Irrational numbers: what is it and what are they used for?
What are irrational numbers? Why are they so called? Where are they used and what are they? Few can answer these questions without hesitation. But in fact the answers to them are quite simple, although not all are needed in very rare situations
The essence and designation
Irrational numbers areinfinite non-periodic decimals. The need to introduce this concept is due to the fact that for solving new emerging problems, there were not enough previously existing concepts of real or real, integer, natural and rational numbers. For example, in order to calculate, by the square of what value is 2, it is necessary to use non-periodic infinite decimals. In addition, many of the simplest equations also have no solution without introducing the concept of an irrational number.
This set is denoted as I. And, as already clear, these values can not be represented in the form of a simple fraction, in the numerator of which there is an integer, and in the denominator - a natural number.
For the first time, one way or another, they encountered this phenomenonIndian mathematicians in the 7th century BC, when it was discovered that square roots of some quantities can not be clearly designated. And the first proof of the existence of such numbers is attributed to the Pythagorean Gippus, who did this in the process of studying an isosceles right triangle. A serious contribution to the study of this set was brought by some other scholars who lived before our era. The introduction of the concept of irrational numbers led to a revision of the existing mathematical system, which is why they are so important.
origin of name
If the ratio in translation from Latin is "fraction", "ratio", the prefix "ir"
gives this word the opposite meaning. Thus, the name of the set of these numbers indicates that they can not be correlated with integer or fractional, have a separate place. This follows from their essence.
Place in the general classification
Irrational numbers along with rationalrefers to the group of real or real, which in turn are complex. There are no subsets, but they distinguish between an algebraic and transcendental variety, which we will discuss below.
Since irrational numbers are part of the set of real numbers, all of their properties are applicable to them, which are studied in arithmetic (they are also called basic algebraic laws).
a + b = b + a (commutativity);
(a + b) + c = a + (b + c) (associativity);
a + 0 = a;
a + (-a) = 0 (the existence of the opposite number);
ab = ba (displacement law);
(ab) c = a (bc) (distributivity);
a (b + c) = ab + ac (distribution law);
a x 1 = a
a x 1 / a = 1 (the existence of an inverse number);
The comparison is also conducted in accordance with general laws and principles:
If a> b and b> c, then a> c (transitivity of the relation) and. and so forth.
Of course, all irrational numbers can be transformed with the help of basic arithmetic operations. No special rules in this case.
In addition, irrational numbersThe action of the axiom of Archimedes extends. It states that for any two quantities a and b, the following assertion holds: taking a as a summand a sufficient number of times, one can exceed b.
Despite the fact that in ordinary life, not so muchoften have to deal with them, irrational numbers do not lend themselves to account. There are a lot of them, but they are almost invisible. We are surrounded by irrational numbers everywhere. Examples familiar to everyone are the number pi, equal to 3,1415926 ..., or e, which is the basis of the natural logarithm, 2,718281828 ... In algebra, trigonometry and geometry, they have to be used constantly. By the way, the famous meaning of the "golden section", that is, the ratio of the majority to the smaller, and vice versa, alsorefers to this set. Less well-known "silver" - too.
On the number line, they are very dense, so that between any two quantities referred to the set of rational ones, one must find an irrational one.
Until now, there are many unsolved problems,connected with this set. There are such criteria as a measure of irrationality and the normality of a number. Mathematicians continue to explore the most significant examples of their belonging to a particular group. For example, it is considered that e is a normal number, that is, the probability of occurrence of different digits in its record is the same. As for pi, research is still underway with regard to it. A measure of irrationality is a quantity indicating how well a number can be approximated by rational numbers.
Algebraic and transcendental
As already mentioned, irrational numbers are arbitrarily divided into algebraic and transcendental. Conditionally, since, strictly speaking, this classification is used to divide the set C.
Under this notation are complex numbers that include real or real numbers.
So, algebraic is called such a value,which is the root of a polynomial that is not identically equal to zero. For example, the square root of 2 will belong to this category, since it is a solution of the equation x2- 2 = 0.
All the rest are real numbers, notThe conditions satisfying this condition are called transcendental. The most famous and already mentioned examples refer to this variety - the number pi and the base of the natural logarithm e.
Interestingly, neither one nor the other wasoriginally derived by mathematicians in this capacity, their irrationality and transcendence were proved many years after their discovery. For pi proof was given in 1882 and simplified in 1894, which put an end to the debate about the problem of quadrature of the circle, which lasted for 2.5 thousand years. It is still not fully understood, so that modern mathematicians have something to work on. By the way, the first accurate calculation of this value was made by Archimedes. Before him, all calculations were too approximate.
For e (the number of Euler or Napier), the proof of his transcendence was found in 1873. It is used in solving logarithmic equations.
Among other examples are the sine, cosine and tangent values for any algebraic non-zero values.