Almost in March/2013, I have enjoyed following entertainments.
[Translated Novel]
(1)☆☆☆The Cobra(Frederick Forsyth)
(2)☆☆☆Avenger(Frederick Forsyth)
[Japanese Novel]
(3)☆☆☆☆Momiji machi eki mae Jisatsu center(Masaki Mitsumoto)
(4)☆☆Taitei no ken 5(Baku Yumemakura) - completed
(5)☆☆Henshin shashin kan(Tomoko Mano)
(6)☆☆☆"Ougon no Bantam" wo yabutta otoko(Naoki Hyakuta)
[Movie]
(7)☆☆☆☆Frozen River(Charlie McDermott, Melissa Leo)
(8)☆☆Nankyoku Ryori nin(Masato Sakai)
(9)☆☆☆Alice in Wonderland（Mia Wasikowska, Johnny Depp）
[Games]
(10)☆☆☆☆Borderlands(ps3)
In this month some works were fine with me. Among them, (3)"Momiji machi" is good. This work describes a extreme grief of losing a child. In (7)"Frozen River", heroine loses her husband and has a small income from a dangerous job and arrests. However, she carries on with her life overcoming the sadness. It is strong and beautiful. (10)"Borderlands" is much fun.
2013/04/27
2013/04/13
real number system 7
We address how to handle $\pm \infty $ in real number system. It is said to be a extended real number system.
In general, $x\in \mathbb{R}$ means $-\infty<x<+\infty$. However, if $x\in \mathbb{R}$ is not bounded above, we understand $x=+\infty$. It is convenient to make the rule of the fictitious number $\infty$.
["$\infty$"] for any $x\in \mathbb{R}, (x\ne 0)$ , that is, $-\infty<x<+\infty, (x\ne 0)$,
$x+\infty=\infty$, $x-\infty=-\infty$, $\infty+\infty=\infty$,
$(\infty)\cdot (\infty)=\infty$, $(-\infty)\cdot (-\infty)=\infty$, $(-\infty)\cdot (\infty)=-\infty$,
If $x>0$, then $x\cdot (+\infty)=\infty$, $x\cdot (-\infty)=-\infty$,
If $x<0$, then $x\cdot (+\infty)=-\infty$, $x\cdot (-\infty)=\infty$,
$\frac{x}{+\infty}=0$, $\frac{x}{-\infty}=0$,
Unfortunately we are not able to define the following forms.
$\infty-\infty$, $\frac{\pm\infty}{\pm\infty}$, $0\cdot\infty$
Therefore, the extended real number system is ordered, but it is not a field. Do not take any notice of these definitions.
In general, $x\in \mathbb{R}$ means $-\infty<x<+\infty$. However, if $x\in \mathbb{R}$ is not bounded above, we understand $x=+\infty$. It is convenient to make the rule of the fictitious number $\infty$.
["$\infty$"] for any $x\in \mathbb{R}, (x\ne 0)$ , that is, $-\infty<x<+\infty, (x\ne 0)$,
$x+\infty=\infty$, $x-\infty=-\infty$, $\infty+\infty=\infty$,
$(\infty)\cdot (\infty)=\infty$, $(-\infty)\cdot (-\infty)=\infty$, $(-\infty)\cdot (\infty)=-\infty$,
If $x>0$, then $x\cdot (+\infty)=\infty$, $x\cdot (-\infty)=-\infty$,
If $x<0$, then $x\cdot (+\infty)=-\infty$, $x\cdot (-\infty)=\infty$,
$\frac{x}{+\infty}=0$, $\frac{x}{-\infty}=0$,
Unfortunately we are not able to define the following forms.
$\infty-\infty$, $\frac{\pm\infty}{\pm\infty}$, $0\cdot\infty$
Therefore, the extended real number system is ordered, but it is not a field. Do not take any notice of these definitions.
2013/04/03
real number system 6
We summarize the properties of real numbers based upon preceding posts.
(1)Real numbers are the set of all decimals. That is, every real numbers has the form as follow.
\[ C.c_1c_2c_3\cdots, \quad \mbox{or} \quad -C.c_1c_2c_3\cdots \]
where $C$ is any nonnegative integer and $c_i$ is a single digit figure between $0$ and $9$ inclusive.
(In this definition, a finite decimal number always has two forms. For example one is $1.000\cdots $ and $0.999\cdots $. After this, in these cases we shall promise that if $x$ is $0.999\cdots$, we make $x$ be $1$. The same applies to another numbers.)
(2)Real numbers consist of rational numbers and irrational numbers.
(3)Real numbers are a field. On a field we can calculate as prescribed.
(4)Real numbers are ordered by a natural magnitude relationship.
(5)Real numbers are complete.
From (1), (3) and (4), we are not able to get (5). Therefore, we must accept Axiom for completeness or Dedekind cut is satisfied in real numbers. If one is accepted, the other can be proved.
(6)Real numbers are dense. Rational numbers have been already dense in real numbers.
(7)Real numbers are uncountable. Although both $\mathbb{R}$ and $\mathbb{N}$ are infinite sets, $\aleph$ is greater than $\aleph_0$.
(8)There is a one to one correspondence between points of the real number line and real numbers. The one to one function from real numbers to points in the square having a side length $1$ also exists.
Essentially we need (1), (3), (4) and (5). However, do you think all things are very interesting?
(1)Real numbers are the set of all decimals. That is, every real numbers has the form as follow.
\[ C.c_1c_2c_3\cdots, \quad \mbox{or} \quad -C.c_1c_2c_3\cdots \]
where $C$ is any nonnegative integer and $c_i$ is a single digit figure between $0$ and $9$ inclusive.
(In this definition, a finite decimal number always has two forms. For example one is $1.000\cdots $ and $0.999\cdots $. After this, in these cases we shall promise that if $x$ is $0.999\cdots$, we make $x$ be $1$. The same applies to another numbers.)
(2)Real numbers consist of rational numbers and irrational numbers.
(3)Real numbers are a field. On a field we can calculate as prescribed.
(4)Real numbers are ordered by a natural magnitude relationship.
(5)Real numbers are complete.
From (1), (3) and (4), we are not able to get (5). Therefore, we must accept Axiom for completeness or Dedekind cut is satisfied in real numbers. If one is accepted, the other can be proved.
(6)Real numbers are dense. Rational numbers have been already dense in real numbers.
(7)Real numbers are uncountable. Although both $\mathbb{R}$ and $\mathbb{N}$ are infinite sets, $\aleph$ is greater than $\aleph_0$.
(8)There is a one to one correspondence between points of the real number line and real numbers. The one to one function from real numbers to points in the square having a side length $1$ also exists.
Essentially we need (1), (3), (4) and (5). However, do you think all things are very interesting?