Number Systems - Binary, Octal, Decimal & Hexadecimal
How computers represent numbers - and how to convert between the bases that matter in CS.
1 The Four Number Systems
Binary (Base 2)
Digits: 0, 1 - used by hardware
Octal (Base 8)
Digits: 0–7 - Unix permissions
Decimal (Base 10)
Digits: 0–9 - human default
Hexadecimal (Base 16)
Digits: 0–9, A–F - colors, memory
Why hex? Each hex
digit represents exactly 4 binary bits. So 1 byte (8 bits) = 2 hex digits. It's a compact
way to write binary:
11111111₂ = FF₁₆ = 255₁₀.2 Positional Value - How Bases Work
In any base, each digit's value = digit × base^position (counting from 0 on the right).
Value = Σ (dᵢ × baseⁱ)
Position 0 = rightmost digit (ones)Position 1 =
next digit (base¹)Position 2 = next (base²), and so on
Worked Example 1
Binary to Decimal Conversion
Problem: Convert 10110101₂ to decimal.
Step 1: Write out place values (powers of 2)
Bit: 1 0 1 1 0 1 0 1 Position: 7 6 5 4 3 2 1 0 Value: 128 64 32 16 8 4 2 1
Step 2: Add positions where bit = 1
128 + 0 + 32 + 16 + 0 + 4 + 0 + 1
10110101₂ = 181₁₀
Answer: 181 in decimal. Tip: memorize the powers of 2 up
to 2⁸ = 256 - you'll use them constantly in CS.
Worked Example 2
Decimal to Binary Conversion
Problem: Convert 200₁₀ to binary.
Step 1: Repeated division by 2 (track remainders)
200 ÷ 2 = 100 remainder 0 100 ÷ 2 = 50 remainder 0 50 ÷ 2 = 25 remainder 0 25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1
Step 2: Read remainders bottom → top
200₁₀ = 11001000₂
Answer: 11001000₂. Verify: 128 + 64 + 8 = 200 ✓.
Alternative method: subtract largest powers of 2 → 200 − 128 = 72, 72 − 64 = 8, 8 − 8 = 0.
Worked Example 3
Hexadecimal Conversions
Problem: Convert (a) 0xB7 to decimal and binary, (b) the
color code #FF6B35 - what are its RGB values?
(a) 0xB7 to decimal
B = 11, 7 = 7
11 × 16¹ + 7 × 16⁰ = 176 + 7
0xB7 = 183₁₀
To binary: split each hex digit into 4 bits:
B = 1011, 7 = 0111
0xB7 = 10110111₂
(b) #FF6B35 RGB values
R = FF = 255, G = 6B = 107, B = 35 = 53
RGB(255, 107, 53) - a vivid orange
Answer: (a) 183₁₀ = 10110111₂. (b) RGB(255, 107, 53).
Every CSS color is 3 hex byte pairs - knowing hex makes web development much faster.
3 Two's Complement - Signed Integers
Computers represent negative numbers using two's complement. The leftmost bit is the sign bit (0 = positive, 1 = negative).
To negate: Flip all bits, then add 1
8-bit range: −128 to +12716-bit range: −32,768
to +32,767n-bit range: −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1
// Example: −42 in 8-bit two's complement 42₁₀ = 00101010₂ Flip: 11010101 Add 1: 11010110 // −42₁₀ = 11010110₂