output: 255

# With formatting
printf "Hex: 0x%X  Octal: 0%o  Binary: " 255 255
# Output: Hex: 0xFF  Octal: 0377  Binary:

bc Calculator

# Set input base (ibase) and output base (obase)
echo "obase=16; 255" | bc
# Output: FF

echo "obase=2; 255" | bc
# Output: 11111111

echo "ibase=16; obase=2; FF" | bc
# Output: 11111111

echo "ibase=2; 11111111" | bc
# Output: 255

# Hex arithmetic
echo "ibase=16; A + B" | bc
# Output: 21 (in decimal)

Bash Built-in Arithmetic

# Hex to decimal
echo $((0xff))
# Output: 255

# Octal to decimal
echo $((0377))
# Output: 255

# Binary to decimal (Bash 4+)
echo $((2#11111111))
# Output: 255

# Arithmetic in different bases
echo $((0xff + 0x10))
# Output: 271

# Base conversion with parameter expansion
decimal=255
printf -v hex "%x" $decimal
echo $hex
# Output: ff

Python One-liners

# Decimal to hex
python3 -c "print(hex(255))"
# Output: 0xff

# Decimal to binary
python3 -c "print(bin(255))"
# Output: 0b11111111

# Decimal to octal
python3 -c "print(oct(255))"
# Output: 0o377

# Hex to decimal
python3 -c "print(int('ff', 16))"
# Output: 255

# Binary to decimal
python3 -c "print(int('11111111', 2))"
# Output: 255

# Interactive calculations
python3 -c "print(0xff & 0xf0)"
# Output: 240

xxd - Hex Dump Utility

# Create hex dump of file
xxd filename

# Reverse: convert hex dump back to binary
xxd -r hexdump.txt > binary_file

# Plain hex output
xxd -p filename

# Binary output (bits)
xxd -b filename

od - Octal Dump

# Octal dump
od filename

# Hex dump
od -x filename

# With ASCII
od -c filename

# Decimal
od -d filename

Practical Applications

File Permissions (Octal)

Special Permission Bits (The Fourth Octal Digit)

Unix permissions actually use 4 octal digits, not 3. The leading digit controls special permission bits:

IP Addresses and Subnet Masks

MAC Addresses (Hex)

Memory Addresses

Color Codes (Hex)

ASCII and Unicode

Quick Reference: Conversion Table (0-31)

Common Hex Values to Memorize

Mental Math Tricks

Quick Binary to Decimal

Use the "doubling" method - start from left, double and add:

Quick Decimal to Binary

Repeatedly divide by 2, read remainders bottom-up:

Hex Mental Math

Think of hex digits as "almost tens":