Real numbers and their properties

Pythagoras claimed that the number lies at the base of the world on a par with the main elements. Plato believed that the number connects the phenomenon and noumenon, helping to cognize, measure and draw conclusions. Arithmetic comes from the word "arithmos" - the number, the beginning began in mathematics. Nim can describe any object - from an elementary apple to abstract spaces.

Needs as a factor of development

At the initial stages of the formation of society, the needs of people were limited to the need to keep score - one bag of grain, two bags of grain, etc. To do this, it was enough to have natural numbers, the set of which is an infinite positive sequence of integers N.

Later, with the development of mathematics as a science, there arose a need for a separate field of integers Z - it includes negative quantities and zero. His appearance at the household level was provoked by the fact that in the primary accounts department it was necessary to somehow fix the debts and losses. At the scientific level, negative numbers made it possible to solve the simplest linear equations. Among other things, now it became possible to display a trivial coordinate system, because a reference point 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 push of growth. So there appeared the field of rational numbers Q.

Finally, rationality ceased to satisfy requests, because all new conclusions required justification. A field of real numbers R appeared, Euclid's works on the incommensurability of certain quantities due to their irrationality. That is, the ancient Greek mathematicians positioned the number not only as a constant, but also as an abstract value, which is characterized by the ratio of incommensurable quantities. Due to the fact that real numbers have appeared, "values" such as "pi" and "e" have "seen 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 disproved the previously introduced postulates. Because of the rapid development of algebra, the outcome was predictable - having real numbers, the solution of many problems was impossible. For example, due to complex numbers, string and chaos theories have been singled out, the equations of hydrodynamics have broadened.

The theory of sets. Cantor

The concept of infinity at all times has been controversial, since it could neither be proved nor disproved. In the context of mathematics, which operated with strictly verified postulates, this manifested itself most clearly, especially since the theological aspect still had a weight in science.

However, thanks to the work of the mathematician Georg Cantor, everything fell into place with the passage of time. He proved that infinite sets exist an infinite set, and that the field R is greater than the field N, let both of them have no end. In the middle of the XIX century, his ideas were loudly called delirium and crime against the classical, unshakable canons, but time put everything in its place.

The basic properties of the field R

Real numbers have not only the same properties as the sub-missions, which are included in them, but are also supplemented by others due to the weight of their elements:

• Zero exists and belongs to the field R. c + 0 = c for any c in R.
• A zero exists and belongs to the field R. c x 0 = 0 for any c in R.
• The ratio c: d for d ≠ 0 exists and is real for any c, d in R.
• The field R is ordered, that is, if c ≤ d, d ≤ c, then c = d for any c, d in R.
• The addition in the field R is commutative, that is, c + d = d + c for any c, d in R.
• The multiplication in the field R is commutative, that is, cx d = dx c for any c, d in R.
• The addition in the field R is associative, that is, (c + d) + f = c + (d + f) for any c, d, f in R.
• The multiplication in the field R is associative, that is (c x d) x f = c x (d x f) for any c, d, f in R.
• For each number from the field R there exists an opposite one, such that c + (-c) = 0, where c, -c from R.
• For every number in the field R there exists an inverse such that c x c ^-1 = 1, where c, c ^-1 of R.
• A unit exists and belongs to R, so that c x 1 = c, for any c in R.
• The distribution law holds, so that c x (d + f) = c x d + c x f, for any c, d, f in R.
• In the field R, zero is not equal to one.
• The field R is transitive: if c ≤ d, d ≤ f, then c ≤ f for any c, d, f in R.
• In the field R, the order and addition are interrelated: if c ≤ d, then c + f ≤ d + f for any c, d, f in R.
• In the field R the 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 in R there is an f of R such that c ≤ f ≤ d.

The module in the field R

Real numbers include such a thing as a module. It is denoted as | f | For any f in R. | f | = F, if 0 ≤ f and | f | = -f if 0> f. If we consider the module as a geometric value, then it represents the distance traveled - it does not matter if you "passed" by zero in the minus or forward to the plus.

Complex and real numbers. What is common and what are the differences?

By and large, complex and real numbers are one and the same, except that the imaginary unit i, whose square is -1, joined the first. The 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. Because the field of complex numbers has the same set of properties as the field of real numbers, f x i = 0, if f = 0.

With respect to practical differences, for example, in the field R the quadratic equation is not solved if the discriminant is negative, whereas the field C does not impose such a restriction due to the introduction of the imaginary unit i.

Results

The "bricks" of the axioms and postulates on which mathematics is based do not change. Some of them, in connection with the increase of 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 the cognition of the man of the world begins.

From a practical point of view, the real numbers look like a straight line. On it you can choose the direction, indicate the origin and step. The line consists of an infinite number of points, each of which corresponds to a single real number, whether rational or not. From the description it is clear that we are talking about a concept on which to build both mathematics in general, and mathematical analysis in particular.