Number Systems

1. Introduction to Number System

The number system is the foundation of mathematics and plays a crucial role in various real-life applications. It encompasses different types of numbers used for counting, measuring, labeling, and performing arithmetic operations.

2. Types of Numbers

2.1 Natural Numbers (ℕ)

The numbers used for counting: ℕ = {1, 2, 3, 4, 5, …}

Smallest natural number: 1

2.2 Whole Numbers (W)

Natural numbers along with zero: W = {0, 1, 2, 3, 4, …}

Smallest whole number: 0

2.3 Integers (ℤ)

All whole numbers and their negatives: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}

2.4 Rational Numbers (ℚ)

Can be expressed as p/q, where p and q are integers and q ≠ 0.

2.5 Irrational Numbers

Numbers that cannot be expressed in the form of p/q. Decimal expansion is non-terminating and non-repeating.

2.6 Real Numbers (ℝ)

Includes both rational and irrational numbers.

3. Representation on the Number Line

Both rational and irrational numbers can be represented on the number line using decimals or geometric constructions.

4. Operations on Real Numbers

• Closed under all operations
• Commutative, Associative, Distributive properties
• Identity and inverse elements

5. Decimal Expansions of Real Numbers

• Terminating
• Non-Terminating Recurring
• Non-Terminating Non-Recurring (Irrational)

6. Converting Decimals to Rational Numbers

Use algebraic methods to convert repeating decimals to fractions.

7. Real Numbers and Their Decimal Expansions

Decimal expansion is terminating if the denominator has only 2 and/or 5 as prime factors.

8. Laws of Exponents for Real Numbers

• am × an = am+n
• am / an = am−n
• (am)n = amn
• a0 = 1
• a−n = 1 / an

9. Surds and Irrational Numbers

Surds are irrational numbers expressed in root form. e.g. √2, ∛7

10. Rationalization

Multiply numerator and denominator by the conjugate to remove a surd from the denominator.

11. Real Numbers and Geometry

All real numbers can be represented as points on the number line. Distance between two points is the absolute difference.

12. Summary of Number System Hierarchy

Real Numbers (ℝ)
  ├── Rational (ℚ)
  │   ├── Integers
  │   │   ├── Whole Numbers
  │   │   │   └── Natural Numbers
  │   └── Fractions / Decimals
  └── Irrational Numbers
  

13. Important Results and Tips

• Every natural number is a whole number.
• Every integer is a rational number.
• √p is irrational if p is prime.

14. Practice Questions

1. Classify 0.123456789…, 22/7, π, √2 as rational or irrational.
2. Represent √3 on the number line.
3. Express 0.272727… as a rational number.
4. Is 7/250 a terminating decimal?
5. Rationalize:
   • 1 / (√5 + 2)
   • 3 / (√2 − 1)
6. Simplify:
   • (2³)⁴ × 2²
   • (4^0.5 × 8^1/3)³