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Numbers, an introduction
In the following quite many terms will be used that might be unfamiliar to you. They will though be defined in the pages following (in due time).
There is several different uses of the word “numbers”. Some use it as a term of all the elements or members of a set on which we do “mathematical” manipulations, some use them for integers only, but I prefer to reserve the word for Natural numbers, and the accepted “natural” extensions you get by trying to solve equations created by operations on the previously defined types of numbers, and to non trivial subsets of these sets of numbers. With “accepted natural extensions” I mean Integers, Rational numbers, Real Numbers, and Complex numbers, and by “non trivial subsets” I mean for example Irrational Numbers, Algebraic Numbers and Prime numbers.
We have one class of numbers not included at first, and that is the transfinite ordinals and cardinals, but in the set theory section we will deal with these as well.
Vectors, tensors, matrices, elements of groups, etc. is not what I would call natural extensions of the Naturals. They are more of “useful elements of useful sets”, and are often invented to make certain type of problems more easy to solve, and in the process often creating even more interesting problems.
The above is not a clear cut definition, but rather an opinion, but an opinion that I think is not totally arbitrary. (Then there are things like p-adic numbers and so on too … but anyhow.)
In the following I will try to give an idea of why the concepts of these types of numbers where thought of to begin with, and give a hunch of how to define arithmetic’s on them. Some of this might seem ridiculous if you have never seen maths at this level and type before. You might think “Of course a+b equals b+a“. But a mathematician, especially if interested in logic, will not settle with this. It needs to be proven from well chosen simpler or more more extensive “facts”.
The proofs may also look a bit silly, but I am trying to only use the tools defined as we go. We can often device “simpler” proofs, but then using more complicated – and “yet not proven” – tools.
In the beginning of the 20:th century this was a big subject, and everything was going along just fine until Gödel punched a hole in the bubble in the 1930’s. He showed that maths can never be completely described in a reductionistic way. There will always be mathematical truths not provable within a finite mathematical system. Fortunately these truths are hard to find, or at least seldom encountered, and we can see most of the maths used safely tucked under the wings of the queen of the sciences.
What then do we mean by proving that a+b=b+a? To prove such a thing is to show that it will be a theorem under the chosen formal system, and the system we will use at first is Peano’s axioms under a simple set theory, and later on, the Zermelo-Fraenkel’s axiom system.
As one is trying to extend maths one is trying to keep as many rules of the previous type as possible. And of course the rules of one type of numbers must be inherited to the “same” numbers of the types above. Operations like +,-– and > have a certain meaning for the natural number 3, and the same meaning must be inherited to the integer 3, the rational number 3, the real number 3 and the complex number 3. This is of course the reason we choose to use the same symbol for all these numbers.
The number 3 will get new qualities at each step. The integer 3 is a positive integer, but the natural number 3 is neither positive nor negative, since that term would not yet defined if we only had natural numbers. The rational number 3 is having a denominator of 1 (3=3/1) , a property neither naturals nor integers have. The real number 3 is larger than e, and e is not even defined using rational numbers only, and finally, the imaginary part of the complex number 3 is equal to 0, but the real number 3 does not even have an imaginary part is seen as a real number only.
At each stage the number 3 is burdened with more and more properties (and more and more information is given by saying “3”), like getting a sign, a denominator, a position in the continuum, a imaginary part. If it was not so, why bother extending the definition of 3?
This property of having the same qualities is examined in the abstract algebra, where sets and operations with similar qualities are put together in groups, fields, modules, rings etc. More about that in the pages about Abstract Algebra (that will be added later) . I will sometimes refer to such topics, and if so I will try to put a link to an appropriate place.
One normally define “=” to be true if and only if RHS (right hand side) and LHS are referring to (or are names to) the same object. But 3 could possibly be a lot of different things. The naming convention (giving the name ‘3’ to all of these objects) is possible because of this inheritance of properties of previous levels. Because of this we extend the property of “=” by defining the natural 3 to be equal to the integer 3, to the rational number 3, to real number 3 and to the complex number 3. By this rule we can talk about the imaginary part of the natural number 3, because of its equality to the imaginary number 3.
We can not blindly do this operation of equalizing these meanings of “3”. We must ensure that the properties of the number remains as we define new types of numbers, and this is partly what we will try to do now.Up a level : Algebra and Arithmetic
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