🔢 Number Systems¶

Understanding how computers count differently than humans

What You'll Learn¶

Why computers use binary (base-2)
How to convert between number systems
Decimal, binary, octal, and hexadecimal
Python's built-in conversion functions

The Story of Counting¶

Humans have ten fingers, so we naturally count in base-10 (decimal). Computers have electrical circuits that are either ON or OFF, so they count in base-2 (binary).

Think of it like different languages: - Decimal: English (what humans speak) - Binary: Computer's native language - Hexadecimal: A compact way to write binary (like shorthand)

Decimal (Base-10) 🧮¶

The number system you're most familiar with. Uses digits 0-9.

Decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...

How it works:

253 = 2×100 + 5×10 + 3×1
    = 2×10² + 5×10¹ + 3×10⁰

Each position represents a power of 10.

Binary (Base-2) 💻¶

Computers only understand 0s and 1s (OFF and ON).

Binary: 0, 1, 10, 11, 100, 101, 110, 111, 1000...

How it works:

1011 (binary) = 1×8 + 0×4 + 1×2 + 1×1
              = 1×2³ + 0×2² + 1×2¹ + 1×2⁰
              = 11 (decimal)

Each position represents a power of 2.

Why Binary?¶

Electrical circuits have two states: ON (1) or OFF (0)
Magnetic storage: North/South polarity
CDs/DVDs: Pits and lands
Simple and reliable!

Octal (Base-8) 🐙¶

Uses digits 0-7. Each octal digit represents 3 binary digits.

Octal: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12...

Conversion:

Decimal 10 = Octal 12 = Binary 1010

Binary:  101 110
Octal:     5   6  → 56 (octal)

Less common today but still used in: - Unix file permissions (e.g., chmod 755) - Legacy systems

Hexadecimal (Base-16) 🔮¶

Uses digits 0-9 and letters A-F. Each hex digit represents 4 binary digits.

Hex: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10...

A = 10    B = 11    C = 12    D = 13    E = 14    F = 15

Conversion:

Decimal 255 = Hex FF = Binary 11111111

Binary:  1101 1011
Hex:        D    B  → DB (hex)

Why Hexadecimal?¶

More compact than binary
Easy conversion (4 bits = 1 hex digit)
Used everywhere in computing:
Color codes: #FF5733
Memory addresses: 0x7FFF5E
MAC addresses
Hash values

Python Conversion Functions¶

Python makes conversions easy with built-in functions:

# To binary
bin(10)      # Returns '0b1010'

# To octal
oct(10)      # Returns '0o12'

# To hexadecimal
hex(10)      # Returns '0xa'

# To decimal (from any base)
int('1010', 2)   # Returns 10
int('12', 8)     # Returns 10
int('a', 16)     # Returns 10

The prefixes: - 0b = binary - 0o = octal - 0x = hexadecimal

Conversion Methods¶

Binary to Decimal¶

Multiply each bit by its place value and sum:

Binary: 1 0 1 0 1
Place: 16 8 4 2 1

Value: 16 + 0 + 4 + 0 + 1 = 21

Decimal to Binary¶

Repeatedly divide by 2, collect remainders (read bottom-up):

13 ÷ 2 = 6 remainder 1
 6 ÷ 2 = 3 remainder 0
 3 ÷ 2 = 1 remainder 1
 1 ÷ 2 = 0 remainder 1

Result: 1101 (read remainders bottom-up)

Quick Reference Table¶

Decimal	Binary	Octal	Hexadecimal
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10

Common Mistakes ⚠️¶

1. Forgetting the prefix¶

# Wrong
int('1010')      # This is decimal 1010, not binary!

# Right
int('1010', 2)   # Specify base=2 for binary

2. Reading binary remainders in wrong order¶

Decimal 5 to binary:
5 ÷ 2 = 2 R 1
2 ÷ 2 = 1 R 0
1 ÷ 2 = 0 R 1

Result: 101 (NOT 011!)
Read from bottom to top!

3. Confusing letter values in hex¶

A = 10, not 11
B = 11, not 12
# Remember: A comes after 9, so A = 10

4. Off-by-one in binary place values¶

Binary: 1 0 0 0
Place:  8 4 2 1  (powers of 2: 2³, 2², 2¹, 2⁰)
        ↑
        Leftmost is 8 (2³), not 4!

Try It Out! 🚀¶

Run examples.py to see number systems in action:

python examples.py

Then try exercises.py to practice:

python exercises.py

Key Takeaways¶

Binary (base-2) is the language of computers (0s and 1s)
Hexadecimal (base-16) is a compact way to write binary
Octal (base-8) is still used for file permissions
Python provides bin(), oct(), hex(), and int() for conversions
Understanding number systems helps you debug and work with low-level code

Next: Bitwise Operations →