Pythagoras claimed that the number lies at the base of the world on a par with the basic elements. Plato believed that the number connects the phenomenon and noumenon, helping to cognize, measure and draw conclusions. Arithmetic comes from the word "arithmos" - a number, the beginning of the beginnings in mathematics. Any object can be described with it - from an elementary apple to abstract spaces.
Needs as a development factor
At the initial stages of the formation of society, the needs of people were limited by the need to keep track of - one bag of grain, two bags of grain, etc. For this, natural numbers were sufficient, the set of which is an infinite positive sequence of integers N.
Later, with the development of mathematics as a science, the need arose for a separate field of integers Z - it includes negative values and zero. His appearance at the household level was provoked by the fact that in primary accounting it was necessary to somehow fix debts and losses. At the scientific level, negative numbers made it possible to solve the simplest linear equations. Among other things, it has now become possible to image a trivial coordinate system, because a reference point has appeared.
The next step was the need to enter fractional numbers, since science did not stand still, more and more new discoveries required a theoretical basis for a new growth impetus. So the field of rational numbers Q appeared.
Finally, rationality ceased to satisfy requests, because all new conclusions required justification. A field of real numbers R, the works of Euclid on the incommensurability of some quantities due to their irrationality, appeared. That is, the ancient Greek mathematicians positioned the number not only as a constant, but also as an abstract quantity, which is characterized by the ratio of incommensurable quantities. Due to the fact that real numbers appeared, such quantities as “pi” and “e” “saw the light”, without which modern mathematics could not have taken place.
The final innovation was the complex number C. It answered a number of questions and refuted the previously introduced postulates. Due to the rapid development of algebra, the outcome was predictable - having real numbers, solving many problems was impossible. For example, thanks to complex numbers, string and chaos theories stood out, and the equations of hydrodynamics expanded.
Set theory. Cantor
The concept of infinity has always been controversial, since it could neither be proved nor disproved. In the context of mathematics, which operated on strictly verified postulates, this manifested itself most clearly, especially since the theological aspect still had weight in science.
However, thanks to the work of mathematician George Cantor, everything fell into place over time. He proved that there are infinite sets of infinite sets, and that the field R is larger than the field N, even though they both have no end. In the middle of the XIX century, his ideas were loudly called nonsense and a crime against the classical, unshakable canons, but time put everything in its place.
The main properties of the field R
Real numbers have not only the same properties as the sub-options that are included in them, but also complemented by others due to the scale of their elements:
- Zero exists and belongs to the field R. c + 0 = c for any c from R.
- Zero exists and belongs to the field R. c x 0 = 0 for any c from R.
- The relation c: d for d ≠ 0 exists and is valid for any c, d from R.
- The field R is ordered, that is, if c ≤ d, d ≤ c, then c = d for any c, d from R.
- Addition in the field R is commutative, that is, c + d = d + c for any c, d from R.
- Multiplication in the field R is commutative, that is, c x d = d x c for any c, d from R.
- Addition in the field R is associative, i.e. (c + d) + f = c + (d + f) for any c, d, f from R.
- Multiplication in the field R is associative, that is, (c x d) x f = c x (d x f) for any c, d, f from R.
- For each number from the field R there is an opposite to it, such that c + (-c) = 0, where c, -c from R.
- For each number from the field R there is an inverse such that c x c -1 = 1, where c, c -1 from R.
- The unit exists and belongs to R, so c x 1 = c, for any c from R.
- The distribution law is valid, so c x (d + f) = c x d + c x f, for any c, d, f from R.
- In the field R, zero is not equal to unity.
- The field R is transitive: if c ≤ d, d ≤ f, then c ≤ f for any c, d, f from R.
- In the field R, order and addition are interconnected: if c ≤ d, then c + f ≤ d + f for any c, d, f from R.
- In the field R, order and multiplication are interrelated: if 0 ≤ c, 0 ≤ d, then 0 ≤ c x d for any c, d from R.
- Both negative and positive real numbers are continuous, that is, for any c, d from R there is f from R such that c ≤ f ≤ d.
Module in field R
Real numbers include such a thing as a module.
It is denoted as | f | for any f from R. | f | = f if 0 ≤ f and | f | = -f if 0> f. If we consider the module as a geometric quantity, then it represents the distance traveled - it doesn’t matter if you “went” for zero to minus or forward to plus.
Complex and real numbers. What is common and what are the differences?
By and large, complex and real numbers are the same thing, except that the imaginary unit i joined the first, the square of which is -1. Elements of the fields R and C can be represented as the following formula:
- c = d + f x i, where d, f belong to the field R, and i is the imaginary unit.
To get c from R in this case, f is simply considered equal to zero, that is, only the real part of the number remains. Due to the fact that the field of complex numbers has the same set of properties as the field of real ones, f x i = 0 if f = 0.
Regarding practical differences, for example, in the field R the quadratic equation is not solved if the discriminant is negative, while the field C does not impose such a restriction due to the introduction of the imaginary unit i.
Summary
The “bricks” of axioms and postulates on which mathematics are based are not replaced. Some of them, in connection with the increase in information and the introduction of new theories, put the following "bricks", which in the future can become the basis for the next step. For example, natural numbers, despite being a subset of the real field R, do not lose their relevance. It is on them that all elementary arithmetic is based, with which man begins to know the world.
From a practical point of view, real numbers look like a straight line. On it you can choose a direction, indicate the origin and step. A straight line consists of an infinite number of points, each of which corresponds to a single real number, regardless of whether it is rational or not. From the description it is clear that we are talking about the concept on which both mathematics in general and mathematical analysis in particular are built.