📘 Number System – Study Notes for WB TET (Quantitative Aptitude)
“The number system is the language of mathematics. Mastering it unlocks the door to all quantitative problems.”
1. Simple Explanation
The number system is a collection of numbers that share common properties and follow specific rules. In the TET exam, you’ll encounter:
- Types of numbers (natural, whole, integer, rational, irrational, real, complex).
- Basic properties (commutative, associative, distributive).
- Divisibility and prime/composite classification.
- Decimal representation (terminating vs. repeating).
- Conversions among bases (decimal ↔ binary, octal, hexadecimal).
Understanding these fundamentals helps you solve a wide range of problems quickly and accurately.
2. Key Points
| Concept | What You Need to Know |
|---|---|
| Natural Numbers | Positive integers: 1, 2, 3 … |
| Whole Numbers | Natural numbers + 0 |
| Integers | Whole numbers + their negatives |
| Rational Numbers | Any number that can be expressed as a fraction a/b (where b ≠ 0 and a, b are integers). |
| Irrational Numbers | Numbers that cannot be expressed as a simple fraction (e.g., √2, π). |
| Real Numbers | Union of rational and irrational numbers. |
| Complex Numbers | a + bi where i² = –1. Not typically used in WB TET Q. |
| Prime Numbers | >1 with only two distinct divisors (1 and itself). |
| Composite Numbers | >1 with more than two divisors. |
| Perfect Square | A number n such that √n is an integer. |
| Perfect Cube | A number n such that ∛n is an integer. |
| Divisibility Rules | Quick checks (e.g., 9: sum of digits, 11: alternating sum, 5: last digit 0/5). |
| Terminating Decimal | Decimal that ends after a finite number of digits. Occurs when the denominator in lowest terms has only 2 and/or 5 as prime factors. |
| Repeating Decimal | Decimal where a group of digits repeats infinitely. |
| Base‑10 Place Value | Units, tens, hundreds, etc. |
| Base Conversion | binary → decimal, decimal → binary, etc. |
| LCM & GCD | Useful for simplifying fractions, solving simultaneous equations, etc. |
3. Worked Example
Problem (Typical WB TET):
*The decimal number 0.375 is given.
- Express it as a fraction in its simplest form.
- Convert the fraction back to a binary number.*
Step‑by‑Step Solution
1️⃣ Express 0.375 as a fraction
- 0.375 = 375/1000 (since 0.375 = 375/1000).
- Reduce the fraction:
- GCD(375, 1000) = 125.
- Divide numerator & denominator by 125:
- 375 ÷ 125 = 3
- 1000 ÷ 125 = 8
- Simplest form:
3/8.
Quick Check: 3 ÷ 8 = 0.375 ✔️
2️⃣ Convert 3/8 to binary
- Any fraction
a/bwherebis a power of 2 (here, 8 = 2³) can be represented exactly in binary. - Write the denominator as 2³ → we need 3 binary digits after the point.
- Compute:
- 3 ÷ 8 = 0.011₂
(Because 0.011₂ = 0×2⁻¹ + 1×2⁻² + 1×2⁻³ = 0 + 0.25 + 0.125 = 0.375)
- 3 ÷ 8 = 0.011₂
Answer:
- Fraction form:
3/8 - Binary form:
0.011₂
4. Quick Revision Summary
| Topic | Quick Fact |
|---|---|
| Natural Numbers | 1, 2, 3, … |
| Whole Numbers | 0, 1, 2, … |
| Integers | …, –2, –1, 0, 1, 2, … |
| Rational | a/b where `a, b ∈ |