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.
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.
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.
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 | |
| Distributive property | a × (b + c) = a×b + a×c |
Must-remember facts:
- Zero is neither positive nor negative; zero is a whole number and an integer
- 1 is neither prime nor composite
- Between any two rational numbers, infinitely many rationals exist (dense property)
- √2 ≈ 1.414, √3 ≈ 1.732, π ≈ 3.14 (for estimation questions)
Worked Examples
Example 1: Integer operation
Evaluate: (−15) + 8 − (−6) × 2
Step 1: Follow BODMAS — multiplication first (−6) × 2 = −12
Step 2: Rewrite expression (−15) + 8 − (−12) = (−15) + 8 + 12
Step 3: Add integers with same sign first 8 + 12 = 20
Step 4: Add integers with different signs (−15) + 20 = 5
Answer: 5
Example 2: Rational number between two fractions
Find a rational number between 3/5 and 4/5.
Method 1 (Average): (3/5 + 4/5) ÷ 2 = (7/5) ÷ 2 = 7/10
Method 2 (Same denominator): Convert to equivalent fractions with larger denominator 3/5 = 6/10, 4/5 = 8/10 Numbers between: 7/10
Answer: 7/10 (or 0.7)
Example 3: Converting repeating decimal to fraction
Express 0.2̄7̄ (0.272727...) as a fraction.
Let x = 0.272727...
Since two digits repeat, multiply by 100: 100x = 27.272727...
Subtract: 100x − x = 27.272727... − 0.272727... 99x = 27 x = 27/99 = 3/11
Answer: 3/11
Common Mistakes
- Wrong sign in subtraction: Students compute (−5) − (−3) as −8 instead of −2. Fix: Subtracting a negative means adding its positive. (−5) − (−3) = (−5) + 3 = −2.
- Confusing "between" with "from...to": When asked for integers between −2 and 3, students include −2 and 3. Fix: "Between" excludes endpoints. Correct answer: −1, 0, 1, 2 (four integers).
- Assuming all square roots are irrational: Students mark √4 or √(9/16) as irrational. Fix: Check if the number under the root is a perfect square. √4 = 2 (rational), √(9/16) = 3/4 (rational).
- Division by zero error: Students write 5/0 = 0 or 5/0 = ∞. Fix: Division by zero is undefined, not zero or infinity.
- Applying closure incorrectly: Claiming integers are closed under division because 6 ÷ 2 = 3. Fix: One example doesn't prove closure. Counter-example: 5 ÷ 2 = 2.5 (not an integer). Integers are NOT closed under division.
Quick Reference
- Integer sign rules: Same signs → positive product; different signs → negative product
- Rational number test: If decimal terminates or repeats, it's rational
- Additive identity = 0; Multiplicative identity = 1
- Between two rationals a and b: (a+b)/2 is always a rational between them
- Absolute value strips the sign: |−7| = 7, |7| = 7
- Zero has no multiplicative inverse (no number multiplied by 0 gives 1)