*The Language of Mathematics*, by Warren W. Esty

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Table of Contents, 19th Edition (for extended comments, see below)

Preface

To the student

ALGEBRA IS A LANGUAGE

1.1. Reexamining Mathematics

Mathematics as a Language

Problem-Solving Methods as Formulas

Problem-Solving Methods as Identities

1.2. Order matters!

Guide to Pronunciation

1.3 Reading Mathematics

1.4. Algebra and Arithmetic

1.5. Reconsidering Numbers

SETS, FUNCTIONS, AND ALGEBRA

2.1. Sets

Equality, Subset, Intersection, Union, Complement

2.2. Functions

Why they are important, Formulas, Notation, Composition of Functions

2.3. Solving Equations

The Rules, Reading the Rules, The Theory of Equation-Solving

RULES OF ALGEBRA (summarized)

2.4. Word Problems

LOGIC FOR MATHEMATICS

3.1. Connectives

3.2. Logical Equivalences

3.3. Logical Equivalences with a Negation

3.4. Tautologies and Proofs

RESULTS FROM CHAPTER 3

SENTENCES, VARIABLES, AND CONNECTIVES

4.1. Sentences with One Variable

4.2. Generalizations and Variables

Generalizations and "True-False" Questions

4.3. Existence Statements and Negation

4.4. Ways to State Generalizations

Conditional Sentences as Defining Conditions

Existence Statements as Defining Conditions

4.5. Reading Theorems and Definitions

4.6. Different Appearance -- Same Meaning

(A summary of all the ways sentences can look different and yet have the same meaning)

PROOFS

5.0. Why Learn to do Proofs?

5.1. Proof

"True" or "Proved"-- The List Approach

5.2. Proofs, Logic, and Absolute Values

5.3. Translation and Organization

5.4. The Theory of Proofs

5.5. Existence Statements and Existence Proofs

5.6. Proofs by Contradiction or Contrapositive

5.7. Mathematical Induction

Students who finish Chapters 1 through 5 are prepared to move on to higher mathematics.

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**Chapter 1 **begins with a section on the definition of abstraction and discusses its importance in written mathematics. Then it discusses *language* and how symbolic language can be used to express the methods of arithmetic. The order conventions of the written language are given. The procedures of arithmetic are described as simple symbolic sentences. The student begins to practice with symbolism which is applied to topics he or she already knows. The context for the language lessons of Chapter 1 is numbers. Students learn that symbolic sentences can express methods. In the last section examples of sentences include properties of absolute values and methods of solving inequalities.

- Give the order in which the operations are to be executed in the expression "-(8 + 4
*x*)";^{2} - State the
*algebraic formulation*of the method used to evaluate these expressions: "12 - (-8)" [answer:*a*- (-*b*) =*a*+*b*] , (3/4)/5 [answer: (*a*/*b*)/*c*=*a*/(*bc*)], etc.

**Chapter 2 **introduces sets and functions as examples of mathematical concepts that can be discussed in the language. Terms such as set "intersection," "union," and "subset" help introduce the key vocabulary words "and," "or," and "if...,then...". Theorems for solving the most basic types of equations also introduce these vocabulary words from logic (these logical connectives become the subject of the next chapter, but you cannot learn about connectives without ideas to connect!)

Section 2.4 on "Word problems" shows how the algebraic concepts emphasized so far (especially the conepts of operations and order = functions) are essential to doing word problems.

From the section on functions here are typical questions of a conceptual nature.

- Give the mathematical expression in terms of "
*x*" which expresses the rule "Add 7 and then multiply by two"; - Give a descriptive imperative name for
*f*(which does not mention "*x*") when f(*x*) = (*x*+ 5)^{2}." [Answer: "Add 5 and then square."] - What is the difference between "
*f*" and "*f*(*x*)"?

From the long section on solving equations:

- [
*E1*and*E2*denote successive equations] If*E1*=>*E2*, how do their solution sets compare? Can they be equal? If one has more solutions than the other, which is it? - To solve the initial equation, would you prefer to have your sequence of equations connected by"=>" or "iff"? Why?
- What is an "extraneous" solution?
- When solving (simple) equations in Chapter 2, the instructions are, "In the following homework the solution is not the only goal. Exhibit every step, exhibit the connective [
*iff*for equivalence, sometimes "=>" is necessary], and cite a rule." A typical problem ranges from a factorable quadratic to a harder problem where a square root must be eliminated:*x*- 1 = sqrt(*x*+ 11) [extraneous solutions may arise]. - Suppose you are asked to solve the equation "(
*x*+ 1)^{2}=*x*(*x*+ 2)." After properly using some of our rules you would find it is equivalent to "1 = 0. " That looks wrong. Did you make a mistake? Now what? What is its solution? Why?

**Chapter 3** introduces truth-table logic, emphasizing a dozen basic logical equivalences and a few tautologies. The logical equivalences selected are those equivalences most often used in mathematics to provide alternative ways of expressing the same thought. The tautologies are those most often used in equation-solving and proofs. Chapter 3 emphasizes the primary patterns of mathematical reasoning. In contrast to truth-table logic as taught in "discrete" or "finite" math courses, the examples are nearly all mathematical and the study of logic continues immediately in **Chapter 4** with its application to the truth of sentences with variables. Open sentences, generalizations, existence statements, and negations are discussed in the context of algebra. A key point is that sentences of different types may appear symbolically similar (e.g. "2(x + 3) = 2x + 6" and "2(x + 3) = 3x"), but require radically different mathematical interpretations. Chapter 4 culminates with a section summarizing four distinct mathematical reasons why two sentences may appear different yet express the same meaning.

In addition to problems which require students to construct truth tables, some questions on logical equivalences and their applications to sentences are:

- Are the following nouns, pronouns, or statements? a) 3(4 + 2) = 18, b) 3(4 + 2) = 20, c) 3(4 +
*x*), d) 3(4 + 2) [Note that b) is a statement a false statement]; - [Use DeMorgan's Law to] Give the negation of "-5 <
*x*< 8." - [Use a stated logical equivalence to] Give another sentence logically equivalent to "If c > 0, then, if c
^{2}> 25, then c > 5." [One answer: "If c > 0 and c^{2}> 25, then c > 5." Another: "If c > 0 and c <= 5, then c^{2}<= 25."] - Here is a sentence: "|x - 5| > 2 => x > 7." Give its contrapositive. Give its negation. Is it true?; [Because it is an implicit generalization, its negation is an existence statement: "There exists x such that |x - 5| > 2 and x <= 7. " Thus the counterexample x = 0 (one of many) proves the negation true and the original statement false.]
- Suppose each of the following sentences is true. Which express mathematical facts, and which express facts which depend upon the particular things represented by the letters? a) 3(x + 5) = 3x + 15, b) 3x = 12, c) |x| <= 0, d) |x| < |x + 1|, e) (A and B) => B, e) (A or B) => B ["A" and "B" represent statements];
- Explain the difference between a sentence with a variable and a statement;
- Explain the difference between an equation and an identity;
- Express this fact using different letters: ab = 0 iff a = 0 or b = 0.
- Use the quadratic formula to solve for x in ax +3x
^{2}= bc; - Give the [logical] form of the sentence: If |x| > 5, then x > 5 or x < -5. [Answer: A => (B or C)]; Restate the assertion in a logically equivalent form.
- State the fact expressed in English as "The product of positive numbers is positive" using proper mathematical notation and connectives." [Answer: If a> 0 and b > 0, then ab > 0.]
- True or false? a) bc > 25 is equivalent to b > 5 or c > 5; b) b < c is equivalent to b + d < c + d; c) a = b is equivalent to a
^{2}= b^{2}. - Discuss this conjecture: The hypothesis of a theorem can be interpreted as describing the cases for which the conclusion is true. [No. It describes some of the cases, but possibly not all.]

**Chapter 5 **examines proof. The roles of prior results, tautologies, reorganization, and definitions in mathematical proofs are discussed.

Some questions on proofs are:

- What is the difference between "prove" and "deduce"?
- Suppose "A => B" is true. Discuss whether A "proves" B.
- Determine whether the steps are sufficient to deduce the conclusion. Steps: H => A. H => B. (A and B) => C. Conclusion: H => C.
- Many examples of, and exercises on, proofs

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For instructors--comments on pace.

In practice, in forty class periods (50 minutes each) with many math-anxious students I could get through Section 5.3 or so, which completes a good introduction to proofs. When I taught school math teachers it was in the summer with longer, but fewer, class periods. Surprisingly, they were unable to go much faster. It seems that this material is accessible to almost everyone, but unavailable elsewhere in the curriculum, so even math teachers had a lot to learn.

**Pace.** The material has not been artificially subdivided into single-day chunks. The inherent unity of some substantial topics produces some sections that are quite long (for example, functions, 1.5; algebraic methods, 4.2, introduction to proofs, 5.2). Here is the pace I use in a Freshman-level class at Montana State University:

Section and number of days (excluding exams)

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 1.1: 1 2.1: 3 3.1: 2 4.1: 1 5.0: O+ 1.2: 1 2.2: 2 3.2: 2 4.2: 1 5.1: 1 1.3: 1 2.3: 3-4 3.3: 1-2 4.3: 1 5.2: 3 1.4: 2 2.4: 2 3.4: 1 4.4: 1 remainder of 1.5: 2 4.5: 2 Chapter 5: 4.6: 1 1 day each --------------------------------------------------------- 7 10-11 6-7 7 4* (totals)

* Chapter 1 through Section 5.2 can be regarded as a complete course which provides some exposure to the theory of proofs.

**Prerequisites:** The prerequisite is some high-school algebra because many of the examples in the text use algebraic symbolism. At Montana State University I have found that many older students have forgotten most of their algebra but do well anyway. Experience shows that many strongly "math anxious" students who cannot complete traditional courses can do well -- even very well -- if they do the work.

A research study showed that this class provides a wide range of students with the opportunity to reenter and succeed in mathematics learning regardless of their entry levels of manipulative skills and conceptual development.

**Conclusion.** It has been traditional to offer abstract courses only to those advanced college students who have shown by their continued success that they are already "good at math." This course has demonstrated that many students who would normally be classified as "math rejects" can succeed and blossom when the reasoning and abstract methods of expression underlying the mathematical material are made explicit. The sequencing and pace of the material provide an opportunity for success with significant mathematics to weak and strong students alike.

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Warren Esty has written another text, *Precalculus*, designed to prepare students for calculus.