Number system

Number System – AWS Certification – Quantitative Aptitude

(Prepared as a quick‑reference study guide for the AWS certification exam)

1. Simple Explanation

A number system is a way of representing numbers using a set of symbols (digits) and a base (radix).

• Base 2 (Binary) – digits 0,1
• Base 8 (Octal) – digits 0–7
• Base 10 (Decimal) – digits 0–9 (our everyday system)
• Base 16 (Hexadecimal) – digits 0–9, A–F (A=10, B=11, …, F=15)

Each digit’s value is multiplied by the base raised to the power of its position (starting from 0 on the right).

Decimal number 345
= 3 × 10² + 4 × 10¹ + 5 × 10⁰
= 300 + 40 + 5 = 345

2. Key Points

Conversion Rules

1. Decimal → Binary/Octal/Hex

   • Repeated division by the target base.
   • Remainders (in reverse order) give the digits.

2. Binary ↔ Octal

   • Group binary digits into 3 from right to left.
   • Convert each group to a single octal digit.

3. Binary ↔ Hex

   • Group binary digits into 4 from right to left.
   • Convert each group to a single hex digit.

4. Octal/Hex ↔ Decimal

   • Use positional value formula.

Practical Tips

• Pad the leftmost group with zeros if it has fewer bits.
• Remember:
  • 2³ = 8, 2⁴ = 16
  • 8 = 2³, 16 = 2⁴
• Binary addition mimics decimal addition but carries at 2 instead of 10.

3. Worked Example

Problem
Convert the decimal number 12345 to binary, octal, and hexadecimal.

Step 1: Decimal → Binary

Repeated division by 2:

Write remainders bottom‑to‑top:
11000000111001₂

Step 2: Binary → Octal

Group into 3 bits from right:
11 000 000 111 001 → pad left group with 0: 011 000 000 111 001

Convert each group:

• 011 → 3
• 000 → 0
• 000 → 0
• 111 → 7
• 001 → 1

Octal number: 30071₈

Step 3: Binary → Hexadecimal

Group into 4 bits from right:
1100 0000 1110 01 → pad left with zeros: 0011 0000 0011 1001

Convert each group:

• 0011 → 3
• 0000 → 0
• 0011 → 3
• 1001 → 9

Hexadecimal number: 3039₁₆

Result Summary

4. Quick Revision Summary

• Bases: Binary (2), Octal (8), Decimal (10), Hex (16).
• Digits: Binary {0,1}; Octal {0–7}; Decimal {0–9}; Hex {0–9, A–F}.
• Conversion
  • Decimal ↔ Binary: repeated division by 2.
  • Binary ↔ Octal: group 3 bits.
  • Binary ↔ Hex: group 4 bits.
  • Octal/Hex ↔ Decimal: positional value formula.
• Common Properties
  • 8 = 2³ → 3 binary digits