NUMBER SYSTEMS

Notes on Number Systems

1. Introduction to Number Systems

Number systems are ways to represent numbers on a number line, categorized into various types:

• Natural Numbers (N): The set of positive integers starting from 1 (1, 2, 3, …).
• Whole Numbers (W): The set of natural numbers plus zero (0, 1, 2, 3, …).
• Integers (Z): The set of whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3,...).
• Rational Numbers (Q): Numbers that can be expressed as the quotient of two integers, where the denominator is not zero (p/q, where p and q are integers, and q ≠ 0).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction, having non-terminating and non-repeating decimal expansions, such as √2, π, etc.
• Real Numbers (R): The combination of rational and irrational numbers, covering all numbers on the number line.

2. Understanding Rational Numbers

Rational numbers can be categorized by their decimal representations:

• Terminating Decimals: These decimals have a finite number of digits (e.g., 0.75).
• Non-Terminating Recurring Decimals: These decimals continue infinitely but begin repeating a sequence of digits (e.g., 1.3333…, represented as 1.3̅).

All rational numbers are included in the set of real numbers, and every rational can be expressed in both fractional and decimal forms.

3. Examining Irrational Numbers

Irrational numbers, unlike rational numbers, do not have a repeating or terminating decimal representation. Notable examples include:

• √2: Approximately 1.414213...
• π: Approximately 3.141592...

Rational numbers can indeed exist between every two rational numbers as they are dense on the number line, indicating an infinite number of them can be found between any two given numbers.

4. Decimal Expansions

The decimal expansion of any real number helps classify it as either rational or irrational:

• Rational Numbers: Have either terminating or non-terminating recurring decimal forms.
• Irrational Numbers: Have non-terminating and non-recurring decimal forms.

For instance, numbers like 0.101101110... are considered irrational since their decimal cannot terminate or repeat.

5. Operations on Real Numbers

Real numbers can be manipulated using various algebraic operations:

• Addition and Multiplication: The sum or difference of a rational number with an irrational number will yield an irrational number.
• Examples:
  • 2 + √3 = Irrational
  • 2 * √3 = Irrational

The combined results of irrational numbers (addition and multiplication) can produce results that are rational or irrational.

6. Laws of Exponents

• The laws of exponents apply to real numbers:
  1. a^m * a^n = a^(m+n)
  2. (a^m)^n = a^(m*n)
  3. a^m / a^n = a^(m-n)
  4. a^(1/n) = n√a

These laws help simplify expressions involving irrational or rational bases and exponents.

7. Conclusion: The Number Line

The chapter concludes with the understanding that every real number can be represented on the number line uniquely, showcasing that both rational and irrational numbers fill the number line, completing our understanding of number systems.

1. Natural Numbers (N): The set of positive integers starting from 1.
2. Whole Numbers (W): Natural numbers plus zero.
3. Integers (Z): Whole numbers and their negatives.
4. Rational Numbers (Q): Numbers that can be expressed as p/q, where q ≠ 0.
5. Irrational Numbers: Numbers that cannot be expressed as a fraction, having non-terminating, non-repeating decimals.
6. Real Numbers (R): The complete set of rational and irrational numbers.
7. Decimal Representation: Rational numbers can be terminating or non-terminating recurring, while irrational numbers are non-terminating non-recurring.
8. Operations on Real Numbers: The sum or product of a rational and an irrational number is irrational.
9. Laws of Exponents: Apply to both rational and irrational numbers; aids in simplifying expressions.
10. Unique Representation: Each real number corresponds to a unique point on the number line.