Beyond real number, let's review some about rational number first. Rational number a.k.a the extension of natural number where we want to separate a integer.
That's we define ration number as ∀q∈Q,q=a/b,a,b∈N,b=0
We would like to use decimal expansion to present rational number. But the interesting thing found here, we have many decimal expansions, how can we found the related one. For the finite decimal expansion, we just simply use 10n where n is the number of digits after dot to times such expansion then we have a integer, and now we can present it as a/b so that all finite decimal expansions are rational. How about infinite decimal expansion? Some of them are eventually periodic, some of them are not. Among those expansion, how can we judge if it's a rational number?
Let's define the eventually periodic first. An infinite decimal x=a0.a1a2⋯ is eventually periodic if there are positive integers n and k such that ai+k=ai for all i>n.
let x=a0.a1a2… be arbitrary eventually periodic infinite decimal where ∃n,k∈N,n≥0,∀i∈N,i>n⟹ai+k=ai
Since 10nx=a0⋯an.an+1…, 10n+kx=a0⋯an⋯an+k.an+k+1… and the decimals are the same (where n+1>n and ai=ai+k,∀i>n) so that we will get a integer for 10n+kx−10nx, and denote such integer as z.
x(10n+k−10n)=z⟹x=10n+k−10nz where z∈N and 10n+k−10n∈N so that x is a rational number.
As we can see the infinite periodic decimal expansion can be present as a/b,a,b∈N,b=0, but we can't find a way to present the other that's why we have real number. Sometimes, we also said R is the extension of Q
Let's look up how we extend it following!
Least Upper Bound Principle (& Greatest Lower Bound)
First, let's take R to describe some properties.
Let S be a subset of real number S⊂R be arbitrary
∃M∈R,∀s∈S,M≥s⟹M is an upper bound for S.
∃M∈R,∀s∈S,M≤s⟹M is an lower bound for S
∃L,∀M∈R,s.t.∀s∈S,M≥s,L≥s∧L≤M⟺L is the supremum (least upper bound) for S. Denote supS=L
∃L,∀M∈R,s.t.∀s∈S,M≤s,L≤s∧L≥M⟺L is the infimum (greatest lower bound) for S. Denote infS=L.
For ∅,inf∅=+∞,sup∅=−∞
When we say a set is bounded⟺ the set is both bounded above and bounded below.
Denote the maximum of S as L, that's max(S)=L. max(S)=L⟹supS=L. Similarly, Denote the minimum of S as L, that's min(S)=L. min(S)=L⟹infS=L.
Notice, supS=L does not imply such maximum exists. Similarly for infimum
Axios of Completeness: Every nonempty subset S of R that is bounded above has a supremum. Similarly, every nonempty subset S of R that is bounded below has an infimum.
Archimedean Property: (proved by Axios of Completeness)
∀x∈R,∃n∈N,n>x
∀y>0∈R,∃n∈N,n1<y
It seems all above already use R to present, it happends due to we can actually use those to all subset of R. Back to Q, now we have following consequence of completeness:
Density of Q in R: ∀x,y∈R,s.t.x<y,∃r∈Q,s.t.x<r<y.
And furthermore, we can write:
∀a∈R,a=sup{r∈Q:r<a}
R={sup(A):A⊆Q,A has upper bound in Q}
N⊂Z⊂Q⊂R
All above show the existance of R is based on N,Z,Q by axiom of completeness, we need to know some definition before look at the proof.
We define an ordered pair (A,B) of nonempty subsets of Q as a Dedekind cut with properties:
y∈A,x<y⟹x∈A
y∈B,x>y⟹x∈B
∀x∈A,∀y∈B,x<y
Now look at the proof:
Let a∈R, let Aa={r∈Q:r<a}⊆Q is bounded by Q, so that sup(Aa)=a.
Let b=sup(Aa) by Axiom of Completeness, we have b∈R.
Assume a=b, then a<b
Case 1: a<b: since b=sup(Aa), then b≤a which is a contradiction.
Case 2: a>b: ∃r∈Q,b<r<a by archimedean property, but b=sup(Aa) which is a contradiction.
Thus, a=b, so that a∈R.
Since we know and prove how real number exists, let's look at a fundemental usage of real number ∃α∈R,α2=2.
proof:
Let A={x∈R:x2<2}, AQ={x∈Q:x2>2}. WTS A=AQ=2 which is equivalent to prove α=sup(A)⟹α2=2.
We can prove the second one by contradiction, assume α2=2, then α2>2 or α2<2.
Case 1: α2>2: ∃n∈N,s.t.(α−n1)2>2 by archimedean property, since α−n1<α∧α=sup(A)⟹α−n1∈A, then (α−n1)2<x2<2∧(α−n1)2>2 which is a contradiction.
Case 2: α2<2: ∃n∈N,s.t.n1<2α+12−α2 by archimedean property, since α2<2⟹α∈A, then (α+n1)2=α2+n2α+n21<2≤α2+n2α+1<α2+2−α2=2 so that α+n1∈A. Since α=sup(A),α+n1∈/A which is a contradiction.
We use the term cardinality to denote the number of elements in a set. If set A has the same number of elements as set B, we say A and B have the same cardinality. We denote A∼B.
Formally, we define the cardinality of a set A∼B if there is a bijection function between A and B.
Cantor Theorem: Q∼R and N∼Z∼Q.
can prove by Schroeder-Bernstein Theorem
N2∼N then we can recursively apply this.
We define a set is finite if ∃n∈N,s.t.X∼{1,2,…,n}. But when we say countbale doesn't mean finite. We define a set is countable if X is infinite and X∼N. And a set X is uncountable if X is infinite and X∼N.
Since real number is a big set, we may find some special number in usage. We define r∈R is algebraic if r is a root of a polynomial with integer coefficients. And we can find that the set of algebraic number is countable. The set of non-algebraic number is dense in R.