Number system

📚 Number System – Quick & Easy Study Notes

For the Nagaland Forest Guard Quantitative Aptitude Test

1️⃣ What is a “Number System”?

A number system is a way of writing numbers.
Think of it as a language that tells us how to read and write numbers.
The most common one is the decimal system (base‑10) that we use every day.

2️⃣ Why Do We Need Other Number Systems?

• Computers use only two lights: on (1) and off (0).
  → That’s the binary system (base‑2).
• Some calculators and digital clocks use octal (base‑8) or hexadecimal (base‑16) because they fit better with the computer’s hardware.

Knowing how to switch between them helps in many quick‑math questions.

3️⃣ The Decimal System (Base‑10)

Place value:

• The rightmost digit is ones (10⁰).
• The next left is tens (10¹).
• Then hundreds (10²), and so on.

Tiny Example

324 =

• 3 × 100 (hundreds)
• • 2 × 10 (tens)
• • 4 × 1 (ones)
    = 300 + 20 + 4 = 324

4️⃣ Binary System (Base‑2)

Place value:
Rightmost is 2⁰ (1), next is 2¹ (2), then 2² (4), 2³ (8), …

Tiny Example

1011₂ =

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

5️⃣ Octal System (Base‑8)

Place value: 8⁰, 8¹, 8², …

Tiny Example

15₈ =

• 1 × 8 (8¹)
• • 5 × 1 (8⁰)
    = 8 + 5 = 13 (decimal)

6️⃣ Hexadecimal System (Base‑16)

Place value: 16⁰, 16¹, 16², …

Tiny Example

2A₁₆ =

• 2 × 16 (16¹)
• • 10 × 1 (16⁰) (because A = 10)
    = 32 + 10 = 42 (decimal)

7️⃣ Converting Between Systems

7.1 Decimal → Binary

Divide by 2, write the remainder, repeat with the quotient.

Write remainders upside‑down → 10100₂ (decimal 20).

7.2 Binary → Decimal

Multiply each digit by its power of 2 and add.

110₁₀ = 1×4 + 1×2 + 0×1 = 6.

7.3 Decimal → Hexadecimal

Divide by 16, write the remainder (use A‑F for 10‑15).

Write remainders upside‑down → 29A₁₆ (decimal 654).

8️⃣ Quick Tips for the Exam

9️⃣ Mini Practice Problems