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2. How to make the definition

Mathematical notions are often defined by detailed descriptions, which usually depends on the shape or appearance. For example, a polynomial is an expression of the form p(t) = a0 + a1t + a2t2 + ... + antn, a matrix is a rectangular array of numbers, the sine function is the ratio of two edges in certain triangle, etc.

We would like to define a concept leading to the phenomenon shared by the three examples. The problem is that the phenomenon occurs to objects that may have very different appearance. Therefore the concept cannot be defined by detailed description of the appearance.

To illustrate the difficulty, think about how you would define the concept of numbers. For example, you cannot define natural numbers simply as the collection N = {1, 2, 3, ...} because that would exclude, by the appearance, the words "one", "two", "three" as numbers (even if you also include these, what about "un/une", "deux", "trois", .., or Chinese, Japanese, binary expression, etc). A more sensible (but not yet mathematical) definition of natural numbers would be any symbolic system used for everyday counting. Note that this is a definition in terms of the property (counting) instead of the appearance.

The natural numbers were found to be inadequate long time ago, so that integers Z, rational numbers Q, real numbers R, complex numbers C, etc., were invented. With the introduction of more sophisticated number systems, the question of the meaning of numbers arises again. A naive definition of numbers may simply list of all the number systems invented so far. This approach has two problems. However, the list will be changing as new number systems are discovered, so that the definition is not absolute or permanent, as is required by mathematics. The right definition, then, still lies in the properties that characterize numbers.

What properties do we expect the numbers to have? Naturally, numbers can be added and multiplied. Moreover, these operations must be reasonable, in the sense that they must have some nice properties such as a + b = b + a. In summary, the definition of numbers may go like the following three steps.

Terminology We have a set (or collection) N, called number system. We also call the elements of N numbers.

Operation Given any two numbers a, b, we can produce numbers a + b, ab, called addition and multiplication of the two numbers.

Usually we denote the operations by

a + b: N×NN, ab: N×NN,

where N×N means that the operations start with pairs of numbers, and → N means that the results of operations are single numbers.

Properties The two operations has the following properties.

The rigorous definition and the theory of numbers can be found here.

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