authored by Premmi and Beguène
Introduction
Number theory involves an extensive study of the properties of integers. Before embarking on this study, it is only natural to ask the question: “What is a number?” since integers are just one type of number.
Intuitively, we often associate numbers with counting objects like books and pencils or measuring quantities such as height, weight and distance.
Counting generally begins with 0, which represents the absence of an object, and then proceeds to 1, 2, 3, \ldots, which are the numbers used to count objects. For example, we denote that we don’t have any books by saying “0 books”, we have one book by saying “1 book”, we have two books by saying “2 books”, and so forth.
Thus far, we have denoted the numbers used in counting as 0, 1, 2, 3 and so on. These numbers, called natural numbers, form a highly non-trivial set because we cannot list every single one of them since they go on forever. Therefore, we would need a different approach to describe them.
There is also another problem with such a concrete representation of numbers. While we can think of numbers like 1, 2 \text{ and } 3 as representing the count of certain objects, how do we reason about numbers like -3, \frac{4}{3} \text{ and } \sqrt{2} ?
To reason about the variety of numbers encountered in mathematics, we need a unified framework that describes numbers abstractly, as mathematical objects satisfying specific properties and having certain operations defined on them, regardless of their concrete representations. Such a framework is known as the axiomatic representation of numbers.
For example, under this framework natural numbers could have various concrete representations like \{0, 1, 2, 3, \ldots \}, \{O, I, II, III, IV, \ldots \} or \{\text{One, Two, Three,} \ldots \} and yet, all these representations would be considered equivalent if they all have the same properties and operations defined on them.
Our axiomatic study of numbers will start with a small list of axioms (an initial set of true statements that cannot be proven) pertaining to natural numbers. These axioms are based on Peano’s Axioms, which were formulated by the Italian mathematician Giuseppe Peano in 1889. A natural number is defined as any mathematical object that satisfies these axioms.
Next, with these axioms as foundation, we will define arithmetic operations such as addition and multiplication on natural numbers and prove all the properties resulting from these operations, for instance, commutativity and associativity, among others.
We will conclude our discussion on natural numbers by introducing the notion of ordering on them.
With the axiomatic definition of natural numbers as our foundation, we will proceed to define integers, rational numbers and finally, real numbers. Similar to our discussion on natural numbers, we will also define operations for each of these number systems.
By the end of our discussion, we will develop a deep appreciation for how the entire edifice of numbers, starting with natural numbers as a base and leading up to other number systems including real and complex numbers, can be constructed from just five axioms pertaining to natural numbers.
Let us now proceed with the axiomatic definition of natural numbers.
Next Topic: An Axiomatic Study of Natural Numbers – Peano’s Axioms