teaching:topics:number:axioms-formal
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teaching:topics:number:axioms-formal [2024/03/14 12:21] simon [number axioms] |
teaching:topics:number:axioms-formal [2024/05/02 09:44] (current) simon [using 'sets' to describe the properties of numbers] |
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| more sophisticated and applying them to the world in new ways. We do some operations with them including | 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`. | + | `quad a+b ,quad a-b ,quad ab ,quad a/b quad` and `quad a^b`. |
| 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. | 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. | ||
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| most exotic kinds of measurement and numbers we have invented at the time. | 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. | + | <WRAP #science/>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. | 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. | ||
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| ===some set notation, as used here ...=== | ===some set notation, as used here ...=== | ||
| * a set is a collection of distinct elements, `a=b` means they both represent the same element | * a set is a collection of distinct elements, `a=b` means they both represent 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` | + | * `\ b in ccN quad` means:`quad`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 | * 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` | + | * `forall b in ccN quad` then means:`quad`for all `b` in `ccN` |
| - | * in other words ... "for every possible `b` that we could choose from `ccN` ..." | + | * in other words:`quad`"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` | + | * `exists c in ccN quad` means:`quad`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). | + | * in other words:`quad`"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" | + | * `quad : quad` means`quad`"such that" |
| * it can often be read as "given" or "where" | * it can often be read as "given" or "where" | ||
| - | * in other words: '... with the following conditions ...' | + | * in other words:`quad`'... with the following conditions ...' |
| - | * so rule \(\eqref{multiply closed}\) ... `quad forall a, b in ccN qquad exists c in ccN : quad c = a × b`\\ reads as:`quad`for every `a` and `b` in `ccN` there is some `c`, also in `ccN`, where `\ c = a × b`. | + | * so rule \(\eqref{multiply closed}\) ...`quad forall a, b in ccN qquad exists c in ccN : quad c = a × b`\\ reads as:`quad`for 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" | * "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" | + | * this is what we mean when we say:`quad`"`ccN` is //closed// under multiplication" |
| * some logic operations: \(\ \therefore\,\implies \land\ \lor\ \neg \quad\)mean: therefore, implies, and, or, not. | * some logic operations: \(\ \therefore\,\implies \land\ \lor\ \neg \quad\)mean: therefore, implies, and, or, not. | ||
| * some more set notation: \(\ \cap\ \cup \subset\ \emptyset\ \notin \quad\)for: intersection, union, subset, the empty set, not in | * some more set notation: \(\ \cap\ \cup \subset\ \emptyset\ \notin \quad\)for: intersection, union, subset, the empty set, not in | ||
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| =====logic, mathematics, the world and philosophical considerations===== | =====logic, mathematics, the world and philosophical considerations===== | ||
| - | Here we have been considering the properties of numbers and the structures of collections of number-like | + | <WRAP #philosophy/>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. |
| - | 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 | The description of numbers presented here is quite distinct from any attempt to formalise mathematics | ||
teaching/topics/number/axioms-formal.1710379295.txt.gz · Last modified: 2024/03/14 12:21 by simon
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