Number System (Class 6–8)

Overview

The number system forms the backbone of all mathematical operations tested in WB TET Paper II. This topic covers the progression from natural numbers through integers, rational numbers, and an introduction to real numbers — exactly mirroring the Class 6–8 NCERT/WBBSE curriculum that upper-primary teachers must master.

For WB TET, expect 3–5 direct questions on number classification, properties of operations, representation on number lines, and word problems involving integers and rationals. A strong grasp here also supports algebra, mensuration, and data handling topics. The key is understanding the "why" behind rules (like why negative times negative is positive) so you can teach conceptually, not just procedurally.

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Key Concepts

• **Number hierarchy**: Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers ⊂ Real numbers. Each set expands to solve previously impossible operations (subtraction needs integers, division needs rationals).

• **Integers (Z)**: Include all positive and negative whole numbers and zero. Every integer can be written as a fraction with denominator 1.

• **Rational numbers (Q)**: Numbers expressible as p/q where p and q are integers and q ≠ 0. Includes terminating and repeating decimals.

• **Irrational numbers**: Cannot be written as p/q. Examples: √2, √3, π. Non-terminating, non-repeating decimals.

• **Real numbers (R)**: Union of rational and irrational numbers. Every point on the number line corresponds to a real number.

• **Closure property**: A set is closed under an operation if the result always stays within that set. Integers are closed under addition, subtraction, multiplication — but not division.

• **Additive inverse**: For any integer a, the additive inverse is −a such that a + (−a) = 0.

• **Multiplicative inverse (reciprocal)**: For any non-zero rational p/q, the multiplicative inverse is q/p such that (p/q) × (q/p) = 1.

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Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Sum of two integers with same sign | Add absolute values, keep the common sign | | Sum of two integers with different signs | Subtract smaller absolute value from larger, take sign of larger | | Product/Quotient sign rule | (+)(+) = +, (−)(−) = +, (+)(−) = −, (−)(+) = − | | Rational number between a and b | (a + b)/2 always lies between a and b | | Decimal to fraction (terminating) | Count decimal places; denominator = 10, 100, 1000... | | Repeating decimal to fraction | For 0.x̄ (single digit repeating): x/9; for 0.x̄ȳ: xy/99 | | Absolute value | |a| = a if a ≥ 0; |a| = −a if a < 0 | | Distributive property | a × (b + c) = a×b + a×c |