Natural numbers are numbers used for counting and numbering objects in nature. They start from 1 and have no upper limit.

- The smallest natural number is one (1).
- There is no largest natural number.
- Zero (0) is not a natural number.

The set of natural numbers is denoted by N = {1, 2, 3, 4, 5, …}.

## Main properties of natural numbers

- Sequential numbering: Natural numbers follow a sequential order. Each successive number in the set of natural numbers is greater than the previous one by one unit.
- Addition and multiplication: Natural numbers can be added, multiplied, and undergo other arithmetic operations. The result of adding two natural numbers is another natural number, and the result of multiplying two natural numbers is also a natural number.
- No lower limit: Natural numbers start from 1 and do not have a lower limit. They cannot be less than 1.
- Increment: Each natural number can be increased by one unit, resulting in the next natural number.

## Operations on natural numbers

Operations on natural numbers include addition, subtraction, multiplication, and division.

Here’s a brief overview of each of these operations:

- Addition: Addition is an operation that combines two or more numbers to obtain their sum. For example, 2 + 3 = 5. When adding natural numbers, carry-overs may occur, which need to be taken into account when adding digits.

- Subtraction: Subtraction is an operation that subtracts one number from another to obtain the difference. Subtraction of natural numbers assumes that the first number is greater than the second. For example, 7 – 3 = 4.

- Multiplication: Multiplication is an operation that combines two numbers to obtain their product. For example, 4 * 3 = 12. When multiplying natural numbers, each digit of one number is multiplied by each digit of the other number, and then the sum of the products is added together.

- Division: Division is an operation that divides one number by another to obtain the quotient. Division of natural numbers can have two results: the quotient and the remainder. For example, 10 / 2 = 5, where 5 is the quotient. If the division is not exact, there can be a remainder, indicating the unallocated portion.

Operations on natural numbers have their rules and properties, such as commutativity (the order of numbers doesn’t matter for addition and multiplication) and associativity (the grouping of numbers doesn’t matter for addition and multiplication). These properties are important for performing operations and simplifying expressions with natural numbers.

## History of natural numbers

The history of natural numbers is reflected in the development of mathematical knowledge and counting systems over many millennia.

Here are a few key moments in the history of natural numbers:

**Prehistoric period:** The earliest records of number usage date back to the Paleolithic period when people began using simple counting forms, such as dividing fingers into groups of five. Rudimentary forms of counting have been found in archaeological discoveries from various cultures.

**Ancient civilizations:** Ancient civilizations such as Mesopotamia, Egypt, India, and China developed their own numeral systems. For example, the Sumerian civilization used a sexagesimal (base-60) numeral system, while the Egyptians used a decimal system with a base of 10.

**Roman numeral system: **The Romans developed their own numeral system, known as the Roman numerals, which used letter symbols to represent numbers. The Roman numeral system was widely used in the Roman Empire and continues to be used in some contexts to this day.

**Development of Arabic numerals:** In the 9th century, Arab mathematicians, particularly Al-Khwarizmi, introduced the use of Indo-Arabic numerals, which included the numbers 0 to 9. This numeral system became known as the decimal numeral system and was widely adopted worldwide.

**Development of mathematics:** With the development of mathematics, natural numbers became the subject of many investigations. Numerous theories were developed, such as prime numbers, division and multiplication of natural numbers, as well as the concepts of even and odd numbers.

Over time, natural numbers became the foundation for the development of other number systems, such as integers, rational numbers, real numbers, and complex numbers. The study of natural numbers continues in modern mathematics, and they are used in various fields, including algebra, geometry, analysis, and number theory.