Mathematics Foundations/3.1 Natural Numbers and Integers

Natural Numbers and Integers Introduction Natural numbers and integers form the foundation of number theory and much of mathematics. This section explores their properties, structures, and the fundamental principles governing their behavior. The Natural Numbers The set of natural numbers, denoted by , consists of the counting numbers: Some texts include 0 in the natural numbers, but we will follow the convention that begins with 1, and denote when we need to include zero. Peano Axioms The natural numbers can be formally defined using the Peano axioms, which provide a rigorous foundation: 1. 1 is a natural number. 2. For every natural number , there exists a unique natural number called the successor of , denoted or . 3. 1 is not the successor of any natural number. 4. If , then (the successor function is injective). 5. If a set contains 1 and contains the successor of every element in , then contains all natural numbers (the principle of mathematical induction). Properties of Natural Numbers Natural numbers possess several important properties: Closure under addition and multiplication: If , then and . Well-ordering principle: Every non-empty subset of has a least element. Division with remainder: For any with , there exist unique integers and such that where . The Integers The set of integers, denoted by , extends the natural numbers to include zero and negative numbers: Formal Construction Integers can be formally constructed as equivalence classes of ordered pairs of natural numbers. The pair represents the result of subtracting from , or . Two pairs and are equivalent if . Under this construction: The natural number corresponds to the equivalence class of for any . Zero corresponds to the equivalence class of for any . The negative number corresponds to the equivalence class of for any . Properties of Integers Integers have several important algebraic properties: Closure under addition, subtraction, and multiplication: If , then , , and . Additive identity: For any , . Additive inverse: For any , there exists such that . Multiplicative identity: For any , . Commutative properties: For any , and . Associative properties: For any , and . Distributive property: For any , . The Division Algorithm One of the most fundamental results in elementary number theory is the division algorithm. Theorem (Division Algorithm): For any integers and with , there exist unique integers (quotient) and (remainder) such that: where Proof: Consider the set . Since , for sufficiently large negative , we have , so is non-empty. By the well-ordering principle, has a least element for some integer . We need to show that : by the definition of If , then , which would be a smaller element in , contradicting the minimality of . Therefore, . For uniqueness, suppose with . Then . If , then , contradicting . Therefore, , which implies . ∎ Integer Representations Integers can be represented in various numeral systems, the most common being the decimal (base-10) system. However, other bases are important in mathematics and computer science. Base-b Representation Any positive integer can be uniquely represented in base (where ) as: where for all , and . This is typically written as . Common bases include: Binary (base-2): digits 0, 1 Octal (base-8): digits 0-7 Decimal (base-10): digits 0-9 Hexadecimal (base-16): digits 0-9, A-F Example: The decimal number 173 in different bases: Binary: because Octal: because Hexadecimal: because (where A represents 10 and D represents 13) Ordering of Integers The integers form a totally ordered set with the following properties: For any , exactly one of the following holds: , , or . If and , then (transitivity). If , then for any . If and , then . If and , then . Applications Natural numbers and integers are fundamental to many areas of mathematics and have numerous applications: Counting and measurement in everyday life Cryptography and secure communications Computer science (data representation, algorithms) Coding theory (error detection and correction) Graph theory (vertex and edge labeling) Exercises 1. Prove that the sum of two consecutive integers is odd. 2. Prove that the product of any three consecutive integers is divisible by 6. 3. Use the division algorithm to find the quotient and remainder when: a) 2025 is divided by 14 b) -137 is divided by 8 4. Convert the decimal number 2023 to: a) Binary b) Octal c) Hexadecimal 5. Prove that for any integer , the value of is always even. 6. Show that if and are integers with the same parity (both even or both odd), then is even. 7. Prove that for any integer , the number is divisible by 6.