Algebra/Theory

The Natural Numbers

Mathematics requires, in order to avoid confusion or absurdity, an unambiguous definition of vocabulary. While this is true of any science, in mathematics this is achieved absolutely through the abstraction of concepts.

However, the full description of math as above requires time, and is nowhere elementary. Also, in order to have a true axiomization, a math built up from the roots, we will have to use the strongest, in the logical sense, statements possible. This makes them powerful for proofs but often sacrifices intuitiveness. Thus, we will concentrate on the later, only rarely using rigour, as necessary.

To do mathematics we must start by finding a way to think about numbers that is as above, unambiguous, while at the same time obvious and natural. We thus come up with the natural numbers.

Intuitively, we define these as the set N = { 1, 2, 3, ... }. We will say a number is a natural number if it belongs to this set. Soon we will find this needs to be expanded even for elementary treatment, but this set alone already has some interesting properties.

If there is a set A such that if we pick an arbitrary number, call it x, and x ∈ N, we say A ⊆ N. In words, A is a subset of N.

By x ∈ A we mean that x is "in" A, or that x is an element of A.

While we have not defined properly a set, membership or inclusion, we have already a feel for what N ought to be like.

Mathematical induction

Principle of mathematical induction

Let A ⊆ N with the following properties:

(i) $1\in A$

(ii) $n\in A$ implies $n+1\in A$

Then A = N

Remark

The principle of mathematical induction is implied by the 'least element principle' (2.1). For, assume that 1.1 holds and let B be the set of all elements of N not in A. If B has any elements (that is, is not empty) then it must have a least one. Call this least element n. Then, by (i), $n\neq 1$. Thus n - 1 is a natural number and is not in B and so is in A. By (ii), $n=(n-1)+1$ is a natural number which is in A. But now n is in both A and B which is impossible. So B must have been empty. That is, A = N.

This serves as a fundamental property of the natural numbers, and we, further defining order, will call these numbers well-ordered, primarily because of this principle. In more rigorous terms, we would have to identify a separate axiom, called the axiom of induction, which encompasses this property. However, for our current purposes, this is sufficient.

Principle of mathematical induction (modified)

Let A be a set of natural numbers which has the following properties:

(i) The natural number $n_{0}$ is in A

(ii) If $n\geq n_{0}$ and $n\in A$, then $n+1\in A$

Then A contains all natural number $n\geq n_{0}$

Numbers

In this section, we shall describe some of the more commonly occurring types of numbers together with some of their properties.

Types of Numbers

Natural Numbers, N

As already mentioned, these are just ordinary counting numbers 1,2,3,4,...,29... . Note that zero has not been included in this set. This varies with different books or mathematicians and may include zero as a natural number.

An important property of the natural numbers is the ordering. Note that natural numbers come with an idea of size so that we can talk about larger and smaller natural numbers.

Integers, Z

By the integers, we mean the natural numbers together with their negatives and zero. Although we all, presumably, have a reasonably clear practical idea of how to work with the integers, there are definite problems as to what they actually are. Referring to such things as the 'number line' does not solve these problems because it relies heavily on our intuition.

One way that mathematicians solve this problem is to make an artificial construction which produces an artificial object with the properties we expect of the integers. Although we do not use this in everyday mathematics, it gives a precise definition which we can use to justify our use of negative numbers and also provides a model we can use when we wish to develop quite new constructions which are not so intuitively reasonable. We shall not describe this for the integers but will give a brief description next of a similar process starting with the integers and producing the rational numbers.

Rational Numbers, Q

With the integers, we have a number system which is closed under addition, multiplication and subtraction. The next step is to produce a collection numbers which is also closed under division (except by zero). This is the rational numbers. Again we should all be able to manipulate rational numbers but there is some problem with what they actually mean. For example, what does it mean to say that ${\frac {2}{3}}={\frac {4}{6}}={\frac {6}{9}}$? We usually want the equals sign to denote the fact that two things are identical and the symbols ${\frac {2}{3}}$ and ${\frac {4}{6}}$ are certainly not identical.

So we redefine what we mean by equality of fractions. Let us consider the set of ordered pairs of integers with the second integer non-zero; that is,

$\mathbb {Q} '=\{(x,y):a,b\in \mathbb {Z}$ and $b\neq 0\}$

(An ordered pair is just a pair in which it is specified which comes first and which second.) We want these ordered pairs to represent rational numbers but on a many-to-one basis. So we redefine equality of these ordered pairs by

$(a,b)\equiv (c,d)$ if and only if ad = bc

(Read this as (a,b) is equivalent to (c,d).) This corresponds to the usual definition of equality of fractions. We can then think of one rational number as being represented by a collection of ordered pairs of integers all equivalent to each other. Two ordered pairs represent the same rational number exactly when they are equivalent. WE can then go on to define, in terms of the ordered pairs, the usual operations of addition, subtraction, multiplication and division. For example,

(a,b) + (c,d) = (ad+bc,bd)

We will define addition of rational numbers by taking an ordered pair for each number and then adding these ordered pairs. There is still a problem however. We need to be sure (without checking each case) that, for example, because $(2,3)\equiv (4,6)\equiv (6,9)$, then also

$(2,3)+(5,6)\equiv (4,6)+(5,6)\equiv (6,9)+(5,6)$

What is order?

Again, as we try to find a way to define, based on the already established ideas of addition, and the intuitive idea of order, in what sense is a number larger than another. In what way can we arrange the numbers?

With N this is easy, we say, if a, b are natural numbers, a < b if there is a natural number c such that a + c = b. From the principle of induction, we can thus order N in the following manner: 1, 2, 3, ... As has been done before with our intuition.

With Z, we run into a problem, where do we start? Since the principle of induction does not apply on Z as a whole, we have to write it as ..., -1, 0, 1, ..., leaving "..." on both sides. However, the definition, as before, applies.

With Q, clearly ${\frac {1}{2}}$ > ${\frac {1}{3}}$, yet ${\frac {1}{6}}$ is not in N and so our definition fails. The new definition would be

Definition: (Order on Q) Let a, b be rational numbers, then a < b if there is a positive rational number c such that a + c = b.

We call a rational c positive if it is equivalent to a rational ${\frac {n}{m}}$ such that n and m are both natural. In other words, if c = ${\frac {p}{q}}$, then c is positive if pq is a natural number. A simple exercise would be to show these are equivalent. (Note: Statements are equivalent if each implies the other.)

Any set such that for each two elements in it we can definitely say a < b, a > b, or a = b is called totally ordered.

The least element property

Any non-empty subset of the natural numbers has a least element. The words 'any non-empty subset' mean that the subset we take should have at least one element in it.

Any set with the least element property is called well-ordered. Thus N is well-ordered as above, while Z and Q are not.

Order properties of natural numbers

(i) There is no largest natural number.

(ii) There are natural numbers which are nearest neighbours in the sense that there is no natural number strictly between them.

Proofs

Definition

A proof is simply a mathematical argument designed to convince the reader of some fact. However an intuitive proof differs from an inductive proof. The former being a fact that can be assumed to be an observable constant from daily experiences (e.g. when two test tubes of milk are poured into one beaker, then if we divide the final volume again into the two beakers, we should get full test tubes again (almost). This explains why ½+½ will still be 1. Since this follows from intuition/observation/experience alone it can be considered an intuitive proof.). The latter, inductive proof, is of the type where many intuitive proofs can be combined to get the desired result. For example, we know that pouring two test tubes of milk gave us two units in the beaker, so if the volume of the test tube can be regarded as a unit volume, then n such test tubes will give us a total of n units of milk in the beaker. Mathematical proof itself is based on natural logic and philosophy. Modern mathematics generally assumes most proofs to be absolute, not without reason though. This allows them to engage in complex problems using purely inductive reasoning. However the basis for all that still remains natural philosophy and logic. This is reflected in Newton's title to a relatively abstract mathematical treatment of his work being titled as 'Principles of natural philosophy'.

Mathematical terms

and - 'A and B' means that both A is true and B is true

not - This has the usual meaning in English; note that not(not(A)) is the same as A

or - This is always what is called 'inclusive or'. That is 'A or B' means that either A is true or B is true or both.

for all - This has the usual meaning in English

there exists - This has the usual meaning in English

implies; if-then - These kinds of statements are at the centre of mathematical reasoning.

if and only if; equivalent - 'A implies B' and 'B implies A'

Theorem, Proposition, Lemma - They all mean roughly the same thing but are in decreasing order of importance.

Corollary - Something which follows easily from a Theorem but for which the statement is not entirely obvious from the statement of the Theorem.

Summation Notation

The summation notation is a convenient abbreviation for sums of several real numbers. If $a_{1}$,$a_{2}$,...,$a_{n}$ are reals, we define

$\sum _{k=1}^{n}a_{k}=a_{1}+a_{2}+...+a_{n}$

The summation index k is often called a dummy index, as it can be replaced by any other letter:

$\sum _{k=1}^{n}a_{k}=\sum _{i=1}^{n}a_{i}=\sum _{j=1}^{n}a_{j}$, etc.

Sometimes it is convenient to start summation from 0 instead of 1, or from some other integral value. For instance,

$\sum _{k=0}^{3}x_{k}=x_{0}+x_{1}+x_{2}+x_{3}$, or $\sum _{j=2}^{5}j^{3}=2^{3}+3^{3}+4^{3}+5^{3}$, etc.

The most important properties of the summation notation can be summarized as

$\sum _{k=1}^{n}(a_{k}+b_{k})=\sum _{k=1}^{n}a_{k}+\sum _{k=1}^{n}b_{k}$ (additive property)

$\sum _{k=1}^{n}(ca_{k})=c\sum _{i=1}^{n}(a_{k})$ (homogeneous property)

$\sum _{k=1}^{n}a_{k}=\sum _{k=0}^{n-1}a_{k+1}=\sum _{k=2}^{n+1}a_{k-1}$ (index translation)

$\sum _{k=1}^{n}(a_{k}-a_{k-1})=a_{n}-a_{0}$ (telescoping)