Number System – NDA Quantitative Aptitude Study Notes
Audience: NDA aspirants
Time‑commitment: ~20 min of focused study
Goal: Master the fundamentals of number systems, conversions, and key properties that appear in NDA quantitative sections.
1. Simple Explanation
A number system is a way of representing numbers using a set of symbols (digits) and rules for combining them.
The most common system we use daily is the decimal (base‑10) system, but the world of numbers contains many others – binary (base‑2), octal (base‑8), hexadecimal (base‑16), and more.
Why does it matter for NDA?
- Many aptitude questions ask you to convert between systems, find divisibility, or work with place values.
- Understanding the hierarchy of systems (binary → octal → decimal → hexadecimal) helps you solve problems quickly and check your answers by back‑conversion.
2. Key Points
| Concept | What It Means | Quick Test Question |
|---|---|---|
| Base | Number of unique digits, including zero, used to represent numbers. | “How many digits does a binary system use?” |
| Place Value | Value contributed by each digit depending on its position. In base‑b, the rightmost digit has weight (b^0), next (b^1), etc. | “What is the value of the digit ‘4’ in 4 × (8^2)?” |
| Conversion | Translating a number from one base to another by repeated division (to the new base) or multiplication (to the old base). | Convert 1101₂ to decimal. |
| Binary | Base‑2, digits 0 & 1. | “What is 1010₂ in decimal?” |
| Octal | Base‑8, digits 0–7. | “What is 345₈ in decimal?” |
| Hexadecimal | Base‑16, digits 0–9 and letters A–F (representing 10–15). | “What is 1A3₁₆ in decimal?” |
| Divisibility Rules (decimal) | e.g., 2 (last digit even), 3 (sum of digits divisible by 3), 9 (sum of digits divisible by 9). | “Is 987,654 divisible by 3?” |
| Parity | Whether a number is even or odd. | “What is the parity of 2^10 + 1?” |
| Prime Numbers | Divisible only by 1 and itself. | “Is 121 a prime?” |
| Powers & Roots | Fast evaluation of expressions like (2^8), (5^2), (\sqrt{64}). | “Compute (2^4).” |
| Rational vs. Irrational | Rational = exact fraction; Irrational = non‑terminating, non‑repeating decimal (π, √2). | “Is (\sqrt{3}) rational?” |
| Floating‑Point | Approximate representation of real numbers in computers (binary). | “Why does 0.1 + 0.2 ≠ 0.3 in binary?” |
| Significance of Leading Zeros | They do not affect the value but can be used in programming to align digits. | “What is 00123₂?” |
| Reversible Conversions | Converting back and forth should give the original number; useful for checking work. | “Convert 45₁₀ to binary and back.” |
3. Worked Example
Problem (Typical NDA Aptitude):
A number in base‑8 (octal) is 345₈. Convert it to decimal and then to hexadecimal.
Step‑by‑Step
-
Octal → Decimal
[ 345_8 = 3 \times 8^2 + 4 \times 8^1 + 5 \times 8^0 ]
[ = 3 \times 64 + 4 \times 8 + 5 \times 1 = 192 + 32 + 5 = 229_{10} ]
-
Decimal → Hexadecimal
Divide 229 by 16 repeatedly, keeping remainders.
Division Quotient Remainder 229 ÷ 16 14 5 14 ÷ 16 0 14 (E) Read remainders from bottom to top → E5₁₆.
-
Verification (Optional)
- Convert E5₁₆ back to decimal:
(E = 14), so (14 \times 16 + 5 = 224 + 5 = 229_{10}). - Convert 229₁₀ back to octal:
(229 ÷ 8 = 28) remainder (5)
(28 ÷ 8 = 3) remainder (4)
(3 ÷ 8 = 0) remainder (3) → 345₈.
✔️ Consistent!
- Convert E5₁₆ back to decimal:
Quick Tips
- Octal to Decimal: Multiply each digit by (8^n) (rightmost digit (n=0)).
- Decimal to Hex: Repeated division by 16; remainders give hex digits.
- Check Work: Reverse conversion.
4. Quick Revision Summary
| Item | What to Remember | Mnemonic / Shortcut |
|---|---|---|
| Base | 2, 8, 10, 16 are the most common. | “Binary, Octal, ** |