Define and classify the following number systems: Natural numbers Example: Natural numbers is a set of positive counting numbers starting from 1 up to \( 1,2,3,4 \) Example: \( 5,10,100 \)

Natural Numbers are the set of positive whole numbers that we naturally use for counting objects. They are sometimes called the counting numbers. For example, 1, 2, 3, 4, … are all natural numbers. Note that while many texts define natural numbers as starting from 1 (i.e., {1, 2, 3, 4, …}), some modern definitions also include 0 (i.e., {0, 1, 2, 3, …}). In the example provided here, the set starts at 1. Definition: • The set of natural numbers, usually symbolized by ℕ, is defined as ℕ = {1, 2, 3, 4, …} (when 0 is not included) or ℕ = {0, 1, 2, 3, …} (when 0 is included). Properties: • They are infinite in number. • They are well-ordered: every non-empty subset of ℕ has a smallest element. • They are closed under addition and multiplication (adding or multiplying two natural numbers produces another natural number). • They are not closed under subtraction (e.g., 2 − 5 does not yield a natural number) or division in general. Classification within the Number System Hierarchy: Natural Numbers form the foundation of the broader number systems. They can be placed in the following hierarchy: 1. Natural Numbers (ℕ) – the basic counting numbers. 2. Integers (ℤ) – ℕ is a subset of ℤ which includes negative numbers and zero. 3. Rational Numbers (ℚ) – integers (and fractions made with integers) are contained within the rational numbers. 4. Real Numbers (ℝ) – rational numbers, along with irrational numbers, make up the real numbers. 5. Complex Numbers (ℂ) – as the most general system, these include all numbers of the form a + bi, where a and b are real numbers and i is the square root of −1. Examples: • 5, 10, and 100 are all natural numbers according to the definition provided. Thus, natural numbers are the simplest and most intuitive set of numbers used for counting and ordering, and they serve as the building blocks for more complex number systems in mathematics.