Intro to Whole Numbers and Number Sets (1.1)

1.1 Introduction to Whole Numbers

• Context and setup

  • The session will cover 1.1, focusing on an introduction to whole numbers.
  • A reminder that there are many extra resources available through MyOpenMath; you can take notes, and the instructor may point out differences across resources.
  • The instructor plans to highlight how the material may differ from other materials, so note-taking is encouraged to capture these nuances.

• Core aims for this unit

  • Begin discussion about numbers by introducing the concept of a set.
  • Provide an overview of what a set is as a foundational idea.
  • Introduce the idea of number sets (e.g., natural numbers, whole numbers, integers, rationals, reals, complex numbers).

• What this implies for learning math

  • Sets organize numbers into meaningful categories that reflect their properties.
  • Understanding sets helps with definitions, proofs, and problem solving across arithmetic and higher-level math.

• Practical notes for study and exam preparation

  • Take advantage of the MyOpenMath resources for practice problems.
  • Pay attention to how different resources present the same concepts (definitions, examples, notation).
  • Keep a running glossary of set-related terms and symbols for quick reference during the exam.

• Ethical and practical implications

  • Clarity in definitions reduces ambiguity when solving problems.
  • Proper notation and consistent conventions aid clear communication in mathematical reasoning.

• Quick preview of what’s ahead

  • We will formalize what a set is, including notation and basic terminology.
  • We will classify numbers into standard sets and discuss their relationships.

• Key takeaway to remember

  • A set is a collection of distinct objects, called elements, and sets provide the language for grouping numbers and objects in mathematics.

1.2 What is a Set?

• Basic definition

  • A set is a collection of distinct objects considered as a whole.
  • Notation: A = {a, b, c} where a, b, c are elements of A.
  • The membership relation is written as a ∈ A if a is an element of A.

• Notation and terminology

  • Elements vs. members: the objects inside a set are called elements or members.
  • Cardinality: the number of elements in a finite set A is denoted by |A|.
  • Finite vs infinite: a set is finite if |A| is a finite number; otherwise, it is infinite.

• Examples

  • The set of even nonnegative integers: E = {0, 2, 4, 6, \dots} (infinite set).
  • The set of letters in the word "SET": S = { 'S', 'E', 'T' } (finite set).
  • The set of natural numbers vs. whole numbers (see 1.3 for number set definitions).

• Foundational concepts related to sets

  • Subsets: if every element of A is also an element of B, then A ⊆ B.
  • Equality of sets: A = B if and only if A ⊆ B and B ⊆ A.
  • The idea of complement, union, and intersection (basic set operations to be explored later).

• Significance of sets in early math

  • Sets provide a precise language for describing collections of numbers and objects.
  • They form the basis for definitions of number systems and the operations defined on them.

1.3 Number Sets Overview

• Hierarchy of common number sets

  • Whole numbers: W = {0, 1, 2, 3, \dots}
  • Natural numbers: \mathbb{N} = {0, 1, 2, 3, \dots} \text{ or } {1, 2, 3, \dots} depending on convention.
  • Integers: \mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}
  • Rational numbers: \mathbb{Q} = \left{ \frac{a}{b} : a \in \mathbb{Z}, b \in \mathbb{Z} \setminus {0} \right}
  • Real numbers: \mathbb{R}
  • Complex numbers: \mathbb{C}

• Containment relationships (typical convention)

  • The natural and whole numbers are contained in the integers: W \subseteq \mathbb{N} \subseteq \mathbb{Z}
  • The integers are contained in the rationals: \mathbb{Z} \subseteq \mathbb{Q}
  • The rationals are contained in the reals: \mathbb{Q} \subseteq \mathbb{R}
  • The reals are contained in the complex numbers: \mathbb{R} \subseteq \mathbb{C}

• Conventions and practical notes

  • The exact definition of natural numbers (whether 0 is included) varies by author; this course will acknowledge both conventions where relevant.
  • When writing elements of these sets, use standard notation: e.g., 5 ∈ \mathbb{Z}, 1/2 ∈ \mathbb{Q} but 1/2 ∉ \mathbb{Z}, \sqrt{2} ∈ \mathbb{R} but \sqrt{2} ∉ \mathbb{Q}, and i ∈ \mathbb{C} where i^2 = -1.

• Basic properties and ideas

  • Closure properties (brief): For example, if a, b ∈ \mathbb{N}, then a + b ∈ \mathbb{N} and a · b ∈ \mathbb{N} (illustrative of how sets support operations).
  • Some sets are finite (e.g., a specific list of digits {0,1,2,3,4,5,6,7,8,9}) while others are infinite (e.g., the set of natural numbers).

• Practical connections to learning and problem solving

  • Understanding these sets helps in classifying numbers for solving equations, understanding divisibility, and framing proofs.
  • Real-world relevance: categorizing data types, counting objects, and modeling quantities.

• Metaphors and hypothetical scenarios

  • Think of a set as a labeled box where each unique item is an element. The box’s contents define the set; adding or removing items changes the set's content and potentially its size.
  • A hierarchy metaphor: different drawers (sets) contain different types of numbers, with smaller drawers nested inside bigger drawers as you move up the hierarchy.

• Summary of key notations to memorize

  • Sets and membership: A = {a, b, c}, a ∈ A, A ⊆ B, |A| for finite cardinality.
  • Standard number sets and their typical inclusions: W \subseteq \mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}
  • Most common special cases: W = {0,1,2,3,\dots}, \mathbb{N} = {0,1,2,3,\dots} \text{ or } {1,2,3,\dots}

• Practice and study tips related to sets

  • Practice identifying elements of each set from given numbers or expressions (e.g., is 0 ∈ \mathbb{N}? is \sqrt{3} ∈ \mathbb{Q}?).
  • Practice writing set-builder notation and simple set operations (union, intersection) as you advance.

• Reflection questions for understanding

  • Why do we separate numbers into different sets? What properties do they share or not share?
  • How do the containment relationships help with solving problems that involve different kinds of numbers?

• Next steps (to be covered in subsequent sections)

  • Define and practice basic set operations (union, intersection, complement).
  • Explore more examples of number sets and their properties in problem contexts.