Number Systems Revision Notes: Binary, Octal, Hexadecimal Conversions and Complements

Number systems questions are quick marks in any Computer Science paper, if you know the methods well. These notes cover every conversion, binary arithmetic and complements with solved examples.

The four number systems#

The value of a digit = digit × base^position, positions counted from 0 at the right.

Any base → Decimal#

Multiply each digit by its place value and add.

Example: 1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11₁₀

Example: 2F₁₆ = 2×16 + 15×1 = 47₁₀

Fractions: places after the point use negative powers. 0.101₂ = 1×2⁻¹ + 0×2⁻² + 1×2⁻³ = 0.5 + 0.125 = 0.625₁₀

Decimal → Any base#

Integer part: divide repeatedly by the base and read the remainders bottom to top.

Example: 25₁₀ to binary

Reading upward: 11001₂

Fraction part: multiply repeatedly by the base and read the integer parts top to bottom.

Example: 0.375₁₀ to binary: 0.375×2 = 0.75 → 0; 0.75×2 = 1.5 → 1; 0.5×2 = 1.0 → 1. Answer: 0.011₂

Binary ↔ Octal ↔ Hexadecimal (the grouping shortcut)#

Because 8 = 2³ and 16 = 2⁴:

• Binary → Octal: group bits in 3s from the point.
• Binary → Hex: group bits in 4s from the point.

Example: 1101011₂

• Octal: 1 | 101 | 011 → 1 5 3 → 153₈
• Hex: 110 | 1011 → 6 B → 6B₁₆

Octal → Hex: go through binary. 153₈ → 001 101 011 → 1101011 → 6B₁₆

Binary arithmetic#

Addition rules: 0+0=0, 0+1=1, 1+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1).

   1 0 1 1   (11)
 + 0 1 1 1   ( 7)
 ---------
 1 0 0 1 0   (18)

Subtraction is usually done with 2's complement (below).

1's and 2's complement#

• 1's complement: flip every bit. 1's complement of 01011 is 10100.
• 2's complement: 1's complement + 1. 2's complement of 01011 is 10100 + 1 = 10101.

Shortcut for 2's complement: from the right, copy bits up to and including the first 1, then flip the rest. 01011000 → 10101000.

Subtraction using 2's complement#

To compute A − B: add A to the 2's complement of B. If there is a carry out, drop it; the result is positive.

Example: 9 − 5 with 4 bits: 1001 + (2's complement of 0101 = 1011) = 1 0100 → drop carry → 0100 = 4 ✔

Range of signed numbers#

With n bits in 2's complement, the range is $-2^{n-1}$ to $2^{n-1} - 1$. For 8 bits: −128 to +127.

Codes you should know#

Binary → Gray: the first bit stays the same; each next Gray bit = XOR of the current and previous binary bits. 1011 → 1, 1⊕0=1, 0⊕1=1, 1⊕1=0 → 1110.

Quick practice#

1. Convert 156₁₀ to hexadecimal.
2. Convert 3A₁₆ to binary and octal.
3. Find the 2's complement of 10110100.
4. What is the range of a 6-bit signed number in 2's complement?

Answers: 1) 9C₁₆ (156 = 9×16 + 12). 2) 0011 1010₂ = 72₈. 3) 01001100. 4) −32 to +31.