Number System (Class 6–8)

UPTET Paper II — Mathematics and Science

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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. Expect 3–5 questions directly from this area, with additional indirect applications scattered across other sections. Focus on properties of operations, representation on the number line, and laws of exponents—these are perennial exam favourites.

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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.

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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) 1. aᵐ × aⁿ = aᵐ⁺ⁿ 2. aᵐ ÷ aⁿ = aᵐ⁻ⁿ 3. (aᵐ)ⁿ = aᵐⁿ 4. aᵐ × bᵐ = (ab)ᵐ 5. a⁰ = 1 6. 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

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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

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### 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

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### 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

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### 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

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Common Mistakes

1. **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.

2. **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.

3. **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).

4. **Assuming √(a + b) = √a + √b:** This is algebraically incorrect. *Fix:* Square-root distributes over multiplication and division, not over addition or subtraction.

5. **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.

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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.**