Natural number addition
is counting times , is counting times first, then counting times
- Associative law:
- Commutative law:
Proof The intuition in the real world is that for counting , no matter how the counting task is manually divided into several subtasks, the result will not be affected, and the total decomposition methods are limited. The associative and commutative laws of addition are just special cases. Just as we recognize natural numbers by counting, we can always recognize the commutative and associative laws by counting. Everything reduces to the case of complete additive decomposition, with only the commutative and associative laws of a large number of s.
There are two definitions of addition, or . By swapping the inputs (midway) in one definition of addition, we can get the other definition of addition. Proving the commutativity of addition means proving that the two definitions give the same output for the same inputs. Intuitively, the result is of course the same, it's just that the amount to be counted is placed in a different "slot position", thus "commutative"
Alternatively, for a function of counting in two positions, if one position is counted first and then used as the "base position," and the other position is counted again and used as the "incrementing position," the result is the "total count" . The commutative property states that swapping the two positions still yields the same resultโthe "total count."
Natural number multiplication
is counting times of counting times
It also satisfies the commutative and associative laws. The intuition in the real world is "two-dimensional and three-dimensional rectangles". For counting , no matter how it is decomposed into subtasks of product decomposition, it will not affect the result. Therefore, the commutative and associative laws of multiplication are just special cases of complete multiplication decomposition i.e. prime factorization, and the total number of decompositions is limited
Distributive law of natural number operations
The intuition is to decompose the side length of a two-dimensional rectangle into sum
Integer
Rational number
equal division operation, the inverse of multiplication
Do not confuse it with the division and remainder of , which is a successive subtraction of a number by another number , rather than equal division
Real number
One intuition of real numbers is length. Or contain rational number + linear order + order completeness
Given the intuitiveness of real numbers, we can assume that it exists and use many axioms to define real numbers i.e. assume a true proposition. But you can also "recover" real numbers from rational numbers
Examples of irrational numbers
Algebraic integer
The "integer" in algebraic integer is because
Proof (p.43 of ref-8)
Take and make them relatively prime. Substitute into the equation, multiply by
The right side is divisible by . But are relatively prime, so or .
So . Thus
Special case . But and
So that is is not a rational number
Algebraic number
Note that is not required, is not restricted, including all rational numbers , some irrational numbers e.g.
Algebraic number is a countable set, real number is an uncountable set
Transcendental number . are transcendental numbers
Decimal, binary vs nested interval vs segmentation
Decimal (nested interval) seems very intuitive
However, decimal cannot natively handle
Many different nested intervals of have the same limit, e.g. vs , which requires a limit-distance-vanish quotient.
let . let and , define the limit-distance-vanish equivalence relation (alias Cauchy convergence) for :=
The nested rational intervals of can be changed to general rational intervals whose length limit approaches zero linear order chain or more general rational intervals (maximal) whose length approaches zero net.
A rational interval is a subset with the property that the order is uninterrupted.
From an operational simplicity perspective, Dedekind-cut should be used. "Operational simplicity" means
- let , one-to-one correspondence
- So and one-to-one correspondence
[Dedekind-cut] irrational number
one-to-one corresponds to
. Record as again
Real number
Logically, we can use only half, for example, any left semi-infinite interval of the rational numbers , and then automatically get the right semi-infinite interval by doing the complement in . But here we use the more symmetrical representation
- [order-real] order
let
- [add-real] addition. let
Because of the existence of , multiplication does not preserve order. But the multiplication of preserves order. First deal with the case of , and then use reflection to get the case of
- [multiply-real] multiplication. let
's all have associativity, commutativity, and distributivity
In fact, for multiplication and its theorems, a possible more convenient approach than using Dedekind partitioning and linear ordering might be to use the limit of a rational interval net (net lies between partial and linear order).
[completeness-real] completeness
[exact-bound] Least-upper-bound property
let have an upper bound
Supremum
[monotone-convergence] monotone bound convergence Proof use exact-bound
[nested-closed-interval-theorem] Nested interval theorem
[closed-interval-intersection-theorem]
In fact, we only need to ensure that the smaller endpoints of closed intervals are all less than or equal to the larger endpoints to obtain a non-empty intersection.
Proof Similarly, applying the supremum to the smaller endpoints and the infimum to the larger endpoints, we obtain that with , the intersection of the family of closed intervals is a closed interval .
[closed-interval-net-theorem] Closed interval net intersection is non-empty
Proof
The reverse statement, "the smaller endpoints of closed intervals are all the larger endpoints ==> closed interval net" does not hold. Consider a family of closed intervals consisting of only two intervals with a non-empty intersection and ; there is no third interval in their intersection. Although after adding finite intersection, it's true
let
def sequence monotone decreasing, monotone increasing
[limsup] Upper limit
[liminf] Lower limit
Example
For the sequence, define
For a 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
==> The limit-distance-vanish sequence is bounded
==> The monotonically increasing or decreasing bounded sequences have limits
limit-distance-vanish property ==>
Thus converges to
For nets, by the upper and lower bounds of closed interval net. Using limit-distance-vanish-net, we obtain that the net converges
Conversely, a convergent sequence is limit-distance-vanish. by the triangle inequality
Sequence or net converges to <==> limit-distance-vanish
[uncountable-real] The real number is uncountable
It has been proved that . cf. cardinal-increase
recall
Proof
According to the nested interval theorem, the binary decimal point representation of real numbers: The -th digit takes or
==> . Where, quotient the possible two equivalent choices in binary
by linear map or affine map
by
Proof
It represents in binary, the first position where appears is , the second position is โฆ
Compare , vs
Distance