Number System (Class 6–8)
UPTET Paper II — Mathematics and Science
Overview
The Number System forms the foundational bedrock of all mathematical learning at the upper-primary level. For UPTET Paper II, this topic tests both your conceptual clarity and your ability to apply operations on integers, rational numbers, and real numbers in problem-solving contexts. Questions typically appear as direct calculations, comparison problems, or word problems involving square/cube roots and exponents.
Mastery here is non-negotiable because every other mathematics topic—algebra, mensuration, commercial mathematics—builds upon number-system fluency. Focus on properties of operations, representation on the number line, and laws of exponents—these are perennial exam favourites.
Key Concepts
- Natural Numbers (N): Counting numbers starting from 1. They do not include zero or negative numbers. N = {1, 2, 3, 4, ...}
- Whole Numbers (W): Natural numbers plus zero. W = {0, 1, 2, 3, ...}
- Integers (Z): Whole numbers extended to include negatives. Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}. Every integer can be represented on a number line; numbers to the left are smaller.
- Rational Numbers (Q): Numbers expressible as p/q where p and q are integers and q ≠ 0. Includes all integers, fractions, and terminating or repeating decimals. Between any two rational numbers, infinitely many rational numbers exist (dense property).
- Irrational Numbers: Numbers that cannot be written as p/q—their decimal expansions are non-terminating and non-repeating (e.g., √2, √3, π).
- Real Numbers (R): The union of rational and irrational numbers. Every point on the number line corresponds to a real number.
- Hierarchy: N ⊂ W ⊂ Z ⊂ Q ⊂ R. Every natural number is a whole number, every whole number is an integer, and so on.
- Closure, Commutativity, Associativity, Distributivity: Operations on each number set obey specific properties—essential for simplification and verification.
Formulas / Key Facts
Operations on Integers
| Operation | Rule |
|---|---|
| (+) + (+) | Positive |
| (−) + (−) | Negative (add magnitudes) |
| (+) + (−) | Subtract smaller from larger; sign of larger magnitude |
| (−) × (−) | Positive |
| (+) × (−) | Negative |
- Additive inverse of a: −a (so a + (−a) = 0)
- Multiplicative inverse of a (a ≠ 0): 1/a
Rational Numbers
- Standard form: p/q with q > 0, HCF(p, q) = 1
- Comparison: Cross-multiply or convert to common denominator
- Between two rationals a/b and c/d: (a + c)/(b + d) lies between them (for positive denominators)
Square Roots and Cube Roots
- √(a × b) = √a × √b
- √(a/b) = √a / √b
- ³√(a × b) = ³√a × ³√b
- Perfect squares end in 0, 1, 4, 5, 6, or 9
- Perfect cubes can end in any digit
Laws of Exponents (a, b ≠ 0; m, n are integers)
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- aᵐ × bᵐ = (ab)ᵐ
- a⁰ = 1
- a⁻ⁿ = 1/aⁿ
Standard Form (Scientific Notation)
A number written as k × 10ⁿ where 1 ≤ k < 10. Example: 5,430,000 = 5.43 × 10⁶
Worked Examples
Example 1: Operations on Integers
Problem: Simplify (−15) + 8 − (−6) + (−3).
Solution: Step 1: Rewrite subtraction of negative as addition: (−15) + 8 + 6 + (−3) Step 2: Group positives and negatives: Positives = 8 + 6 = 14; Negatives = 15 + 3 = 18 Step 3: Result = 14 − 18 = −4
Answer: −4
Example 2: Comparing Rational Numbers
Problem: Which is greater: −5/6 or −7/8?
Solution: Step 1: Find LCM of denominators 6 and 8 → LCM = 24 Step 2: Convert: −5/6 = −20/24; −7/8 = −21/24 Step 3: On the number line, −20/24 is to the right of −21/24.
Answer: −5/6 > −7/8
Example 3: Exponents
Problem: Simplify (2³ × 2⁵) ÷ 2⁴ and express the answer as a power of 2.
Solution: Step 1: Apply product rule: 2³ × 2⁵ = 2⁸ Step 2: Apply quotient rule: 2⁸ ÷ 2⁴ = 2⁴
Answer: 2⁴ = 16
Example 4: Square Root by Prime Factorisation
Problem: Find √1764.
Solution: Step 1: Prime factorise 1764 = 2² × 3² × 7² Step 2: √1764 = 2 × 3 × 7 = 42
Answer: 42
Common Mistakes
- Sign errors in integer subtraction: Students treat a − (−b) as a − b instead of a + b. Fix: Remember subtracting a negative is adding a positive—visualise movement on the number line.
- Forgetting q ≠ 0 in rational numbers: Writing expressions like 5/0 as a valid rational number. Fix: Division by zero is undefined; always verify the denominator before simplifying.
- Misapplying exponent laws: Writing aᵐ × bⁿ = (ab)ᵐ⁺ⁿ. Fix: Bases must be the same to add exponents (aᵐ × aⁿ = aᵐ⁺ⁿ), or exponents must be the same to multiply bases (aᵐ × bᵐ = (ab)ᵐ).
- Assuming √(a + b) = √a + √b: This is algebraically incorrect. Fix: Square-root distributes over multiplication and division, not over addition or subtraction.
- Confusing terminating and non-terminating decimals: Believing all fractions give terminating decimals. Fix: A rational p/q (in lowest terms) terminates only if q has no prime factors other than 2 and 5.
Quick Reference
- Number hierarchy: N ⊂ W ⊂ Z ⊂ Q ⊂ R
- Product of two negatives is positive; product of a positive and a negative is negative.
- Additive inverse of a is −a; multiplicative inverse of a is 1/a (a ≠ 0).
- aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1.
- √(a×b) = √a × √b; but √(a+b) ≠ √a + √b.
- Standard form: k × 10ⁿ where 1 ≤ k < 10.