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teaching:topics:number:axioms-formal

## a more formal way to define numbers

Mathematicians have used different collections of numbers for thousands of years, gradually making them more sophisticated and applying them to the world in new ways. We do some operations with them including

quad a+b ,quad a-b ,quad a×b ,quad a/b quad and quad a^b quad.

We make sure these operations that we define with numbers are consistent as we extend the kinds of numbers we use. Much later in mathematics (especially beyond school) you might study these ideas very closely. Some school students — those who like to think abstractly and have that particular kind of imagination and creativity — find these ‘Pure Maths’ ideas a very good way to understand the ideas in school math. Many (most!) students find it all a bit scary and prefer more practical examples. This means lots of practice is needed to replace thinking deeply and eventually it all clicks. We hope.

It is important to remember that historically these very formal definitions came a long time after the ideas they define were invented and you do not need them to do the routine maths, but after you get the general idea it can help you understand what is happening in a much deeper way — and so make much better use of that maths in the world (and in the more difficult problem questions you get in tests and exams). This is the way that those original ideas are extended and become a powerful tool, and a way of thinking and communicating.

### using 'sets' to describe the properties of numbers

We can propose rules to construct a set which we will call cc N of number-like elements with the operations + and × between them. Next we can show that some, or all, of the 14 properties listed below are true whenever those rules are true. We need some other important properties including what equality and order mean — that is what we mean by =, < and >. But essentially we need to draw a line somewhere — it is actually logically impossible to give a complete set of self-contained rules like this, we must leave some things as ‘just understood intuitively’ and we can always pose unanswerable questions within any system like this. This conclusion was proved almost 100 years ago.

We can make interesting and useful sets of things where only some of these properties hold or we can make finite sets of things that otherwise have all the properties of, say, the integers or rationals. At school we are mostly looking at either whole numbers (dealing with discrete things — that we can count) or fractions and real numbers (dealing with continuous things — quantities we can measure like length or weight). The difference between these two kinds of number is profound, and starting to understand that difference is a crucial step in school maths.

In this formal definition we see that by including reciprocals (rule (12)) along with the other rules we get the Rational Numbers (all the fractions), and that with these new numbers we can always divide any quantity into smaller parts. With these numbers we can describe any measurement as accurately as we want by making the denominator of our number big enough.

Whole fields of Mathematics work on these ideas, and applying them in new situations. Much of Physics involves applying this mathematics to measurements in the world around us, usually the wildest and most exotic kinds of measurement and numbers we have invented at the time.

Often the behaviour of these abstract, mathematical number-sets seems to match patterns we observe in the world around us. Then we work with those observations assuming the measurements behave like that set, like one of our number systems — that means we use all the familiar methods and theorems that we know about that kind of number.

We use our familiar algebra techniques to help solve problems or make predictions about that aspect of the world. We can compare our calculated results with what we observe — perhaps straightforward measurements of a particular property, or perhaps something much more exotic. If those predictions are correct then the idea that these things we are observing really do behave like the mathematics we are using as a model gets stronger. Science is mostly about exploring, testing and extending these models and seeking more observations in a wider range of situations. Technology, and analysis in a vast range of fields, depends on a working familiarity with these mathematical models.

Here is a very formal summary of some of the most basic rules, using notation that might be new to you …

#### some set notation, as used here ...

• a set is a collection of distinct elements, a=b means they are both names for the same element
• \ b in ccN quad means: we are giving the name b to “an element of” (that is something “in”) the set which we are calling ccN
• this is exactly what we do when we use a pronumeral in algebra
• forall b in ccN quad then means: for all b in ccN
• in other words … “for every possible b that we could choose from ccN …”
• exists c in ccN quad means: there exists something we will call c in ccN
• in other words … “we can always find some suitable element c in this particular collection called ccN …” (usually with some property that will be given given next).
• quad : quad means “such that”
• it can often be read as “given” or “where”
• in other words: '… with the following conditions …’
• so rule (5) … quad forall a, b in ccN qquad exists c in ccN : quad c = a × b
reads as:quadfor every a and b in ccN there is some c, also in ccN, where \ c = a × b.
• “whenever we multiply two of these numbers the result is also one of these numbers”
• this is what we mean when we say: “ccN is closed under multiplication”

## logic, mathematics, the world and philosophical considerations

Here we have been considering the properties of numbers and the structures of collections of number-like things. This helps us build and work with mathematical models of systems and patterns we see around us. Numbers, and mathematical representations of certain ideas, are part of language. They have been part of human communication, reasoning and decision making since well before any recorded history.

The description of numbers presented here is quite distinct from any attempt to formalise mathematics within some form of abstract, symbolic logic. It is also not especially concerned with the details of implementing calculating, reasoning or decision-making machines. It does present a structure which any symbolic representation or machine implementation must provide if it is to be able to work with this idea we call number, an abstraction we have worked with and developed over many millenia.

Strategies to make a symbolic logical base for mathematics were pursued especially vigorously around 1900. Taking the idea of a Set as the starting point then building everything including numbers from there was an influential example. Another powerful strategy is to start with a Function and construct everything from there. A more recent path has developed the idea of Categories as a way of representing mathematical structures.

The philosophical questions “What is the relationship between truth and mathematics?” then “What is mathematics?” and “What is truth?” have been driving some of this quest. It is a fascinating topic, but not the one addressed here.