Number System – AFCAT Quantitative Aptitude
Why it matters – AFCAT problems often ask you to convert numbers between different bases, find the value of a digit in a given base, or check if a number is divisible by a base‑specific rule.
1. Simple Explanation
| Base | Prefix | Typical use |
|---|---|---|
| 2 | binary | Digital electronics, computer memory |
| 3‑6 | – | Rare in exams |
| 7 | – | Can appear in puzzles |
| 8 | octal | Older computer systems |
| 9 | – | Rare |
| 10 | decimal | Everyday numbers |
| 12 | – | “Duodecimal” – sometimes used for time (12‑hr clock) |
| 16 | hexadecimal | Modern computing, memory addresses |
How a number is written
- Digits allowed in base b are
0, 1, …, b‑1. - For bases >10 we use letters:
10→A,11→B,12→C, …15→F(hex).
Value of a digit
If a number in base b is written as
…d₂ d₁ d₀ (rightmost digit d₀ is the 1’s place), its decimal value is
[ N = d_0 \times b^0 + d_1 \times b^1 + d_2 \times b^2 + \dots ]
2. Key Points to Remember
| Topic | Quick Tips |
|---|---|
| Decimal to Other Base | Repeated division by the target base. Keep remainders (in reverse order). |
| Other Base to Decimal | Multiply each digit by base power, sum up. |
| Binary ↔ Octal/Hex | Group bits in 3 for octal, 4 for hex. No division needed. |
| Hexadecimal to Binary | 0→0000, 1→0001, … F→1111. |
| Octal to Binary | 0→000, 1→001, … 7→111. |
| Divisibility in Base b | Use rules similar to decimal but with base‑specific digit sums. |
| Place‑Value | 1, b, b², b³, … (right to left). |
| Common Mistakes | Forget that digits in base >10 use letters; mis‑order remainders. |
| Power of 2/5/8/16 | Powers of 2 are easy in binary; powers of 8/16 easier in octal/hex. |
| Length of Repeating Fraction | In base b, a fraction with denominator d repeats after at most the order of b modulo d. |
| Use of Logarithms | For quick magnitude checks: log_b N ≈ log_10 N / log_10 b. |
3. Worked Example
Problem
Convert the decimal number
987to:
- Binary
- Octal
- Hexadecimal
Step‑by‑Step
3.1 Decimal → Binary
| Division | Quotient | Remainder |
|---|---|---|
| 987 ÷ 2 | 493 | 1 |
| 493 ÷ 2 | 246 | 1 |
| 246 ÷ 2 | 123 | 0 |
| 123 ÷ 2 | 61 | 1 |
| 61 ÷ 2 | 30 | 1 |
| 30 ÷ 2 | 15 | 0 |
| 15 ÷ 2 | 7 | 1 |
| 7 ÷ 2 | 3 | 1 |
| 3 ÷ 2 | 1 | 1 |
| 1 ÷ 2 | 0 | 1 |
Collect remainders from bottom to top → 1111011011₂
3.2 Decimal → Octal
| Division | Quotient | Remainder |
|---|---|---|
| 987 ÷ 8 | 123 | 3 |
| 123 ÷ 8 | 15 | 3 |
| 15 ÷ 8 | 1 | 7 |
| 1 ÷ 8 | 0 | 1 |
Collect → 1733₈
3.3 Decimal → Hexadecimal
| Division | Quotient | Remainder |
|---|---|---|
| 987 ÷ 16 | 61 | 11 → B |
| 61 ÷ 16 | 3 | 13 → D |
| 3 ÷ 16 | 0 | 3 |
Collect → 3DB₁₆