Mathematics stood out from general philosophy around the sixth century BC. e., and from that moment began its victorious march around the world. Each stage of development introduced something new - an elementary account evolved, transformed into differential and integral calculus, centuries changed, formulas became more complicated, and the moment came when “the most complicated mathematics began - all numbers disappeared from it”. But what was the basis?
The beginning of time
Natural numbers appeared along with the first mathematical operations. Once a spine, two spines, three spines ... They appeared thanks to Indian scientists who developed the first positional number system.
The word "positionality" means that the location of each digit in a number is strictly defined and corresponds to its category. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundred, while the second only 4. The Arabs picked up the Indians, who brought the numbers to the form that we know now.
In ancient times, numbers were given a mystical meaning, the greatest mathematician Pythagoras believed that number lies at the heart of the creation of the world along with the basic elements - fire, water, earth, air. If we consider everything only from the mathematical side, then what is a natural number? The field of natural numbers is denoted as N and represents an infinite series of numbers that are integers and positives: 1, 2, 3, ... + ∞. Zero is excluded. It is used mainly for counting objects and indicating order.
What is a natural number in math? Axioms of Peano
Field N is the basic field on which elementary mathematics is based. Over time, the fields of integer, rational, complex numbers were distinguished .
The work of the Italian mathematician Giuseppe Peano made it possible to further structure arithmetic, achieved its formality and set the stage for further conclusions that went beyond the field field N.
What is a natural number, it was clarified earlier in plain language, below we will consider a mathematical definition based on Peano's axioms.
- The unit is considered a natural number.
- The number that comes after the natural number is natural.
- There is no natural number in front of the unit.
- If the number b follows both the number c and the number d, then c = d.
- The axiom of induction, which in turn shows what a natural number is: if some statement that depends on the parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is also true for n = 1 from the field of natural numbers N.
Basic operations for the field of natural numbers
Since the field N was the first for mathematical calculations, it is precisely to this field that both the definition areas and the ranges of values of a number of operations below are related. They are closed and not. The main difference is that closed operations are guaranteed to leave a result within the set N, regardless of which numbers are involved. It’s enough that they are natural. The outcome of the remaining numerical interactions is no longer so straightforward and directly depends on what kind of numbers are involved in the expression, since it can contradict the basic definition. So, closed operations:
- addition - x + y = z, where x, y, z are included in the field N;
- multiplication - x * y = z, where x, y, z are included in the field N;
- exponentiation - x y , where x, y are included in the field N.
The remaining operations, the result of which may not exist in the context of the definition of "what is a natural number", are as follows:
- subtraction - x - y = z. The field of natural numbers admits it only if x is greater than y;
- division - x / y = z. The field of natural numbers admits it only if z is divisible by y without a remainder, that is, completely.
Properties of numbers belonging to the field N
All further mathematical considerations will be based on the following properties, the most trivial, but from this no less important.
- The translational property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or everyone knows, "the sum does not change from a change in the places of the terms."
- The translational property of the multiplication is x * y = y * x, where the numbers x, y are included in the field N.
- The combination property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the field N.
- The combined property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the field N.
- distribution property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the field N.
Pythagoras table
One of the first steps in the students' understanding of the whole structure of elementary mathematics after they have figured out for themselves which numbers are called natural numbers is the Pythagorean table. It can be considered not only from the point of view of science, but also as a most valuable scientific monument.
This multiplication table has undergone a number of changes over time: zero was removed from it, and numbers from 1 to 10 denote themselves, excluding orders (hundreds, thousands ...). It is a table in which the titles of rows and columns are numbers, and the contents of the cells of their intersection are equal to their product.
In the practice of teaching over the past decades, there has been a need to memorize the Pythagorean table "in order", that is, memorization first went on. Multiplication by 1 was excluded, as the result was equal to 1 or a larger factor. Meanwhile, in the table with a naked eye you can notice the pattern: the product of numbers grows by one step, which is equal to the title of the line. Thus, the second factor shows us how many times you need to take the first in order to get the desired product. This system is not an example more convenient than the one that was practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their daily count by using a system that was based on powers of two.
Subset like the cradle of math
At the moment, the field of natural numbers N is considered only as one of the subsets of complex numbers, but this does not make them less valuable in science. A natural number is the first thing a child learns by studying himself and the world around him. Once a finger, two fingers ... Thanks to him, a person develops logical thinking, as well as the ability to determine the cause and display the effect, preparing the ground for great discoveries.