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Contents
This is not a first introduction to number, but a discussion and interpretation with links to further resources and an emphasis on algorithmic aspects ({{!chutil.a("r.cp")}}).
The natural numbers are the real numbers in the sense that they represent a count. All other sets of numbers have additional information or are quite different altogether
The count is a real variable, which specifies the cardinality of a set. A value of this variable, like three, means three things. Further properties are not considered, they are abstracted away. That is why every natural number can be seen as equivalence class ({{!chutil.a("r.co")}}).
Note
In mathematics one makes a further distinction: cardinal numbers as above and ordinal numbers to specify the order.
The intuitive idea of \(\mathbb{N}\) is formalized with the Peano Axioms.
Note
Essentially \(\mathbb{N}\) is a construction of an ordered multitude (0 with successors), to address values of other variables (like count), just like words address concepts.
Zero was a great invention for the number representation, which the roman system did not yet have. In general one can now include variables, even if they are not there every time. This often helps to generalize description. As an example in 103 the position coded tens variable is there, even if there is no ten grouping in the number ({{!chutil.a("r.cn")}}). Another example are vectors ({{!chutil.a("r.cg")}}).
\(\mathbb{Z}\) is more than the count.
As motivation for the integers one can add to every element of \(\mathbb{N}\) a process or a direction. 2 is then not only the count 2, but has the additional information to add the 2 things (\(2 = {2,+} = +2\)).
If you understand \(\mathbb{N}\) already associated with a process or direction, then it is a good idea to extend this to include the same values but the undoing process or counter direction to get back to the original situation. This way one intuitively comes to the integers \(\mathbb{Z}\).
\(+\) means to add and \(-\) means to subtract, but that can change. The \(+\) is often dropped, but it must be implicitly assumed, because an integer is not the same thing as a natural number. It has additionally the reversible process or direction.
If you understand only count with \(\mathbb{N}\), then \(\mathbb{Z}=\mathbb{N}\times\{+,-\}\). Then \(\mathbb{N}\) is not a subset of \(\mathbb{Z}\), but an isomorph embedding.
Note
An formal introduction of \(\mathbb{Z}\), starting from \(\mathbb{N}\) with as few new concepts as possible (no \(+\) und \(-\)), is via equivalence classes ({{!chutil.a("r.co")}}) of number pairs \((n,m)\) from \(\mathbb{N}\) with the equivalence relation: \((n_1,m_1)\sim(n_2,m_2)\equiv n_1+m_2=n_2+m_1\). In the canonical representation one number is 0. \(+2 = (2,0)\) und \(-2 = (0,2)\).
With \(+\) and \(-\) as opposite processes one has encoded this process in the number. The \(Addition\) itself is then algorithmically a sequential execution or sequence: \(+2+(-2)\) means: add 2 then(=+) subtract 2. If you swap the numbers \(((-2)+(+2))\) you get the same result (commutative law). With more numbers you can choose freely which to calculate first \((-2)+((+2)+(+3))=((-2)+(+2))+(+3)\) (associative law).
Note
The subtraction 2-2 is an abbreviation for +2+(-2).
The result of +2+(-2)= 0, the neutral element of addition. +2 is the opposite number (additive inverse) of -2 and vice versa. \((\mathbb{Z},+)\) is an abelean Group ({{!chutil.a("r.cl")}}).
Note
\(+\) as part of the number and \(+\) as binary operation are not the same. Similarly for \(-\). \(-\) in addition can be a unary operation that returns the opposite number (additive inverse).
A process can be repeated and multiplication says how often addition (+2) or subtraction (-2) is repeated. Algorithmically multiplication is a loop:
\(3\cdot(-2) = (-2)+(-2)+(-2)\)
Multiplication with 1 means the thing itself. 1 is the neutral element of the multiplication.
The multiplication with -1 means: revert the process, i.e. make plus (+) to minus (-).
\((-1)\cdot(-2) = +2\)
\((-1)\cdot(-1)\cdot(-2) = -2\)
With this one can multiply every integer with every other integer and one gets an integer again.`(mathbb{Z},cdot)` is closed and the assoziative law holds. This makes \((\mathbb{Z},+,\cdot)\) to an integrity ring ({{!chutil.a("r.cm")}}). \((\mathbb{N},+,\cdot)\) alone is only a semiring ({{!chutil.a("r.cm")}}) .
Analogous to \(\mathbb{Z}=\mathbb{N}\times\{+,-\}\) one can think the repeating process united with the number and it is a good idea to include the inverse process (dividing).
Which subtraction do I need to repeat 3 times in order to get a (-6) subtraction?
(-6)/3 = -2
Analogous to \(\mathbb{Z}=\mathbb{N}\times\{+,-\}\) one can unite \(\mathbb{N}\times\{\cdot,\div\}\) with the count multiplication and division.
The binary operations \(\cdot\) and \(+\) must be handled separately, only the distributive law ties them together.
\(a\cdot(b+c) = a\cdot b + a\cdot c\)
e.g. \(2\cdot(3+4)=2\cdot 3+2\cdot 4=14\)
If you look for the part that repeated (multiplied) yields no change, i.e. 1, then we get to the reciprocal, which is the inverse element of multiplication.
While with (-6)/3 we still get a whole number, i.e. a multiple of 1, this is not the case for the reciprocal in general.
Therefore the set is extended by these reciprocals to make it closed. This is analogous to the extension from \(\mathbb{N}\) to \(\mathbb{Z}\).
There the process “add” was united to form a tuple (count,add). “add” has a reverse process “away”. One has extended by (count, away).
With \(\mathbb{Q}\) one extends (count,repeat) with the reciprocal (count,divide).
Note
In analogy to \(\mathbb{N}\times\{+,-\}\) one could write \(\mathbb{Q}\)-elements as:
\(\cdot 2\) corresponds to +2
\(\div 2\) corresponds to -2
The binary operation \(\cdot\) then is only a successive processing and can be dropped.
\((\cdot 2)\cdot(\div 2) = \cdot 2\div 2 = 1\)
But actually we write
\(2\cdot 2^{-1} = 1\) or
\(2\cdot \frac{1}{2} = 1\)
the first is because one can add the exponent for the same basis and so we have \(2\cdot 2^{-1}=2^1\cdot 2^{-1}=2^{1-1}=2⁰=1\).
\((\mathbb{Q},\cdot)\) is a abelean Group with neutral element 1.
Because the multiplication in \((\mathbb{Q},\cdot)\) shall yield an element of \((\mathbb{Q},\cdot)\) again (closure), one takes all fractions \(p/q=pq^{-1}\) into \((\mathbb{Q},\cdot)\). 3/2 means to first do \(\cdot 3\) and then \(\div 2\) (reciprocal of 2).
\(\frac{3}{2}=3\cdot 2^{-1}=3\frac{1}{2}=\frac{1}{2}\cdot 3=2^{-1}\cdot 3\)
\(pq^{-1}\) means to copy/repeat p times then divided q times. To additionally multiply r times and undo that by dividing r times, one doesn’t change a thing.
\(pq^{-1}=rr^{-1}pq^{-1}=rp(rq)^{-1}=\frac{rp}{rq}\)
All such pairs of numbers are equivalent and the canonical representation is with p and q without common divisor.
Hinweis
\(\mathbb{Q}\) formally is introduced as set of equivalence classes of such equivalent number pairs: \((n_1,n_2)\sim(n_2,m_2)\equiv n_1m_2=n_2m_1\).
Count (\(\mathbb{N}\)) with addition (+) and subtraction (-) is \(\mathbb{Z}\). \(\mathbb{Z}\) with repetition (\(\cdot\)) and division (\(\div\)) is \(\mathbb{Q}\). If we stay with \(+,-,\cdot,\div\), then we can do with \(\mathbb{Q}\).
But if we want the power operation to be reversible, then we must extend again. There is for example no \(p/q\) in \(\mathbb{Q}\), for which \(p^2/q^2=2\). (Proof: p/q shall have no common divisor. If \(p^2\) is even, so is p (p=2n). \(p^2=4n^2=2q^2\) means that q is even, but that is a contradiction).
There are though algorithms that make rationale Numbers (sequences), whose square gets arbitrarily close to 2. All such algorithms are combined into a equivalence class and this is then the new number \(\sqrt{2}\)
The irrational numbers \(\mathbb{I}\) are equivalence classes of number sequences. By naming the algorithm, and \(\sqrt{}\) refers to such an algorithm, the irrational number is determined. One cannot write an irrational number as decimal number. One can also not run the algorithm to an end, because it does not terminate. So the irrational number is really the algorithm itself.
The irrational numbers get further classified as algebraische irrationals, which are those that are roots of polynomials, and the transcendental irrationals. The latter exist, because there are functions beyond finite polynomials, like Sin, Cos, … most of which can be expressed with infinite polynomials (series), though. \(\pi\) and \(e\) are transcendental.
New operations/functions lead to new numbers. But the definition equivalence classes of sequences is so general that it includes algebraic and transcendental numbers and \(\mathbb{Q}\) itself.
This is \(\mathbb{R}\):
\(\mathbb{R} = \mathbb{Q} \cup \mathbb{I}\)
Another very useful and exciting extension are the complex numbers \(\mathbb{C}\)).
Note
Since \(\mathbb{R}\) includes all never ending number sequences, one could include \(\infty\) and \(-\infty\), which are also never ending sequences of numbers, if it weren’t for \(\infty+1=\infty\) and the like. Still in complex analysis (function theory) the complex number set is extended with \(\infty\) fruitfully.