real-numbers - note-math

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.

It seems difficult for computers to express this intuition, but it seems that all finite results must be correct. Similar to what is done in natural numbers, in order for a computer to express the property that holds for all natural numbers using finite characters, memory, and finite time (and potentially infinite time), it is necessary to define (assume, axiom) that it is a true proposition.

The usual proof is to use minimal assumptions (axioms), the associativity of or the definition of addition , and then derive others.

To an extreme, if we always set conclusions that can be proven with a few axioms as axioms, then we will have no proofs at all. So we may use minimal axiom, but at least let us choose the more symmetric assumption .

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

The number axis has two directions

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

[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

completeness [completeness-real]

[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

Whether it is nested intervals or linearly ordered chain nested intervals, linear order means the monotonicity of interval endpoints, use supremum and infimum for the endpoints with to get the intersection of nested closed intervals is a closed interval . can be understood as the minimal element of linear order chain nested closed sets

[closed-interval-net-theorem] Closed interval net intersection is non-empty

Proof

Supplement the net with all finite intersections

Take a maximal linearly ordered chain . By the nested interval theorem, its intersection is a non-empty closed interval

By the linear order maximality of , intuitively, the closed interval will be smaller than all closed intervals of , so

, we prove

Define the closed interval linear ordered chain . . We prove

Proof by contradiction. Assume . Then the small/large endpoint of is larger/smaller than the small/large endpoint of

The linear ordered chain satisfies

If the closed interval , then use the exact bound principle for the endpoints

There exists that belongs to , which contradicts that is a maximal linear ordered chain

If the closed interval , the same contradiction applies

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

Unbounded ==>

==> 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 closed interval net theorem, let . 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

Proof

It represents in binary, the first position where appears is , the second position is …

Compare , vs

Distance