Fundamental Concepts of Natural Numbers

Basic Laws of Natural Numbers

• Law of Closure for Addition: If a and b are natural numbers, then their sum a + b is also a natural number. Example: 2 + 5 = 7.
• Commutative Law for Addition: The order of addition does not affect the sum, a + b = b + a. Example: 5 + 7 = 7 + 5 = 12.
• Associative Law for Addition: Grouping does not affect the sum, a + (b + c) = (a + b) + c. Example: 12 + (13 + 22) = (12 + 13) + 22 = 47.
• Law of Closure for Multiplication: If a and b are natural numbers, then their product a × b is also a natural number. Example: 2 × 4 = 8.
• Commutative Law for Multiplication: The order of multiplication does not affect the product, a × b = b × a. Example: 12 × -3 = -3 × 12 = -36.
• Associative Law for Multiplication: Grouping does not affect the product, a(b × c) = (a × b) × c. Example: 6 × (-4 × -3) = (6 × -4) × -3 = 72.
• Distributive Law: This law links addition and multiplication, a(b + c) = ab + ac. Example: 3(5 + 8) = 15 + 24 = 39; 3(5 - 8) = 15 - 24 = -9.

Basic Laws of Equality

• Reflexive Property: Any quantity is equal to itself, a = a.
• Symmetric Property: If a = b, then b = a.
• Transitive Property: If a = b and b = c, then a = c; things equal to the same thing are equal to each other.
• Addition of Equals: If a = b and c = d, then a + c = b + d.
• Multiplication of Equals: If a = b and c = d, then ac = bd.

Inequalities

• Definition: An inequality expresses a relationship between two quantities, indicating one is greater or less than the other using symbols like >, <, ≥, and ≤.
• Compound Inequalities: Can indicate a range of values using "and" (e.g., 1 < x ≤ 5 means x is between 1 and 5) or "or" (e.g., x < 1 or x > 5 means x is outside the interval (1, 5)).

Theorems of Inequalities

• Theorem 1: Adding the same number to both sides of an inequality will not change the sense of the inequality. (If a > b, then a + c > b + c).
• Theorem 2: Multiplying both sides of an inequality by a positive number preserves the inequality. (If a > b and c > 0, then ac > bc). Example: For 8 > -5, multiplying both sides by 3 gives 24 > -15.
• Division of Inequalities: Similar to multiplication, dividing by a positive number maintains the inequality's sense. (If a > b and c > 0, then a/c > b/c).