Addition of natural numbers
is counting times, is counting times first, then counting times
- Associativity:
- Commutativity:
Proof The intuition in the real world is that for counting , no matter how the counting task is manually divided into several sub-tasks, the result will not be affected, and the total number of decomposition methods is limited, and the final decomposition is the commutativity and associativity of a large number of s
Multiplication of natural numbers
is counting times, times
It also satisfies commutativity and associativity. The intuition in the real world is โtwo-dimensional rectangle (commutativity)โ and โthree-dimensional rectangle (associativity)โ. No matter how the product decomposition sub-tasks are decomposed, the result will not be affected. And the total number of decompositions is limited, and the final decomposition is the commutativity and associativity of a large number of prime numbers
The intuition is to decompose it by the sum of the side lengths of a two-dimensional rectangle
Integers
Rational numbers
-division operation, the inverse of multiplication
Do not confuse it with division and remainder of , which is successive subtraction of one number by another number , not equal division
Real numbers
One intuition for real numbers is length. Or it contains rational numbers + linear order + order completeness
Given the intuitiveness of real numbers, it can be considered that they exist, and many axioms can be used to define real numbers i.e. assume true propositions. But real numbers can also be โrecoveredโ from rational numbers
Examples of irrational numbers
We prove that is irrational, or
- Every natural number can be uniquely factorized into prime factors
- are coprime := have no common prime factors
- If are coprime, then are coprime.
- can be uniquely represented as a fraction of coprime .
- If , then . Proof by contradiction: If then thus thus .
Specifically, when , , but and . This indicates that there is no such that .
Therefore is not a rational number.
This method of determination can be extended to algebraic integers .
The โintegerโ in algebraic integer is because .
Proof (p.43 of ref-8)
Take coprime. Substitute into the equation, multiply by .
The right side is divisible by . But are coprime, so or .
Thus . Thus .
Algebraic numbers .
Note that is not required, is not restricted, including all rational numbers , some irrational numbers e.g.
Algebraic numbers are a countable set, real numbers are an uncountable set
Transcendental numbers . are transcendental numbers
Decimal and binary systems can both define real numbers, both are special nested interval methods
Rational number intervals are subsets with the property of uninterrupted order
Note that interval endpoints can be irrational numbers. Defining real numbers using nested intervals also requires handling Cauchy property or limit-distance-vanish property
From the perspective of operational simplicity, Dedekind-cut should be used. โOperational simplicityโ means
- let , is a one-to-one correspondence
- So and are in one-to-one correspondence
[Dedekind_cut] Irrational numbers
is in one-to-one correspondence with
. Relabel as
Real numbers
Logically equivalently, only one half can be used, for example, any left-half infinite interval of rational numbers , and then the right-half infinite interval is automatically obtained by taking the complement in . But here a more symmetrical representation is used
- [order_real] order
let
- [add_real] addition. let
Due to the existence of , multiplication does not preserve order. However, multiplication in preserves order. First handle the case , then use reflection to obtain the case
- [multiply_real] multiplication. let
have associativity, commutativity, distributivity for
[completeness_real] completeness
[exact_bound] Supremum Principle
let have an upper bound
Supremum
[monotone_convergence] Monotone bounded convergence Proof use Supremum Principle
[nested_closed_interval_theorem] Nested Closed Interval Theorem
[closed_interval_intersection_theorem]
In fact, itโs sufficient that the smaller endpoints of the closed interval family are all the larger endpoints to conclude the intersection is non-empty
Proof Similarly, for the smaller endpoints use supremum , for the set of larger endpoints use infimum , obtaining with that the intersection of the closed interval family is the closed interval
[closed_interval_net_theorem] Closed interval net intersection non-empty
Proof
The reverse direction โthe small endpoints of the closed interval family are all large endpoints ==> interval netโ is not true. Consider a closed interval family with only two intervals where their intersection is non-empty and , and there is no third interval belonging to their intersection. Although if we supplement the intersection, it can be satisfied
let
def sequence is monotonically decreasing, is monotonically increasing
[limsup] Upper limit
[liminf] Lower limit
Example
For sequences, define
For general net, define
[limit_distance_vanish_sequence] := . i.e. tail distance vanish
[limit_distance_vanish_net] :=
[Cauchy_completeness_real] limit-distance-vanish sequence or net converges
Proof
limit-distance-vanish ==>
let , then . We can take
==> limit-distance-vanish sequence is bounded
==> Monotonically increasing/decreasing bounded sequences have limits
limit-distance-vanish property ==>
Thus converges to
For the net, similarly, it is proved that the tail of the net is bounded, and then, it has been proved that the intersection of bounded interval nets is non-empty. Take a point from it, and use limit-distance-vanish-net to get the net convergence
Conversely, a convergent sequence is limit-distance-vanish. by triangle inequality
Prop A sequence or net converges to <==> limit-distance-vanish
[uncountable_real] Real numbers are uncountable
It has been proved that . cf. cardinal_increase
recall
Proof
According to the nested interval theorem, the binary decimal representation of real numbers: The -th digit takes or
==> . Where, the two possible equivalent choices in binary are quotiented
by linear mapping or affine mapping
by
Proof
It represents that in binary, the first position where appears is , the second position is โฆ
Compare , vs
Distance