🔢 Number System – Quick & Easy Study Notes
(For UPSC Geo‑Scientist – Quantitative Aptitude)
What you’ll learn – All the basic types of numbers, how to read different “ciphers” (decimal, binary, octal, hexadecimal), how to do simple operations, and a few tricks that help solve practice questions fast.
Why it matters – These concepts show up in almost every numeric problem on the paper – from simple maths to data interpretation and even some computer‑science type questions.
Goal – Keep it short, use everyday language, and practice a tiny example for each idea.
1️⃣ Types of Numbers
| Type | Everyday analogy | Simple sentence |
|---|---|---|
| Natural Numbers | Numbers you count with (1, 2, 3…) | “I have 3 apples.” |
| Whole Numbers | Natural numbers + 0 | “I have 0 oranges.” |
| Integers | Whole + negatives (…‑2, ‑1, 0, 1, 2…) | “My temperature is –5 °C.” |
| Rational Numbers | Numbers that can be written as a fraction a/b | “I drank ½ cup of water.” |
| Irrational Numbers | Numbers that can’t be written as a simple fraction | “π ≈ 3.14159…” |
| Real Numbers | All rational + irrational | “The length of the board is √2 meters.” |
| Complex Numbers | a + bi (includes the imaginary unit i) | “A wave can be represented as 3 + 4i.” |
Tip – When you see a number with a decimal or a fraction, it’s rational. If it keeps going forever without repeating, it’s irrational.
2️⃣ Number Systems (Ciphers)
| System | Base | Digits you can use | Example |
|---|---|---|---|
| Decimal | 10 | 0–9 | 7 × 6 = 42 |
| Binary | 2 | 0, 1 | 10₂ × 11₂ = 110₂ |
| Octal | 8 | 0–7 | 12₈ + 5₈ = 17₈ |
| Hexadecimal | 16 | 0–9, A–F | 1A₁₆ + 5₁₆ = 1F₁₆ |
Why bother?
- Computers use binary.
- Octal and Hex are handy shortcuts for long binary strings.
Quick Conversion
| Convert | Binary | Hex |
|---|---|---|
| 13 (decimal) | 1101₂ | D₁₆ |
| 42 (decimal) | 101010₂ | 2A₁₆ |
Tiny example – Convert 7 to binary:
7 ÷ 2 = 3 R 1
3 ÷ 2 = 1 R 1
1 ÷ 2 = 0 R 1 → 111₂.
3️⃣ Basic Operations
| Operation | What it does | Everyday rule |
|---|---|---|
| Addition | Puts numbers together | “Add 2 apples + 3 apples = 5 apples.” |
| Subtraction | Takes one away | “Take 3 oranges from 5 oranges = 2 oranges.” |
| Multiplication | Repeated addition | “3 × 4 = 3 added 4 times.” |
| Division | Splits evenly | “12 ÷ 4 = 3 pieces of 4.” |
Order – Do multiplication/division before addition/subtraction (PEMDAS).
Example – 2 + 3 × 4 = 2 + 12 = 14.
4️⃣ Properties of Operations
| Property | What it means | Quick test |
|---|---|---|
| Commutative | Order doesn’t matter | 3 + 5 = 5 + 3 |
| Associative | Grouping doesn’t matter | (2 + 3) + 4 = 2 + (3 + 4) |
| Distributive | Multiply over addition | 4 × (2 + 3) = 4×2 + 4×3 |
Why useful – Lets you rearrange terms to get a simpler calculation.
5️⃣ Fractions & Decimals
| Format | How it looks | Example |
|---|---|---|
| Fraction | a / b | ⅓ |
| Decimal | digits after dot | 0.75 |
Convert Fraction → Decimal
- Divide numerator by denominator.
- Stop when the remainder is 0 or repeats.
Tiny example – ½ = 0.5 (divide 1 by 2).
Convert Decimal → Fraction
- Count digits after decimal.
- Put the number as numerator, 10ⁿ as denominator, reduce.
Tiny example – 0.6 = 6/10 = 3/5.
6️⃣ Powers & Roots
| Concept | Simple word | Example |
|---|---|---|
| Power (Exponent) | Repeated multiplication | 2³ = 2 × 2 × 2 = 8 |
| Root | “How many times do you multiply to get the number?” | √9 = 3 |
Quick rule – The square root of a perfect square is an integer.
Tiny example – 4² = 16 → √16 = 4.