Understanding Mathematics: From Counting to Calculus

Book Information

Understanding Mathematics: From Counting to Calculus
352 pages
ISBN: 978-0-9657300-1-3


Understanding Mathematics


An easy-to-read, non-technical discussion covering a broad range of math topics ranging from Counting to Calculus. Imparts intuition and understanding while explaining both the "why" and "how" of math.

Why Anyone Needs This Math Book


Math is arguably the most important school subject for success in today's society. Most people are taught mathematics from the time they learn to talk through high school. To some mathematics comes easy, but to others it is extremely difficult. Obviously, people have different abilities. However, math books, math teachers, real-world applications, and a variety of other factors often affect one's perspective on the subject of mathematics. This is a subject which is often either loved or hated. If the subject is hated, it is not math, but the failure to understand the math that gives considerable frustration to the student.


The process of understanding math should not be frustrating and difficult. Mathematics (including Algebra) can be clearly understood with some effort and practice. The goal of this book is for the reader to obtain a clear understanding of mathematics, from counting to Calculus.

What This Book Provides

Following this book step-by-step will lead from counting to a brief introduction of Calculus, hence the book's title. Therefore, this one text covers all of the math fundamentals normally taught from first grade to college. It is most directly written as a supplement to existing math texts. Therefore, a student could use this book along with his own math text for additional explanation and clarity. However, this book can also stand independently with no outside references.


Concepts and real-world applications are emphasized. This book is not intended to be rigorous. This means that you will not be given detailed proofs or complicated math theory.

This book was written to be read as simply as possible. You won't be given cute stories, long histories, or complex scenarios. The author will be making no effort to impress you with his vocabulary, but will give popular mathematical notation. There will be no leaps of faith in moving from one topic to the next. Each topic will build on the previous topic, creating a smooth flow of mathematical progress.


There are many math help books available. However, the extreme mathematical details and lack of explanation contained in most math books often cloud understanding. Even a thorough review of "made simple" math books reveals that math is not made simple, but is merely abbreviated. Shortening explanation of material is not the answer. Rather, expanding explanation is necessary.


Some math help books only consist of examples. These books are useful for solving particular book problems, but they fail to explain the basic theories and concepts of math which are required to solve real-world problems and to establish a foundation for further learning. Examples are wonderful, and many are given herein. However, without understanding concepts, an infinite number of examples will not prepare the student for the future, or help him understand why problems are solved in a certain manner.


In this text, the examples will make sure that you are on the right track. The formulas will ensure that you are always prepared. But most importantly, math will be explained. The answer to "Why do I have to do it like that" will be answered along with the usual "How do I do it." This will enable the reader to remember what needs to be done to solve a variety of problems. It will also provide an overall outline of math from the very basics to advanced techniques.


A section on using calculators or computers is not included. These are wonderful and necessary tools, but the goal of this text is to provide understanding. The use of these devices, without a firm math foundation, reduces intuition and the important ability to determine if an answer is reasonable. Even today, standardized tests such as the SAT, ACT, or GMAT do not allow the use of calculators. There is good reason for this policy.


Many examples and detailed answers are provided throughout each chapter. However, there is not a list of problems for the reader to work on his own at the end of each chapter. Numerous problems (and solutions) require much valuable space and can be found in almost every other math book.


Some of the final chapters of this text contain concepts that are seldom explained in any standard math book. Specifically, chapters 18 through 20 provide some of the key derivations that tie different mathematical disciplines together. In addition to practicality, these chapters demonstrate a glimpse at some of the amazing (and even beautiful) qualities of mathematics.


The Layout of This Text

This text is divided into three parts. Part I (chapters 2 through 9) consists of mathematics normally taught in grade school and are math essentials for everyone. Part II (chapters 10 through 17) consists of mathematics normally taught at the high school level including a thorough coverage of Algebra, the essentials of Geometry, and basic Trigonometry. Part III (chapters 18 through 24) consists of advanced Algebra and an introduction to Calculus.


STEP-BY-STEP boxes give the recipe for solving standard problems.


Bold type highlights important definitions.


Pictures are used to emphasize important concepts and support surrounding text.


Introductions and summaries are provided to keep the reader aware of topics already learned and explain goals for future learning. Notepads are also included at the end of every chapter. Notepad topics include math laws, important definitions, and topics presented in the chapter. The reader may use these notepads to ensure that all important concepts have been reviewed and are understood.


An index is provided for easy reference and encourages the reader to learn a definition in its appropriate context.


Who Would Benefit from This Book

This book would be especially helpful as a supplemental guide for parents who are home-schooling their children.


However, it does not matter if you are a motivated grade school student, a curious high school student, a struggling non-technical graduate student (or "poet"), a working professional, or a parent attempting to help his child with mathematics. All would greatly benefit from this text.


A motivated grade school student could use this book to reach levels of mathematics normally only obtained in college. An early head start in mathematics will certainly yield dividends in the future. Early starts in mathematics are a common thread with many successful people, from billionaire Bill Gates to Nobel Prize winning physicist Richard Feynman. This fact was a great motivation for the writing of this text.


Why Buy One Book that Contains 12 Years of Mathematics?

As already stated, this book is divided into three parts. You may wonder why three separate texts are not available rather than one. Does a parent with a fourth grader really need a book that contains Calculus? Does a graduate student need a book that shows how to add and multiply?


Listed below are a few of the benefits of including so much math in one text.


  • It eliminates the need to assume any prior level of mathematical understanding.
  • It ensures that the reader has a complete overview and understanding of mathematical concepts. No concepts are missed in the reader's educational development.
  • One book is very efficient, eliminating unnecessary repetition of explanation.
  • Concepts that the reader already knows well (e.g. numbers) may be explained from a new perspective.
  • A step-by-step methodical progression may be made to give the reader a solid mathematical framework. The importance of this point is represented by the pyramid on the cover.
  • If a new concept is encountered in an advanced class (e.g. matrices in junior high), the reader does not have to find a new supplemental text that contains the subject.
  • Finally, the price of this one text is comparable to other math help books which cover far less material. The goal of this text is not for the author to make money. The goal is to help readers understand math.

This Math is Really Used in Practice

Students often hear that "advanced" math is hardly used anyway, so they should not be concerned with learning it. This statement could not be farther from the truth.


Actuaries use complex statistics to determine insurance rates. Physicists use mathematical models to develop nuclear energy. Engineers use math to build all types of electronics, including computers and communications equipment. Math is used to predict the stock market, calculate weather trends, control automobiles, distribute electricity, design roads and determine marketing strategies. Techniques to perform all of these tasks and more, rest on the fundamentals of math which are presented.


Math fundamentals are tested on college entrance exams such as the SAT and ACT. Furthermore, graduate entrance exams such as the GRE and GMAT also test math skills. The math tested on these and other standard entrance exams is explained in this text.


What the Reader Should Already Know


Every book assumes the reader already knows something about the book's subject. This book assumes that the reader has a basic understanding of the definition of a number, and has already seen addition, subtraction, and basic multiplication and division. However, each of these topics will be explained thoroughly. With parental guidance, students as young as the third or fourth grade may benefit from this text. Unlike most math texts, math concepts do not become increasingly difficult with each ensuing chapter. However, each chapter assumes that previous material is clearly understood.


If the reader already knows much of the material presented, it would still be most beneficial to start at the beginning of the book and confidently read until the area of new material. It is very likely that the reader will obtain a different perspective on even the most basic mathematics.


Now, onto learning...