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This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high-school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one.

Readers will see how various mathematical concepts are tied and will see that mathematics is not a pile of formulas and facts; rather, it has an orderly and beautiful edifice.

The author begins with basic rules of logic and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader.

**Reviewer Comments:**

I find the book very impressive. The choice and sequence of topics is excellent, and it is wonderful to have all of these things together in one volume. Theorems are clearly stated, and proofs are accurate. – *Michael Comenetz*

The thoroughness of the narrative is one of the main strengths of the book. The book provides a perfect illustration of mathematical thinking. Each step of a given derivation is precise and clear. – *Julie Gershunskaya*

Draganov’s book stands out from the many competing books. Draganov’s goal is to show that mathematics depends on the notion of proof. Unlike other transition books, he addresses mathematical topics at an accessible level, rather than topics studied later in the university curriculum. – *Ken Rosen*

List of Figures

Preface

1 A Few Rules of Logic

1 1.1 True and False Statements

1.2 General and Particular Cases

1.3 “If... then” Statements

1.4 Combining “or”, “and”, and Negation

1.5 Logic Lingo

1.6 No Contradictions Are Allowed

1.7 The Need for Existence

1.8 What is Typically Proved?

1.9 Types of Proofs

Exercises

2 Numbers

2.1 Natural Numbers and Primes

2.2 Integers

2.3 Rational Numbers

2.4 The Decimal Representation

2.4.1 Decimal Representation of Integers

2.4.2 Decimal Representation of Rational Numbers

2.5 Irrational Numbers

2.5.1 Mixing Rational and Irrational Numbers

2.6 Two Theorems about Real Numbers

2.7 Complex Numbers

2.8 How Many Numbers Are There?

2.8.1 Sets

2.8.2 How Many Rational Numbers Are There?

2.8.3 How Many Real Numbers Are There?

2.8.4 Cardinality of Real Numbers in an Interval

2.8.5 Points on a Plane

2.8.6 Numbers That We Can Define

2.8.7 The Completeness of Real Numbers

Exercises

3 Inequalities

3.1 Basic Properties

3.2 Several Theorems

3.3 Several Inequalities

3.3.1 A Sum of Absolute Values

3.3.2 The Arithmetic Mean

3.3.3 The Geometric Mean

3.3.4 The Harmonic Mean

3.3.5 The Quadratic Mean

3.3.6 Inequalities for the Four Means

3.3.7 Bernoulli’s inequality

3.3.8 The Cauchy-Schwarz Inequality

Exercises

4 Functions

4.1 Definition and Examples

4.2 Odd and Even Functions

4.3 Composite Functions

4.4 Monotonic Functions

4.5 Inverse Functions

4.5.1 Inverse of an Inverse

4.5.2 Inverse of a Composite Function

4.6 Applying a Function to both Sides of an Inequality

4.7 Concave and Convex Functions

Exercises

5 Polynomials

5.1 Definition and Examples

5.2 Binomial Expansion

5.3 Pascal’s Triangle

5.4 Adding and Multiplying Polynomials

5.5 When Are Two Polynomials Equal?

5.6 Roots

5.7 The Polynomial Remainder Theorem

5.8 The Fundamental Theorem of Algebra

5.9 Vieta’s Theorem

Exercises

6 Power Law, Exponents, and Logarithms

6.1 Integer Exponents

6.2 Radicals as Inverse Exponents

6.3 Rational Exponents

6.4 From Rational to Real Exponents

6.5 The Exponential Function

6.5.1 The Number e

6.6 Properties of the Exponential Function

6.7 Is the Exponent Monotonic?

6.8 Logarithms

6.9 The Base of the Logarithmic Function

Exercises

7 Trigonometry

7.1 How to Use Algebra for Solving Problems in Geometry

7.2 Measuring Angles

7.3 Adding and Subtracting Angles

7.4 The Sine and Cosine Functions

7.5 Most Common Trigonometric Identities

7.6 Inverse Trigonometric Functions

7.7 Other Trigonometric Functions

7.8 Polar Coordinates

7.9 Cosine of the Difference of Two Angles

7.10 Back to the Identities for Complementary Angles

7.11 Sine of a Sum of Two Angles

7.12 Sine and Cosine of a Double Angle

7.13 One Way to Compute Trigonometric Functions

7.14 More Trigonometric Identities

7.14.1 Tangent of a Sum of Angles

7.14.2 Sine of a Half-Angle

7.14.3 Cosine of a Half-Angle

7.14.4 Sums and Differences of Trigonometric Functions

7.14.5 Products of Trigonometric Functions

7.15 Multiplication of Complex Numbers

7.16 Back to the Fundamental Theorem of Algebra

7.17 Euler’s Formula

7.18 Three Trigonometric Inequalities

7.19 Analytical Geometry

Exercises

8 Conic Sections

8.1 Cone and Plane Definitions

8.2 Metric Definitions

8.3 Focus and Directrix Definitions

8.4 Algebraic Definitions

8.5 Equivalency of Definitions 1 and 2

8.6 Equivalency of Definitions 2 and 4

8.7 Equivalency of Definitions 3 and 4

8.8 Conics in Polar Coordinates

8.9 Ray Reflections by Conics

8.10 The Design of X-Ray Telescopes

Exercises

9 Sequences and Sums

9.1 Arithmetic Sequence

9.2 Geometric Sequence

9.3 Infinite Sequences

9.4 Limits: Definition

9.5 Does the Geometric Sequence Converge?

9.6 Arithmetic Operations for Sequences

9.7 Monotone and Bounded Sequences

9.8 The Bolzano-Weierstrass Theorem

9.9 More on the Ratio of Two Sequences

9.10 A Sequence with Nested Radicals

9.11 More Sequences with Nested Radicals

9.12 The Limit for the Base of Natural Logarithms

9.13 Partial Sums and Infinite Series

9.14 The Harmonic Series

9.15 The Harmonic Sequence and Prime Numbers

9.16 Intuition May Fail Us for Infinite Series

9.17 Sometimes Neglecting Rigor Is a Good Thing

Exercises

10 The Fibonacci Sequence

10.1 Cassini’s Identity

10.2 The Golden Ratio

10.3 The Golden Ratio via Nested Radicals

10.4 Successive Powers

10.5 A Proof of Convergence

10.6 More about Successive Powers

10.7 Integers as Sums of Fibonacci Numbers

10.8 The Partial Sum of Fibonacci Numbers

10.9 Continued Fractions

10.10 Linking Geometric and Fibonacci Sequences

10.11 Two Related Sequences

10.11.1 Lucas Numbers

10.11.2 Pell Numbers

Exercises

Conclusion

Further Reading

Bibliography

Index

### Biography

**Alexandr Draganov** holds a PhD in Electrical Engineering from Stanford. After a career in high-tech, he pivoted to teaching and writing.