An Elementary Transition to Abstract Mathematics

Gove Effinger, Gary L. Mullen

• Publisher: CRC Press
• Year: 2019
• Print ISBN:
  9780367336936
• eBook ISBN:
  9781000702279
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role.

The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part introduces key objects from abstract algebra; and the final part focuses on polynomials.

Features:

• The material is presented in many short chapters, so that one concept at a time can be absorbed by the student.
• Two "looking back" chapters at the outset (pre-calculus and calculus) are designed to start the student's transition by working with familiar concepts.
• Many examples of every concept are given to make the material as concrete as possible and to emphasize the importance of searching for patterns.
• A conversational writing style is employed throughout in an effort to encourage active learning on the part of the student.

Chapter 1

A Look Back: Precalculus Math

We begin our transition to abstract mathematics by looking backwards in your mathematics education to some ideas from algebra, combinatorics and geometry/trigonometry. The emphasis here will be not on the “what does it say?”, but rather on the “why is it true?”, as we discussed in the Preface for the Student. By looking at a few familiar ideas from this different point of view, we can begin to develop the kind of abstract thinking required to really understand more advanced mathematics.

In what follows, you are expected to have paper and pencil ready. Mathematics is not a spectator sport; the only way to fully understand an idea is to work with it yourself. In fact, in the whole text, but especially here and in the following two introductory chapters, you should think of the examples really as exercises for you to work your way through.

Example 1.1. (The Quadratic Formula) Most algebra students have memorized, or at least know how to quickly look up, the Quadratic Formula, which says:

If a ≠ 0, the solutions of the equation ax² + bx + c = 0 are

x=−b±b2−4ac2a.

That’s the “what”; let’s work out the “why?” now. The derivation of this famous formula comes from the technique for solving quadratic equations known as completing the square. This technique is an example of an algorithm, which is a finite list of steps to achieve a desired outcome. Here are the steps in this algorithm, given a quadratic equation in standard form (with a ≠ 0), i.e., set equal to 0 with the 0 on the right-hand side.

1. Divide through by the coefficient of x² and move the resulting constant term to the right-hand side of the equation.

2. Take half of the coefficient of x, square this, add this result to both sides of the equation, and put the right-hand side over a common denominator.

3. The left-hand side is now the square of a linear polynomial. Rewrite the left-hand side in this form.

4. Take the square root of both sides, remembering the plus/minus on the right.

5. Solve for x, hence displaying our two solutions.

Use this algorithm now to solve the equation 3x² + 5x − 1 = 0. Did you get x=−5±376? If so, good.

Now use this algorithm to solve ax² + bx + c = 0, hence deriving the Quadratic Formula. Having done so, we now know both what this formula is and why it always works.

Before moving on from the Quadratic Formula, we mention here a very surprising result which was proved by the famous mathematicians Galois and Abel in the early 19th century. It had been known previously that there existed algorithms to solve in radicals every quadratic, cubic (i.e., degree 3) and quartic (i.e., degree 4) polynomial equation with integer coefficients, but no one had been able to find algorithms to solve every such polynomial equation of degree 5 or higher. Galois and Abel proved, using advanced mathematics, that no such algorithms can exist for degrees ≥ 5. So, it’s not that no one has as yet discovered them; it’s that they are impossible to discover. You might well ask “why?”; to find out, take a course in abstract algebra, more specifically in “field theory” (see Chapter 29).

Next, we move to the area of mathematics called combinatorics, or simply “counting theory”.

Example 1.2. (Binomial or “Choose” Coefficients) Suppose you have n distinct objects and you wish to select (or “choose”) som...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. A Look Back: Precalculus Math
• 2. A Look Back: Calculus
• 3. About Proofs and Proof Strategies
• 4. Mathematical Induction
• 5. The Well-Ordering Principle
• 6. Sets
• 7. Equivalence Relations
• 8. Functions
• 9. Cardinality of Sets
• 10. Permutations
• 11. Complex Numbers
• 12. Matrices and Sets with Algebraic Structure
• 13. Divisibility in Z and Number Theory
• 14. Primes and Unique Factorization
• 15. Congruences and the Finite Sets Zn
• 16. Solving Congruences
• 17. Fermat’s Theorem
• 18. Diffie-Hellman Key Exchange
• 19. Euler’s Formula and Euler’s Theorem
• 20. RSA Cryptographic System
• 21. Groups - Definition and Examples
• 22. Groups - Basic Properties
• 23. Groups - Subgroups
• 24. Groups - Cosets
• 25. Groups - Lagrange’s Theorem
• 26. Rings
• 27. Subrings and Ideals
• 28. Integral Domains
• 29. Fields
• 30. Vector Spaces
• 31. Vector Space Properties
• 32. Subspaces of Vector Spaces
• 33. Polynomials
• 34. Polynomials - Unique Factorization
• 35. Polynomials over the Rational, Real and Complex Numbers
• Suggested Solutions to Selected Examples and Exercises
• Bibliography
• Index