Pythagoras claimed that the number is the foundation of the world on an equal basis with the basic elements.Plato believed that the number of links the phenomenon and noumenon, helping to know, to be weighed and draw conclusions.Arithmetic comes from the word "arifmos" - the number, the starting point in mathematics.It is possible to describe any object - from elementary to apple abstract spaces.
needs as a factor of
In the initial stages of the society needs people limited by the need to keep score - one bag of grain, two sacks of grain, and so on. D. To do this, it was natural numbers, the set of which is an infinite sequence of positive integersN.
Later, with the development of mathematics as a science, it was necessary to separate the field of integers Z - it includes negative values and zero.His appearance at the household level was triggered by the fact that the initial accounting had to somehow fix the debts and losses.On the scientific level, negative numbers made it possible to solve simple linear equations.Among other things, it is now possible to image trivial coordinate system, ie. A. Appeared benchmark.
The next step was the need to enter fractional numbers, because science does not stand still, more and more new discoveries demanded a theoretical framework for a new push growth.So there was a field of rational numbers Q.
finally ceased to meet the demands of rationality, because all new findings require justification.There the field of real numbers R, the works of Euclid's incommensurability some variables because of their irrationality.That is, the number of Greek mathematics positioned not only as a constant, but as an abstract value which is characterized by the ratio of incommensurable magnitudes.Due to the fact that there are real numbers, "saw the light" quantities such as "pi" and "e", without which modern mathematics would not have taken place.
The final innovation was a complex number C. It responded to a number of issues and denied previously entered postulates.Due to the rapid development of algebra outcome was predictable - with real numbers, the decision of many problems was not possible.For example, with complex numbers stood out string theory and chaos expanded the equations of hydrodynamics.
Set Theory.Cantor
concept of infinity has always caused controversy since it was impossible to prove or disprove.In the context of mathematics, which is operated strictly verified postulates, it manifests itself most clearly, especially as theological aspects still weighed in science.
However, through the work of mathematician Georg Cantor all time fell into place.He proved that there is an infinite set of infinite set, and that the field R is greater than the field N, let both of them and have no end.In the middle of the XIX century, his ideas loudly called nonsense and a crime against classical immutable canons, but time will put everything in its place.
basic properties of the field R
Actual numbers not only have the same properties as the podmozhestva that they include, but are supplemented by other effect masshabnosti its elements:
- Zero exists and belongs to the field R. c + 0 =c for any c of R.
- Zero exists and belongs to the field R. c x 0 = 0 for any c of R.
- ratio of c: d if d ≠ 0 exists and is valid for any c, d of R.
- Golf R is ordered, that is, if c ≤ d, d ≤ c, then c = d for all c, d of R.
- Addition in R is commutative, that is, c + d = d + c for any c,d of R.
- multiplication in R is commutative, that is c x d = d X c for any c, d of R.
- Addition in R is an associative, that is, (c + d) + f = c+ (d + f) for any c, d, f of R.
- Multiplication in R is associative i.e. (c x d) x = f x c (d x f) for any c, d, f of R.
- For each number of the field R, there exists its opposite, such that c + (-c) = 0, where c, -c from R.
- For each number of the field R there opposite him, so that c x c-1 = 1 where c, c-1 of R.
- Unit exists and belongs to R, so that c 1 = c x, c for each of R.
- Valid distributive law, so that c x (d + f) = c d x + c x f, for any c, d, f of R.
- in R not equal to zero to unity.
- field R is transitive: if d ≤ c, d ≤ f, then f ≤ c for any c, d, f of R.
- field R and the order of addition of interrelated: if d ≤ c, then c + f ≤d + f for all c, d, f of R.
- The R field multiplication procedure and linked: if 0 ≤ c, d ≤ 0, then 0 ≤ c x d for any c, d of R.
- As negativeand positive real numbers are continuous, that is, for any c, d of R there exists f in R, such that c ≤ f ≤ d.
module in the R
Real numbers include such a thing as a module.It denotes both | f | for all f in R. | f | = f, if 0 ≤ f and | f | = -f, if 0 & gt;f.If we consider the module as a geometric value, it represents the distance traveled - whether "passed" you as zero in the negative to the positive or forward.
Complex and real numbers.What are the similarities and differences?
By and large, complex and real numbers - is the same, except that the first has joined the imaginary unit i, whose square is -1.Elements fields R and C can be represented by the following formula:
- c = d + f x i, where d, f belong to the field R, and i - imaginary unit.
To obtain c of R in the case f simply assumed to be zero, then there is only the real part of the number.Because the complex field has the same feature set as the real field, f x i = 0 if f = 0.
regards practical differences, for example in R quadratic equation can not be solved if the discriminant negativewhereas the field C does not impose such a limitation due to the introduction of the imaginary unit i.
Results
"bricks" of axioms and postulates upon which the mathematics do not change.On some of them due to the increase of information and the introduction of new theories placed the following "bricks" that could potentially be the basis for the next step.For example, natural numbers, despite the fact that they are a subset of the real field R, do not lose their relevance.It is on the basis of all of them elementary arithmetic, which begins the knowledge of a man of peace.
From a practical point of view, the real numbers look like a straight line.It is possible to choose the direction, to designate the origin and pitch.Direct consists of an infinite number of points, each of which corresponds to a single real number, regardless of whether or not it is efficient.From the description it is clear that we are talking about the concept, which is based on mathematics in general, and mathematical analysis in particular.
