Mathematics separated from the general philosophy about the sixth century BC.e., and from that moment it began its triumphal march around the world.Each stage of development brings something new - an elementary account of evolved, transformed into the differential and integral calculus, alternated century, the formula became more confusing and it's a time when "the beginning of the most difficult math - it disappeared from all the numbers."But what was the basis?
Getting started
Natural numbers were on par with the first mathematical operations.Once back, two back, three back ... They have appeared thanks to an Indian scientist who first brought the positional number system.The word "positional" means that the location of each digit in the number of strictly defined and corresponds to its category.For example, numbers 784 and 487 - the numbers are the same, but the numbers are not equivalently, because the first includes seven hundred, while the second - only 4. Innovation Indians picked up by the Arabs, who brought up the number of species that we knowNow.
In the ancient mystical significance attached to numbers, the greatest mathematician Pythagoras believed that the number is the basis of creation of the world on an equal basis with the basic elements - fire, water, earth, air.If we consider all the only mathematical side, that is a positive integer?Field of integers is denoted as N and is an infinite number of integers that are positive integers and 1, 2, 3, ... ∞ +.Zero is excluded.Mainly used for counting items and specify the order.
What integer math?Axioms Peano
field N is a base, which is based on elementary mathematics.Over time, the isolated field of integers, rational, complex numbers.
by Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, made its formal and paved the way for further conclusions that go beyond the area of the field N. What is a natural number, it has been found previously in simple language, the following will be considered on the basis of a mathematical definition of axiomsPeano.
- unit is considered to be a natural number.
- number that goes beyond the natural number, is a natural.
- Before the unit, there is no natural number.
- If the number b must be like for a number c, and the number of d, then c = d.
- axiom of induction, which in turn suggests that a positive integer, if a claim is dependent on the parameter is true for the number 1, then we assume that it is working and the number n of the field of natural numbers N. Then the statement is true andfor n = 1, from the field of natural numbers N.
Basic operations for the field of natural numbers
Since N field was the first to mathematical calculations, it is to be treated as the domain and the range of the number of operations below.They are closed and no.The main difference is that the closed operations guaranteed leave the result in the framework of N, regardless of what numbers are involved.It is enough that they are natural.The outcome of the rest of numerical interactions is not so straightforward and depends on the fact that for those involved in the expression, as it may conflict with the basic definition.So, closed operations:
- addition - x + y = z, where x, y, z are included in the N;
- multiplication - x * y = z, where x, y, z is from field N;
- exponentiation - xy, where x, y are included in the box N.
remaining operations, the results of which may not exist in the context of the definition of "what is a natural number", the following:
- subtraction - x - y = z.Field integers allows it only if x is greater than y;
- division - x / y = z.Field integers allows it only if z is divided by y no remainder, that is divisible.
numbers properties belonging to the field of N
All further mathematical reasoning will be based on these properties, the most trivial, but no less important.
- commutative property of addition - x + y = y + x, where the numbers x, y included in the N. Or the well-known "by the relocation of sum does not change."
- commutative property of multiplication - x * y = y * x, where the numbers x, y are included in the N.
- associative property of addition - (x + y) + z = x + (y + z), where x, y, z is from field N.
- associative property of multiplication - (x * y) * z = x * (y * z), where the numbers x, y, z are included in the N.
- distributive property - x (y +z) = x * y + x * z, where the numbers x, y, z are included in the box N.
Table Pythagoras
One of the first steps in the knowledge of the students of the entire structure of elementary mathematics after they have understood for himself,which numbers are called natural, it is a table of Pythagoras.It can be seen not only from the point of view of science, but also as a valuable scientific monument.
This multiplication table has undergone a number of changes over time: it removed from zero, and numbers from 1 to 10 stand for themselves, excluding orders of magnitude (hundreds, thousands ...).It is a table in which the title rows and columns - the number and contents of the cells of their intersection is equal to the product of their own.
In practical training the last few decades there was the need to memorize the table of Pythagoras "in order", that is, first went memorization.Multiplication 1 is excluded because the result is equal to 1 or greater factor.Meanwhile, in the table can be seen with the naked eye pattern: the product of the numbers increases by one step, which is equal to the line title.Thus, the second factor shows us how many times you need to take the first, in order to obtain the desired product.This system is unlike the more convenient one that was practiced in the Middle Ages: Even knowing that is a positive integer and how it is trivial, people managed to complicate yourself everyday by using a system that was based on a power of two.
subset as the cradle of mathematics
At the moment, the field of natural numbers N considered only as one of the subsets of the complex numbers, but that does not make them less valuable to science.A positive integer - the first thing a child learns by studying ourselves and the world around us.Every finger, two finger ... Thanks to him, a man formed by logical thinking and the ability to determine the cause and conclusions of the investigation, setting the stage for greater openness.
