NATURAL NUMBERS: HISTORY, PROPERTIES, OPERATIONS, EXAMPLES - MATHS - 2025

The natural numbers are those that serve to count the number of elements of 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.

Figure 1. Natural numbers are those used for counting and ordering. Source: Pixabay.

In addition to being used for counting and ordering, natural numbers are also used as a way to identify and differentiate 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 to say that the set has no end.

No matter how big a natural number may be, you can always get the next highest.

History

Before the natural numbers appeared, that is, the set of symbols and names to denote a certain amount, 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 number.

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 first they were marks on a bone or a stick.

Cuneiform engravings on clay panels, representing numerical symbols and dating from 400 BC, are known from Mesopotamia, which is currently the nation of Iraq.

Symbols were evolving, so the Greeks and later the Romans used letters to denote numbers.

Arabic 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.

We have ten symbols to express any numerical quantity, one symbol for each finger of the hand.

These symbols are:

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 in some way or another used by humans.

Properties of natural numbers

The set of natural numbers is:

ℕ + = {0, 1, 2, 3, 4, 5, ………}

And with them you can count the number of elements in another set or also order these elements, if each one is assigned a natural number.

It is infinite and countable

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.

It's a neat set

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 such as these:

7> 3 means that seven is greater than three

2 <11 is read two is less than eleven

They can be grouped together (addition operation)

3 + 2 = 5 means that if you join three elements with two elements, you have five elements. The symbol + denotes the addition operation.

Operations with natural numbers

- Sum

1.- The addition is an internal operation, in the sense that if two elements of the set ℕ of natural numbers are added, another element that belongs to said set will be obtained. Symbolically it would read like this:

2.- The sum operation on the naturals is commutative, which means that the result is the same even if the addends are inverted. Symbolically it is expressed like this:

If a ∊ 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 you add in this way you also get the same result:

3 + 6 + 8 = (3 + 8) + 6 = 11 + 6 = 17

4.- There is the neutral element of the sum and this element is zero: a + 0 = 0 + a = a. For example:

7 + 0 = 0 + 7 = 7.

- Subtraction

-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

-Multiplication is denoted by a ⋅ by means to add to itself b times. For example: 6 ⋅ 4 = 6 + 6 + 6 + 6 = 24.

- Division

The division is denoted by: a ÷ by means how many times is b in a. For example, 6 ÷ 2 = 3 because 2 is contained in 6 three times (3).

Examples

Figure 2. Natural numbers allow you to count how many apples a box has. Source: pixabay

- Example 1

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?

Reply

15 + 22 = 37 apples.

- Example 2

If in the box of 37 apples 5 are removed, how many will be left in the box?

Reply

37 - 5 = 32 apples.

- Example 3

If you have 5 boxes with 32 apples each, how many apples will there be in all?

Reply

The operation would be to add 32 with itself 5 times what is denoted like this:

32 ⋅ 5 = 32 + 32 + 32 + 32 + 32 = 160

- Example 4

You want to divide a box of 32 apples into 4 parts. How many apples will each part contain?

Reply

The operation is a division that is denoted like this:

32 ÷ 4 = 8

That is, there are four groups of eight apples each.

References

1. Set of natural numbers for fifth grade of primary school. Recovered from: activitieseducativas.net
2. Mathematics for children. Natural numbers. Recovered from: elhuevodechocolate.com
3. Martha. Natural numbers. Recovered from: superprof.es
4. A teacher. The natural numbers. Recovered from: unprofesor.com
5. wikipedia. Natural number. Recovered from: wikipedia.com