1.1

Chapter 1: Speaking Mathematically

Introduction

  • The aim of this book is to introduce a mathematical way of thinking applicable in various situations.

  • Problem-solving often starts with a vague idea of how to proceed, evolving into clearer thought through various methods like examples, diagrams, and notation.

  • The goal is to familiarize students with the specific language essential for mathematical thinking, including variables, sets, relations, and functions.

  • This chapter serves as a mental warm-up for better mathematical performance.

1.1 Variables

  • Definition: A variable acts as a placeholder, representing an unknown or general quantity.

    • It allows discussion about numbers without specifying values, or to express truths applicable to all elements within a set.

  • Example of Variable Usage:

    • Problem: Is there a number such that doubling it and adding 3 results in its square?

    • Using a variable: Is there a number x such that 2x + 3 = x²?

Role of Variables

  • Placeholders in Computation:

    • Temporarily name unknown numbers to perform calculations.

    • Example: Can use a box (☐) to represent an unknown variable, aiding in imagining different values.

  • General Statements:

    • Variables help maintain generality for statements applicable to various numerical values.

    • Example statement: "No matter what number n is chosen, if n > 2, then n² > 4."

Writing Sentences Using Variables

  • Variables can formally rewrite sentences.

    1. Are there numbers a and b such that a² + b² = (a + b)²?

    2. Given any real number r, r² is nonnegative (i.e., r² ≥ 0).

Important Mathematical Statements

  • Types of Statements:

    • Universal Statements: A property true for all elements in a set. (Example: All positive numbers are greater than zero.)

    • Conditional Statements: If one thing holds true, then another must also hold true. (Example: If 378 is divisible by 18, then it is divisible by 6.)

    • Existential Statements: Asserts that at least one element satisfies a given property. (Example: There is a prime number that is even.)

Universal Conditional Statements

  • A universal conditional statement combines both universal and conditional aspects.

    • Example: "For every animal a, if a is a dog, then a is a mammal."

  • Can be rewritten to emphasize universal or conditional aspects:

    • If a is a dog, then a is a mammal.

    • All dogs are mammals.

Rewriting Universal Conditional Statements

  • Example exercise: "For every real number x, if x is nonzero then x² is positive."

    • Various forms:

      • If a real number is nonzero, then its square is positive.

      • For every nonzero real number x, x² is positive.

Universal Existential Statements

  • A statement asserting that something holds for all objects and simultaneously exists.

    • Example: "Every real number has an additive inverse."

  • Variations:

    • All real numbers have an additive inverse.

    • For every real number r, there is an additive inverse s for r.

Existential Universal Statements

  • Existential first, universal second.

    • Example: "There is a positive integer that is less than or equal to every positive integer."

  • Such a statement can be formalized in several ways, showcasing variances in structure.

Exercises: Transformation and Rewriting

  • Include exercises that involve transforming statements using variables and determining their truthfulness.

  • Example Exercise:

    1. Is there a real number whose square is -1?

    • Use variable x to express it formally as: x² = -1.

Summary of Key Concepts

  • The chapter emphasizes the significance of variables in mathematical language, aiding in expressing complex ideas clearly and succinctly while allowing for generality and specificity in statements.