axiomatic sets 27 (real numbers 3)

In preceding posts, we have seen the properties of real numbers.

(1)arithmetics of addition
(2)arithmetics of multiplication
(3)distributive law
(4)ordered relation
(5)the completeness

Let us look at one of 'Dedekind cut' $\left\{ x\in\mathbb{Q} : x\lt a \right\}$  as one number $a$  ,and the set of all 'Dedekind cuts' as a set of numbers.
(you may remember the natural number $a+1=\left\{ a\cup\left\{ a \right\} \right\}$ . )

The set of all 'Dedekind cuts' $\mathcal{C}$ will satisfy from (1) to (4). (do you agree?)

Axiom for completeness (5) means that if a set of numbers is bounded above,
then it has a supremum. (a supremum must be in all numbers.)

We can not express $\sqrt{2}$ as a rational number and there is not $\sqrt{2}$ in $\mathcal{C}$ .

However, in $\mathcal{C}$  we are able to get a rational number close to $\sqrt{2}$
as much as you want. Because in $\mathcal{C}$  there is all rational numbers.

$1.4=\frac{14}{10}\in \mathcal{C}$
$1.41=\frac{141}{100}\in \mathcal{C}$
$1.414=\frac{1414}{1000}\in \mathcal{C}$
$1.4142=\frac{14142}{10000}\in \mathcal{C}$
$\cdots\cdots\cdots$
$1.41421356237=\frac{141421356237}{10000000000}\in \mathcal{C}$
$\cdots\cdots\cdots$

Here, we want you to remember that $0.99999\cdots=1$ in the preceding posts.
That is, the rational number series which becomes $1.41421356237\cdots$ will go to $\sqrt{2}$ .
Althogh $\sqrt{2}$  is not a rational number, there exists a 'Dedekind cut' $\left\{x\in\mathbb{Q} : x\lt\sqrt{2} \right\}$ .

Thus, $\mathcal{C}$ is almost enough for real numbers.

The problem is that there are many numbers which are not able to be expressed by rational numbers.

We call the numbers which are not rational numbers irrational numbers.

Thus, real numbers are constructed rational numbers and irrational numbers.


(Correctly, you must understand, all above is needed proofs. )

coffee break 13 (the graphic card)

I bought a new graphic card nvidia geforce gtx1070 recently.

I have used amd radeon r9-280x.
Unfortunately on r9-280x Batman arkham knight crashes almost surely,
if I make the graphic level of my display monitor cheapest.

Many people says different types of the error reason on web and
my pc does not have a newest hardware configuration.  
I thought one of reasons was a lack of memories in the graphic card.
(the memories of r9-280x is 3gb and my PC is 16gb.)

Despite I worried about errors caused by the graphic driver (changing from amd to nvidia),
the change of the card went successfully.

The new card gtx1070 gives me playing the game comfortably.
(the memories of gtx1070 is 8gb.)

I remember that a few years ago, when I could not play TorchLight 2 by intel HD graphics on mother board, I bought the garaphic card radeon HD7750.



***
I have heard that gtx1070 being configured micron memories has OC issues.
By GPU-Z, my gtx1070 has micron memories.(orz)