What are irrational numbers? Why are they called that? 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 everyone needs them and in very rare situations
The essence and designation
Irrational numbers are infinite non-periodic decimal fractions. The necessity of introducing this concept is due to the fact that the previously existing concepts of real or material, integer, natural and rational numbers were not enough to solve new emerging problems. For example, in order to calculate which square is 2, it is necessary to use non-periodic infinite decimal fractions. In addition, many simple equations also have no solution without introducing the concept of an irrational number.
This set is denoted as I. And, as is already clear, these values cannot be represented as a simple fraction, in the numerator of which there will be an integer, and in the denominator - a natural number.
For the first time, one way or another, Indian mathematicians encountered this phenomenon in the 7th century
BC, when it was discovered that the square roots of some quantities cannot be designated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this in the process of studying an isosceles right triangle. A serious contribution to the study of this multitude was made by some other scientists who lived before our era. The introduction of the concept of irrational numbers entailed a revision of the existing mathematical system, which is why they are so important.
origin of name
If the ratio in Latin is “fraction”, “relation”, then the prefix “ir"
gives the word the opposite meaning. Thus, the name of the set of these numbers suggests that they cannot be correlated with integer or fractional, have a separate place. This follows from their essence.
Place in the general classification
Irrational numbers, along with rational ones, belong to the group of real or real ones, which in turn belong to complex ones. There are no subsets, however, an algebraic and transcendental variety is distinguished, which will be discussed below.
The properties
Since irrational numbers are part of the set of real ones, all of their properties that are studied in arithmetic (they are also called basic algebraic laws) apply to them.
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 (relocation law);
(ab) c = a (bc) (distributivity);
a (b + c) = ab + ac (distribution law);
ax 1 = a
ax 1 / a = 1 (the existence of the inverse number);
Comparison is also carried out in accordance with general laws and principles:
If a> b and b> c, then a> c (transitivity of the relation) and. etc.
Of course, all irrational numbers can be converted using basic arithmetic. There are no special rules.
In addition, the action of the axiom of Archimedes extends to irrational numbers. It states that for any two quantities a and b, the statement is true that, taking a as a term a sufficient number of times, we can exceed b.
Using
Despite the fact that in ordinary life it is not so often necessary to deal with them, irrational numbers can not be counted. There are a lot of them, but they are almost invisible. We are surrounded by irrational numbers. Examples familiar to all are the pi number, equal to 3.1415926 ..., or e, which in essence 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 both the larger to the smaller, and vice versa, is also
refers to this set. The lesser-known silver one, too.
On the numerical line they are located very densely, so that between any two quantities assigned to the set of rational, irrational is necessarily met.
There are still many unsolved problems associated with this set. There are criteria such as a measure of irrationality and the normality of a number. Mathematicians continue to explore the most significant examples of their belonging to one or another group. For example, it is believed that e is a normal number, that is, the probability of the appearance of different digits in his record is the same. As for pi, research is still being conducted on it. A measure of irrationality is called a quantity that shows how well a particular number can be approximated by rational numbers.
Algebraic and Transcendental
As already mentioned, irrational numbers are conditionally divided into algebraic and transcendental. Conditionally, since, strictly speaking, this classification is used to divide the set C.
Under this designation are complex numbers, which include real or real.
So, algebraic is called such a value that is the root of a polynomial that is not identically equal to zero. For example, the square root of 2 will fall into this category because it is a solution to the equation x 2 - 2 = 0.
All other real numbers that do not satisfy this condition are called transcendental. This variety includes the most famous and already mentioned examples - pi and the base of the natural logarithm e.
Interestingly, neither one nor the second was originally deduced by mathematicians in this capacity, their irrationality and transcendence were proved many years after their discovery. For pi, the proof was given in 1882 and simplified in 1894, which put an end to the debate about the problem of squaring the circle, which lasted for 2.5 thousand years. It is still not fully understood, so modern mathematicians have something to work on. By the way, the first fairly accurate calculation of this value was carried out by Archimedes. Before him, all the calculations were too rough.
For e (Euler or Nepher numbers), evidence of its transcendence was found in 1873. It is used in solving logarithmic equations.
Other examples include sine, cosine, and tangent values for any algebraic nonzero values.