Number System – NMAT Quantitative Ability
Goal – Understand how numbers are represented in different bases, how to convert between them, and how to use these concepts in typical NMAT questions.
1. Simple Explanation
A number system (or numeral system) is a way of representing numbers using a set of symbols (digits) and a base (radix).
| Base | Symbol set | Example | Common name |
|---|---|---|---|
| 2 | 0,1 | 1011₂ | Binary |
| 3 | 0,1,2 | 102₂ | Ternary |
| 4 | 0,1,2,3 | 321₄ | Quaternary |
| 5 | 0–4 | 213₅ | Quinary |
| 8 | 0–7 | 156₈ | Octal |
| 10 | 0–9 | 1234₁₀ | Decimal (our everyday system) |
| 16 | 0–9, A–F | 1A3₁₆ | Hexadecimal |
- Base b representation of a number
(N = d_k d_{k-1}\dots d_1 d_0_b)
means
(N = d_k b^k + d_{k-1} b^{k-1} + \dots + d_1 b + d_0).
2. Key Points for NMAT
| Topic | What to Know | Why it matters in NMAT |
|---|---|---|
| Decimal to other bases | Multiply by base and keep remainders (or use division‑by‑base method). | Many questions ask for binary/hex equivalents. |
| Other bases to Decimal | Sum of digit × (base^position). | Quick calculations for powers and patterns. |
| Hex digits | A=10, B=11, …, F=15. | Common in computer‑related questions. |
| Power in different bases | 2ⁿ in binary = 1 followed by n zeros. | Useful for speed‑yields of large powers. |
| Remainder properties | For base‑b, remainder on division by (b−1) equals sum of digits mod (b−1). | Divisibility tests, quick checks. |
| Conversion tricks | 8↔2 (group bits) and 16↔2 (group 4 bits). | Reduces time in exam. |
| Negative numbers in binary | Sign‑bit, two’s complement, or use “−x = (2ⁿ−x)”. | Useful for subtraction problems. |
| Decimal fractions to binary | Multiply fractional part by base, keep integer part. | Rare but may appear in “convert 0.375 to binary”. |
| Patterns in digits | Repeating sequences in base‑b expansions (e.g., 1/3 in decimal is 0.333…). | Recognise when a number has a repeating binary/hex expansion. |
| Common pitfalls | Forgetting to convert hex digits to decimal when adding/subtracting. | Avoid mis‑calculations. |
3. Worked Example
Question
Convert the hexadecimal number 3F9₁₆ to its decimal equivalent, then find its binary representation.
Step‑by‑step
-
Convert to Decimal
Hex digits: 3, F(=15), 9
Positions (from right, starting at 0):[ \begin{aligned} N &= 3 \times 16^2 + 15 \times 16^1 + 9 \times 16^0 \ &= 3 \times 256 + 15 \times 16 + 9 \ &= 768 + 240 + 9 \ &= 1017_{10} \end{aligned} ]
-
Convert Decimal to Binary (using division by 2)
Quotient Remainder 1017 ÷ 2 = 508 1 508 ÷ 2 = 254 0 254 ÷ 2 = 127 0 127 ÷ 2 = 63 1 63 ÷ 2 = 31 1 31 ÷ 2 = 15 1 15 ÷ 2 = 7 1 7 ÷ 2 = 3 1 3 ÷ 2 = 1 1 1 ÷ 2 = 0 1 Reading remainders bottom‑to‑top gives:
(111111001_2) -
Answer
[ 3F9_{16} ;=; 1017_{10} ;=; 111111001_{2} ]
Tip: For quick binary conversion, notice that 3F9₁₆ has 3 hex digits → 12 binary bits. Pad with leading zeros if needed.
4. Quick Revision Summary
| Concept | Quick Tip | Example |
|---|---|---|
| Binary from Decimal | Divide by 2, stack remainders. | 13 → |