A New Year

George E. Hrabovsky, President, MAST

Introduction to the Fourth Year of this Column

Hello and welcome to our fourth year. It has been strongly suggested that I take a more dedicated project approach to this column. To that end I propose the following: Each column will address an entire project, or at least a significant part of such a project. I will include very little background material within the body of the column. Instead I will supply links to existing background material that can be found elsewhere.

The intent is to model, in mathematics, what we see in the write-ups for experiments. Just as each experiment that we are able to perform increases our skill in experimentation, so, too, each project in mathematics increases our skill there. Experiments also provide valuable tools in terms of apparatus that can be used in other experiments, just as mathematical principles can be applied in other math projects.

In each project, just as in a good experimental write-up, I will begin with a statement of the problem being considered. I will then include a list of all the "materials" needed (usually mathematical principles and tools that will be needed, along with links to relevant materials). I will then proceed, step-by-step, through the project. This will be followed by several additional questions raised by the project so that you can strike out on your own, but with some guidance as to where you can go with what has been done.

In a previous column, "The Return of the Mathematics Corner," I described how to develop a mathematics library (including many freely available resources). Keep these resources in mind as you proceed through these projects.

Many, if not most, of the projects presented will be pure mathematics. That is, mathematics for the sake of mathematics. If, on the other hand, an interesting project in applied mathematics occurs to me, that will be explored. If you can think of an interesting math project (either pure of applied) write me.

If, for any particular project, you can think of another way of doing it, send that to me, too. It is likely that I will include your solution in a column, or suggest you submit it to the editor and include a link to it from my column. It is possible that fundamental results could be discovered in this way. If this happens, I might suggest that you submit your work to a peer-reviewed journal. In this way significant collaborations could be developed.

Unless I get suggestions for projects from other people, and I am open to suggestions, I will tend to write about projects of personal interest. I am interested in all aspects of mathematics, but my principle interests involve geometry, topology, analysis, differential equations, and dynamical systems.

Of course, there is a great temptation to dive into the open problems at the cutting edge of mathematics. After all, that's where the great contributions are hiding, and it is always a great desire to attempt to discover the answer to one of these. Of course, it is extremely unlikely that an amateur will ever solve one of these problems, but it could happen. Even if we are unable to crack any of these nuts, an intermediate result could be found. Even if that does not happen, we can revel in the attempts and learn a lot of math at the same time.

The Sorts of Questions to be Considered

In order to do mathematics, rather than simply study it, there are a number of questions that may be asked that go well beyond the basics of What? When? Where? Why? and How? Each of these questions is important in understanding the bigger question. We will consider the questions: What can we infer from a number of cases? What can we deduce from a fundamental principle? How are the mathematical objects/structures under study related in such a way that they are equivalent? What results can be calculated? Can we formulate a conjecture or definition? Can we prove a theorem?

Inference is the process of taking several cases and abstracting a general principle. For example, by adding lots of natural numbers we realize that the sum always seems to be a natural number, so we make the conjecture (an unproven assertion) that the sum of any two natural numbers is itself a natural number (this is the property of closure under addition).

Deduction is the process of using a theorem (a proven assertion), an axiom (an assumed assertion) or a definition to justify some procedure or statement. For example, the definition of a group is:

1. A set together with a binary operation (an operation between any two elements of the set, such as addition) that is closed (thus, the application of the operation produces another element of the set).

2. Is associative (using addition we have 1 + (2 + 3) = (1 + 2) + 3).

3. Has an identity (any element of the set combined by the operation to the identity leaves you with the original element, 1 + 0 = 1 tells us that the identity element for addition is 0), and an inverse (by combining an element with its inverse we get the identity element, for addition we have 2 + -2 = 0).

We can apply each part of this definition to a set and its binary operation, one-at-a-time to find out if the set and operation form a group.

Equivalence relations have a special place in mathematics. They tell us that a relationship is reflexive (that something is always equivalent to itself), symmetrical (if is equivalent to then is equivalent to , thus we can make substitutions of symbol combinations that are equivalent), and transitive (if is equivalent to and is equivalent to , then is equivalent to ). Finding two things to be equivalent shows us that two branches of mathematics can be related in new ways.

Calculations are important for obtaining solutions to problems such as equations or determination of specific properties. The most interesting calculations are those where the most extreme values are considered.

Formulation of conjectures or definitions is an important part of creating new mathematics. Conjectures are the life blood of mathematics, they point to fundamental properties. Definitions are important, but care must be taken; a definition is a construction that is designed to meet a criteria. In this way you can create definitions that hold only for extremely exceptional cases.

Proving a conjecture to be true creates a theorem. Theorems are the glue that, together with definitions, hold mathematics together. Theorems are always preferable to axioms or definitions, since they are neither assumed true nor constructed for special cases.

This, then, is the arena of mathematics.

Math Project #1: Is the number system self-consistent?

Background

In order to understand the question we need to know two things: What do we mean by the number system? What do we mean by self-consistent?

A good place to define the number system is the natural numbers. We denote these with the symbol and define it as the set of all counting numbers beginning with zero. Note that some definitions of natural numbers exclude zero. So, listing the elements of would like this: .

Self constancy, in terms of numbers, is the  property that any arithmetic operation can be performed on any pair of elements of our set and yield another element of the set. This is called closure and any set having this property is said to be closed under the operation.

Necessary Principles

The notion of a set as a collection of elements each having a specific property that allows their inclusion in the set:

Definition 1: is the set of natural numbers, listed as .

The seven binary arithmetic operations (recall that a binary operation on a set is any operation that combines any two elements of the set):

Definition 2: + is the symbol for the binary operation of addition. I assume the reader is familiar with addition.

Definition 3: I assume the reader is familiar with multiplication.

Definition 4: is the symbol for binary operation of taking the th power of . I assume the reader is familiar with exponents and powers.

Definition 5: - is the symbol for the binary operation of subtraction. I assume the reader is familiar with subtraction.

Definition 6: is the symbol for the binary operation of division. I assume the reader is familiar with division.

Definition 7: is the symbol for the binary operation of extracting the th root of . I assume the reader is familiar with root extraction.

Definition 8: is the symbol for the binary operation of extracting the th-base logarithm of . I assume the reader is familiar with logarithms.

Finally,

Definition 9: Closure is the property that a set has when the result of the application of a particular binary operation between any two elements of a set is itself an element of the set. We can write this as if , and we apply a binary operation symbolized by , then .

Theorem 1: Principle of Induction. This can be found at this link http://mathworld.wolfram.com/PrincipleofMathematicalInduction.html.

Addition, Multiplication, and Powers

We can begin by altering the wording for the definition of . The set of natural numbers is the number zero and all larger counting numbers.

Beginning with 0 + 1 = 1, we see that this is clearly an element of .

We can call the result of increasing a natural number by one its successor.

Naturally, 0 is not a successor of any natural number.

Given an arbitrary natural number, , (or, ), we can see that its successor is also a counting number greater than 0. Thus, if , then . This satisfies Theorem 1.

In this way we can define any natural number greater than 0 to be equal to .

Now, if we have two natural numbers, we can make the definition,

Then we can write,

Thus, by applying Theorem 1 again, we have proven that the sum of any two natural numbers is itself a natural number. Thus is closed under addition.

Since  multiplication can be defined as repeated addition of a natural number, that too will result in a natural number. Thus is closed under multiplication. This is looking pretty good.

Also, taking a power is repeated multiplication of a natural number, thus the result must also be a natural number. Thus is closed under exponentiation.

Subtraction

Here is where everything comes to a screeching halt and a disaster occurs. So long as we confine ourselves to subtractions of the form, , where , we are fine. As soon as we allow we have trouble. There are no natural numbers that allow this subtraction to take place. is NOT closed under subtraction. In order for the number system to be consistent, it must be closed under the operation. is not self-consistent.

The Answer

We must either accept this disturbing lack of closure, or we must invent new numbers. Eventually we must create negative numbers. When we add the negative numbers to the natural numbers, we get a new set called the set of integers. We write it like this, . In this set, subtraction is simply addition by a negative integer. The result will always be another integer. Thus, we see that is closed under addition, subtraction, multiplication, and exponentiation.

Division

Another disaster occurs here. Once again, if as we confine ourselves to divisions of the form, where we are fine. We can even allow for remainders. As soon as we allow we have more trouble. There are no integers that allow this division to take place. is NOT closed under division, and is thus not self-consistent.

The Answer

Just as before, we need to create a new set of numbers. Here we create the rational numbers. We denote the set by and define it as, . Often rational numbers are called fractions. We can see that is closed under addition, subtraction, multiplication, division, and exponentiation. Is it also closed under extraction of roots or logarithms? Since this column is now quite long, I will have to wait until next time to complete our story. I will say that what we are doing here is constructing the number system from first principles. This is a fascinating example of a class of problems called preimage problems.

Created by Mathematica  (August 1, 2005)