Number System
Integers, Rational Numbers, Exponents and Powers
Overview
The Number System forms the bedrock of upper-primary mathematics and is a consistently tested area in UTET Paper II. This topic builds upon whole numbers (Classes I-V) and extends into negative numbers, fractions expressed as ratios, and the compact notation of exponents. Mastery here directly supports success in algebra, mensuration, and data handling.
For UTET, expect questions that test conceptual clarity—properties of integers, placement on the number line, comparison of rational numbers, and simplification using laws of exponents. Students must be comfortable with both computational accuracy and the reasoning behind rules (e.g., why a negative times a negative is positive, or why any non-zero number raised to zero equals 1).
Key Concepts
- Integers (Z) include all positive whole numbers, zero, and their negatives: ...−3, −2, −1, 0, 1, 2, 3... They extend the natural/whole number system to handle situations like temperature below zero or debt.
- Rational Numbers (Q) are numbers expressible as p/q where p and q are integers and q ≠ 0. Every integer is rational (e.g., 5 = 5/1). Between any two rational numbers, infinitely many rationals exist (density property).
- Additive Identity is 0 (a + 0 = a); Multiplicative Identity is 1 (a × 1 = a). These hold for integers and rationals alike.
- Additive Inverse of a is −a; Multiplicative Inverse (reciprocal) of p/q is q/p (provided p ≠ 0).
- Closure Property: Integers are closed under addition, subtraction, and multiplication (result is always an integer) but NOT under division. Rationals are closed under all four operations (excluding division by zero).
- Exponent notation: aⁿ means a multiplied by itself n times. Here a is the base and n is the exponent (or power).
- Standard Form (Scientific Notation): A number written as k × 10ⁿ where 1 ≤ k < 10. Used for very large or very small quantities.
Formulas / Key Facts
| Concept | Formula / Rule | Quick Note |
|---|---|---|
| Product of integers with same sign | (+)(+) = + ; (−)(−) = + | Two negatives make positive |
| Product of integers with different signs | (+)(−) = − ; (−)(+) = − | Mixed signs give negative |
| Division rule (signs) | Same as multiplication | Sign rules identical |
| Equivalent rational numbers | p/q = (p×k)/(q×k) for any k ≠ 0 | Multiply/divide top & bottom by same number |
| Comparison of rationals | Convert to common denominator or cross-multiply | a/b ? c/d → compare ad and bc |
| Product of powers (same base) | aᵐ × aⁿ = aᵐ⁺ⁿ | Add exponents |
| Quotient of powers (same base) | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Subtract exponents |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | Multiply exponents |
| Power of a product | (ab)ⁿ = aⁿ × bⁿ | Distribute exponent |
| Power of a quotient | (a/b)ⁿ = aⁿ / bⁿ | Distribute exponent |
| Zero exponent | a⁰ = 1 (a ≠ 0) | Any non-zero base to power 0 is 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | Flip to reciprocal |
Worked Examples
Example 1: Integer Operations
Problem: Evaluate (−8) × (−5) + (−12) ÷ 4
Solution:
- (−8) × (−5) = +40 (same signs → positive)
- (−12) ÷ 4 = −3 (different signs → negative)
- 40 + (−3) = 40 − 3 = 37
Example 2: Comparing Rational Numbers
Problem: Which is greater: −3/5 or −7/10?
Solution (Cross-multiply method):
- Compare −3/5 and −7/10
- Cross-multiply: (−3) × 10 = −30 and (−7) × 5 = −35
- Since −30 > −35, we have −3/5 > −7/10
Alternate: Convert to common denominator 10 → −6/10 vs −7/10. Clearly −6/10 > −7/10.
Example 3: Simplifying Exponents
Problem: Simplify: (2³ × 2⁵) ÷ 2⁴
Solution:
- Numerator: 2³ × 2⁵ = 2³⁺⁵ = 2⁸
- Division: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16
Example 4: Negative Exponent
Problem: Express 5⁻³ as a fraction.
Solution: 5⁻³ = 1/5³ = 1/125 = 1/125
Example 5: Standard Form
Problem: Write 0.00047 in standard form.
Solution: Move decimal 4 places right → 4.7 So, 0.00047 = 4.7 × 10⁻⁴
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Believing −5 − (−3) = −8 (adding the magnitudes) | Subtracting a negative means adding: −5 − (−3) = −5 + 3 = −2 |
| Thinking 0 is not a rational number | 0 = 0/1, hence it is rational. It is simply neither positive nor negative. |
| Applying aᵐ × bⁿ = (ab)ᵐ⁺ⁿ | This law works only for the same base. Different bases cannot have exponents added. |
| Writing 2³ × 2⁴ = 2¹² (multiplying exponents) | When multiplying same bases, add exponents: 2³ × 2⁴ = 2⁷. Exponents multiply only in power-of-a-power: (2³)⁴ = 2¹². |
| Claiming a⁰ = 0 | a⁰ = 1 for any a ≠ 0. Remember: 5⁰ = 1, not 0. |
| Forgetting sign when converting negative exponent | a⁻ⁿ = 1/aⁿ, not −aⁿ. The negative exponent indicates reciprocal, not a negative value. |
Quick Reference
- Integers closed under +, −, × but NOT ÷.
- Rationals closed under +, −, ×, ÷ (except ÷ by 0).
- Same-sign product/quotient → positive; different signs → negative.
- aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents); aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract exponents).
- a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ.
- Standard form: k × 10ⁿ where 1 ≤ k < 10.