Natural numbers arose from the need for ordering of objects or beings. For example, we should count how many trees are in a certain place.
Oral representation of natural numbers is by
words;
written representation is made by
symbols.
People have invented several types of numbering systems (counting). The best known are
 The decimal positional system.
 The Roman unnumbered system.
Decimal numeration system is used today worldwide. The basis of this system are the numbers: 0, 1, 2,3, 4, 5, 6, 7, 8, 9. Any natural number can be written using these numbers.
For instance, in above table, first number are written like 34 128 could be readed as È™i se citeÈ™te thirtyfour thousand one hundred twenty and eight.
Any natural number with 2 or more digits can't have 0 as first digit.
Any natural number with 2 digits is written as , where a, b are digits and .
If is a natural number and , then is its overturned.
For instance, for 568 the overturned is 865, and for 20369 the overturned is 96302.
Definition. Any natural number which has the units digit 0, 2, 4, 6 or 8 is called even number.
Definition. Any natural number which is the number of units 1, 3, 5, 7 or 9 is called odd.
There is other positional numbering systems. For instance, binar system. Or hexadecimal system.
Romans digits
The numbering system used by ancient Romans is nonpositional. Here we have a table with some of the numbers used by Romans:
Romans

Arabs

I
V
X
L
C
D
M

1
5
10
50
100
500
1000

Example: XV = 10 + 5 = 15
XVI = 10 + 5 +1 = 16
MDIX = 1000 + 500 + 10 – 1 = 1509
Rules: When a figure lower value is written to the right one digit higher value, gather their values.
Example: XI = 10 + 1 = 11
When a figure value is written bottom left one digit higher value, less the amount of value lower annual figure following.
Example: IX = 10 – 1 = 9
When a figure lower value is between two figures higher values, the difference shall be firstly according to the above rule.
Example: XIX = 10 + (101) = 19
Representation of natural numbers axle
Definition. On a axe (d) A fixed point called origin, it sets a direction (from origin to right) called positive sense and choose a unit (a certain segment length MN). The axe (d) these three characteristics is called axis numbers.
This line is endless (infinite). Regarding the points on the axis, we can say that's a corresponding number 1, number 2 corresponds to point B etc.
Comparing and ordering of natural numbers
Natural number (positive integers) that are written as: 0, 1, 2, 3, 4, ..., 39, 40, 41, 42, 43, 44, 45, ... form the string of natural number. Any of natural number has a succesor. E.g., 112 has as a successor the number 113. The natural numbers's string doesn't have a last number, this means that the natural number's string is endless. Any natural number different from 0 (zero) has a predecessor. For example, 11 is the predecessor of 12.
Number 0 is not a successor and has a predecessor. Natural predecessor and the other numbers are successor.
The string of natural number starting with the number 0.
Relations between the natural numbers
For any two integers a, b there is only one of the following relationships:
1. a<b ( a is less than b );
2. a=b ( a equals b );
3. a>b ( a is greater than b ).
Exemples:
5 < 9; 7 < 23 ; 10256 < 11000
2 = 2; 10 = 10;
6 > 1; 12 > 8; 11202 > 9989
We can write which means less than or equal to b. Example: .
To compare two integers with the same number of digits, compared digit by digit, from left, until two figures of the same order are different. The higher the number the higher the respective figure. For example, 23 569> 23 469.
If two integers have the same number of digits exceeds the number that has more digits. For example, 236> 46.
Approximation of natural numbers
Where should not accurately express a natural number, we can approximate or rounded number. For example, a notebook cost 2.86 dollars. When we express we use 3 dollars. I did so by adding an approximation.
Number 61 is closer to 60 than 70, so that when we approximate it, we will approximate to 60. We did so by default approximation.