The natural numbers they are used to count the number of elements in a certain set. For example, natural numbers are those that are used to find out how many apples are in a box. They are also used to order the elements of a set, for example the first graders in order of size.
In the first case we speak of Cardinal numbers and in the second of ordinal numbers, in fact, "first" and "second" are ordinal natural numbers. On the contrary, one (1), two (2) and three (3) are cardinal natural numbers.
In addition to being used for counting and ordering, natural numbers are also used as a way of identifying and differentiating the elements of a certain set..
For example, the identity card has a unique number, assigned to each person who belongs to a certain country.
In mathematical notation the set of natural numbers is denoted like this:
ℕ = 1, 2, 3, 4, 5,…
And the set of natural numbers with zero is denoted in this other way:
ℕ+ = 0, 1, 2, 3, 4, 5, ...
In both sets, the ellipses indicate that the elements continue consecutively to infinity, the word infinity being the way of saying that the set has no end.
No matter how big a natural number may be, you can always get the next largest.
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Before natural numbers appeared, that is, the set of symbols and names to denote a certain quantity, the first humans used another set of comparison, for example the fingers of the hands..
So, to say that they found a herd of five mammoths, they used the fingers of one hand to symbolize that amount.
This system could vary from one human group to another, perhaps others used instead of their fingers a group of sticks, stones, necklace beads or knots in a rope. But the safest thing is that they used their fingers.
Then symbols began to appear to represent a certain amount. At the beginning they were marks on a bone or a stick.
Cuneiform engravings are known on clay boards, representing numerical symbols and dating from 400 BC, found in Mesopotamia, which is currently the nation of Iraq..
Symbols evolved, so the Greeks and later the Romans used letters to denote the numbers.
Arabic numbers are the system we use today and they were brought to Europe by the Arabs who occupied the Iberian Peninsula, but they were actually invented in India, which is why they are known as the Indo-Arabic numbering system..
Our numbering system is based on ten, because there are ten fingers of the hands.
We have ten symbols to express any numerical quantity, one symbol for each finger of the hand.
These symbols are:
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
With these symbols it is possible to represent any quantity using the positional system: 10 is a ten zero units, 13 is a ten and three units, 22 two tens two units.
It must be made clear that beyond the symbols and the numbering system, natural numbers have always existed and were always used by humans in one way or another..
The set of natural numbers is:
ℕ+ = 0, 1, 2, 3, 4, 5, ...
And with them you can count the number of elements of another set or also order these elements, if each one is assigned a natural number.
The set of natural numbers is an ordered set that has infinite elements.
However, it is a countable set in the sense that it is possible to know how many elements or natural numbers there are between one number and another..
For example, we know that between 5 and 9 there are five elements, including 5 and 9..
Being an ordered set, you can know which numbers are after or before a given number. In this way, it is possible to establish, between two elements of the natural set, comparison relationships like these:
7> 3 means that seven is greater than three
two < 11 se lee dos es menor que once
3 + 2 = 5 means that if you join three elements with two elements, you have five elements. The symbol + denotes the addition operation.
1.- The addition is an internal operation, in the sense that if two elements of the set are added ℕ from the natural numbers, another element that belongs to said set will be obtained. Symbolically it would read like this:
Yes a∊ ℕ and b∊ ℕ, then a + b ∊ ℕ
2.- The sum operation on the natural ones is commutative, which means that the result is the same even if the addends are inverted. Symbolically it is expressed like this:
Yes to ∊ ℕ and b ∊ ℕ , then a + b = b + a = c where c ∊ ℕ
For example, 3 + 5 = 8 and 5 + 3 = 8, where 8 is an element of the natural numbers.
3.- The sum of natural numbers fulfills the associative property:
a + b + c = a + (b + c) = (a + b) + c
An example will make it clearer. We can add like this:
3 + 6 + 8 = 3 + (6 + 8) = 3 + 14 = 17
And in this way also:
3 + 6 + 8 = (3 + 6) + 8 = 9 + 8 = 17
Finally, if it is added in this way, the same result is also reached:
3 + 6 + 8 = (3 + 8) + 6 = 11 + 6 = 17
4.- There is the neutral element of the sum and that element is zero: a + 0 = 0 + a = a. For example:
7 + 0 = 0 + 7 = 7.
-The subtraction operator is denoted by the symbol -. For example:
5 - 3 = 2.
It is important that the first operand is greater than or equal to (≥) than the second operand, because otherwise the subtraction operation would not be defined in the naturals:
a - b = c, where c ∊ ℕ if and only if a ≥ b.
-Multiplication is denoted by a ⋅ b and means adding to yourself b times. For example: 6 ⋅ 4 = 6 + 6 + 6 + 6 = 24.
The division is denoted by: a ÷ b and means how many times is b in a. For example, 6 ÷ 2 = 3 because 2 is contained in 6 three times (3).
In one box, 15 apples are counted, while in another, 22 apples are counted. If all the apples from the second box are placed in the first, how many apples will there be in the first box??
15 + 22 = 37 apples.
If in the box of 37 apples 5 are extracted, how many will be left in the box?
37 - 5 = 32 apples.
If you have 5 boxes with 32 apples each, how many apples will there be in all??
The operation would be to add 32 with itself 5 times what is denoted like this:
32 ⋅ 5 = 32 + 32 + 32 + 32 + 32 = 160
You want to divide a box of 32 apples into 4 parts. How many apples will each part contain?
The operation is a division denoted like this:
32 ÷ 4 = 8
That is, there are four groups of eight apples each.
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