SIMPLE INTRODUCTION TO MUNBER THEORY

2. Other Sets of Munbers

2.1 Review of real munbers

Recall that in Section 1.1, we defined real munbers as those munbers which contain at least one real number. The intent of this chapter is to show that any other set may be used to define a munber.

DEFINITION. Let S be a set. If a₁, a₂, ... a_n ∈ S, the object «a₁, a₂, ... a_n» is a munber of S. In particular, M_S denotes the set of all possible munbers of S.

In short, any conceivable set can have munbers constructed from it. Here, we shall review some basic properties of real munbers. They are equal to one another if and only if they share at least one common member, and they are identical to one another if and only if every member of one is a member of the other and vice versa. They may be summed by adding each member of the first to each member of the second, and likewise, the product is identical to the munber containing the products of each pair of members. Intersections and unions may be taken in the same way as for sets. The operations of addition and multiplication satisfy the six field axioms, but they do not satisfy any order axioms. In the following Sections, we will examine some other sets of munbers that use the same definitions for addition and multiplication.

2.2 Integer munbers

In Exercise 6 of Section 1.4, we defined integer munbers. M_Z, the set of all integer munbers, is continuous through addition, subtraction, and multiplication. That is to say, if we add, subtract, or multiply any two integer munbers, we will obtain more integer munbers. Subsets of M_Z can be defined such that they have certain properties.

DEFINITION. Let a ≡ «a₁, a₂, ... a_n» be an integer munber. Then we say a is closed from a₁ to a_n iff ∀ k ∈ Z⁺, k < n, a_k + 1 = a_{k + 1}.

If a closed integer munber is graphed, it exhibits a point at every integer from its minimum to its maximum, such as «-2, -1, 0, 1».

THEOREM 2.1. Let a ≡ «a₁, a₂, ... a_n» be a closed integer munber. Then a is symmetric about the arithmetic mean of a₁ and a_n.

The proof of this theorem will be left as an exercise for the reader. As a final note on closed integer munbers, note that by their nature, they may be expressed with only the endpoints; ie, the above example becomes «-2 ... 1».

In later chapters, we shall see how integer munbers may be used with sequences and serieses [in munber theory, the plural of series is serieses]. For now, however, we proceed to another type of munber.

2.3 Rational munbers

Just a few paragraphs ago, we stated that integer munbers are continuous through addition, subtraction, and multiplication. So what happens when an integer munber is divided by another? Clearly, it must be analogous to dividing integers by one another. This introduces us to M_Q, the set of all rational munbers. The same facts that hold for numbers also hold for munbers, ie, M_Z ⊂ M_Q ⊂ M_R. Rational munbers add the capability of continuity through division, permitting us to do more computations involving munbers. In the terminology introduced in the last chapter, the reciprocal of a given integer munber is probably not an integer munber, but it is a rational munber.

On the whole, rational munbers are not all that interesting in comparison with other sets of munbers. This is mainly because anything that can be accomplished with rational munbers can just as easily be accomplished with the full blown set of real munbers; like their numeric counterparts, rational munbers are not worth very much. Thus, we proceed right on into the next, more fun set of munbers.

2.4 Complex munbers

Complex numbers were developed in order to solve equations such as x² = -1. As we all know, they have been remarkably successful at this task. Sometimes, though, there are times when we want to be able to manipulate more than one complex number simultaneously, and sets just won't get the job done. Thus, we are saved by complex munbers, M_C.

As Gauss proved, any polynomial has a distinct number of solutions, all of which are elements of C. So, it is very easy to represent the solutions of any real or complex polynomial as a complex munber containing those solutions.

DEFINITION. Let a be a complex munber, and let p(x) be a polynomial. Then we say a is a muneric root of p iff ∀ z ∈ C, z = a iff p(z) = 0.

THEOREM 2.2. All polynomials have exactly one muneric root. In particular, the degree of that muneric root is bounded above by the degree of the polynomial.

Proof. We shall prove this theorem in our favorite method, pretending there are two and then showing that they're identical. Suppose a and b are both muneric roots of p(x). We shall call the numeric roots of p z₁, z₂, ... z_n. It's easy to show that z_k = a and z_k = b for any positive integer k ≤ n. Again by the definition, any other complex number y satisfies y ≠ a and y ≠ b. Thus a ≡ b, meaning that any polynomial can indeed have only one muneric root. To prove that its degree is less than or equal to the polynomial's degree, we note that the number of roots of a polynomial is at most its degree, making it evident that the muneric root's degree is the number of roots.

Complex munbers may be used most handily with the roots of unity problem. The muneric fourth root of one, for example, is «1, i, -1, -i». In fact, munbers define a new way to express polynomials. Any polynomial may be written as its muneric root, although this method has its shortcomings. This topic will be addressed both in the following set of Exercises and in a later chapter.

2.5 Exercises

1. Of M_Z, M_Q, M_R, and M_C, determine to which set or sets the given munbers must belong.
   

   (a) «0, 1, 2, 2.5»	(c) «e, e + 1»
   (b) «-1, 1, -i, i»	(d) «e-πi, e2πi»

2. Let y, z ∈ M_Z, p, q ∈ M_Q, r, s ∈ M_R, and c, d ∈ M_C Determine to which set or sets the given munbers must belong.
   

   (a) y/z	(e) r + 2s
   (b) pr	(f) r + p
   (c) yz/z	(g) c/(sy)
   (d) yz + c	(h) z + 1

3. Repeat Exercise 2, but determine to which set or sets the given munbers might belong.
4. Let m and n be integer munbers, and define a complex integer munber as a munber whose members can all be expressed in the form x + yi, where x, y ∈ Z.
   (a) Show that m + ni is a complex integer munber.
   (b) Suppose that for integers a, b, c, and d, we have a + bi = m + ni and c + di = m + ni. Does this imply that a = c or that b = d? Explain.
5. Using the definitions of symmetry, integer munbers, and closed integer munbers, prove Theorem 2.1.
6. Find the muneric roots for the following polynomials.
   

   (a) x² - 2x	(c) x² + 1
   (b) x³ - 12x² + 48x - 64	(d) x4 - 16

7. Rookie munber theorists often make the mistake of thinking that the muneric root of a sum of polynomials is equal to the sum of the muneric roots. Why is this not the case? Give a counterexample and explain in general.