NUMBERS Numbers are divided into classes by a system called the number system. According to this system formation (respectively): 1) Natural Numbers2) Integers3) Rational Numbers4) Real (Real) Numbers. 1) Natural Numbers Considering in general, they are the most commonly used number type. Every person may want to count something in some situations. Here is the number that almost every person uses […]


Numbers are divided into classes with a system called the number system. According to this system formation (respectively):

1) Natural Numbers
2) Integers
3) Rational Numbers
4) Real (Real) Numbers


1) Natural Numbers

Generally speaking, they are the most frequently used number type. Every person may want to count something in some situations. These are the types of numbers that almost every person uses.

We all know intuitively what natural numbers are; 0,1,2,…, m,… All other number systems are created with the help of natural numbers. So let’s first start by defining these numbers:

We can create the first few natural numbers like this:

0 = Ø
1 =
2 =
3 =

In other words, these numbers are 13 as can be shown.

So we can use a notation like this:

The first natural number is Ø and a defined for the natural number 32. Now let’s make a few definitions:


A sequence with A being a cluster


“ Let’s say /05/52.png “alt =” 52 “width =” 23 “height =” 18 “/> set.

By this definition, natural numbers are 62.

However, this method is not sufficient to construct a set of natural numbers. The existence of such a cluster cannot be demonstrated in this way. Because with what we have done so far, when it contains the empty set and covers an A set 52? question remains unanswered for now.


82 92 A set X that fulfills the condition 101 is called a successor set.

After giving this definition, we can give a positive answer to the above question with the following” axiom of infinity “:

Axiom (Axiom of Infinity)

There is a successor set.

According to this axiom, we understand that a successor set contains an infinite number of elements.

Before making this definition, let’s look at the following theorem:


The intersection of all successive sets is again a successor set.

Proof :



which is the intersection of all successive sets The set of N is called the set of natural numbers. Elements of the set N are also called natural numbers.

2) Integers

We have explained the natural numbers above. Solution sets of some equations defined on the set of natural numbers do not exist. For example, if we want to determine the unknowns x that satisfies x + 3 = 8, where x is an unknown number, it is clear that the solution of the equation above N is x = 5.

On the other hand, we cannot solve the equation x + 4 = 6 in N because The unknowns x, which is the solution of this equation, is a negative number and since there is no negative number in the set N, the equation we have has no solution in the set N. For this reason, the necessity of expanding the set N has arisen so that the aforementioned and similar equivalents can be found.

We can write each natural number as the difference of two natural numbers in various ways.

If we correspond to these expressions from which the second term is removed from the first term with ordered pairs such as

(4,0), (5,1), (6,2), …

, the common its feature is

a + d = b + c

for any two ordered binaries such as (a, b), (c, d). It is clear that this operation for 4 can be repeated for all other natural numbers as well. If we take this method for natural numbers one step further, we need to define a relation like this:


A 121 relation 131.png

(a, b) (c, d) a + d = b + c

for = “131” width = “136” height = “19” /> . above is an equivalence relation.

An equivalence division of Z like [(a, b)] is also called an integer and the representation

[a, b] = [(a, b)]

for brevity.

3) Rational Numbers

The two number systems we have defined so far are still insufficient to solve some equations. For example, consider the equation 5x = 3. The solution of this equation is 14 with 15 obtained from the joint solution of the inequalities. If we say the solution for the first of these is Ç * and the other is Ç **

. The cross section of these two solution sets found is 17, there is no solution to the equation 5x = 3 in Z.


< With Z * = Z

On Z x Z * for (m, n) ~ (p, r) mr = np The ~ relation defined by is a parity relation.



If (m, n) ~ (p, r) then mr = np and pn = rm (p, r) ~ (m, n).
3. If (m, n) ~ (p, r) and (p, r) ~ (m, n) then m.r = n.p and p.s = r.k. If p = 0, then m = 0 = r. In this case ms = nk 19.png (m, n) ~ (k, s). If p is 0 m. (ps) = m. (rk) = (mr) .k = (np) .k (ms) .p = (nk) .p 19 ms = nk 19 (m, n) ~ (n, r) as desired.


The set of (Z x Z *) / ~ equivalence parts is called the set of rational numbers and denoted by Q. Any element of the set Q is also called a rational number.

For example with [(m, n)],

[(0,1)], [(1,3)], [(0, -5)] will be rational numbers.

4) Real Numbers 211 let’s show that there is no rational number:

Conversely, a 221.



. Since it is 2 | 2.n, we have 2 | m and from there 2 | m. So m = 2.p 25. 26 If we use the equation, 27. Thus, the equations m = 2.p, n = 2.q conflict with the above thought. So no 211 there is no rational number.

As seen in the example above, 29 for the equation, before Considering the number systems defined, we could not find a solution.

So let’s define Real (Real) Numbers.


A 311 set of rational numbers



Dedekind segments are called real numbers and the set of real numbers is denoted by R. < /.

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