What irrational numbers?Why are they called?Where they are used, and that represent?Few can without hesitation to answer these questions.But in fact, the answers are pretty simple, though not all are needed, and in very rare situations
essence and designation
Irrational numbers are infinite non-recurring decimal.The need to introduce this concept due to the fact that in order to address new emerging challenges have been insufficient before existing concepts of actual or real, whole, natural and rational numbers.For example, to calculate the square of a variable is 2, you must use a non-periodic infinite decimals.In addition, many simple equations also have no solution without the introduction of the concept of irrational numbers.
This set is referred to as I. And, as is clear, these values can not be represented as a simple fraction, the numerator of which is an integer, and the denominator - a natural number.
first anyway this phenomenon faced Indian mathematicians in the VII century BC, when it was discovered that the square roots of certain quantities can not be identified clearly.A first proof of the existence of such numbers is credited Hippasus Pythagorean who made it in the study of an isosceles right triangle.A serious contribution to the study of this set have brought even some scientists who lived before Christ.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 the name
If the ratio in Latin - is "shot", "attitude", the prefix "ir"
gives this word of opposite meaning.Thus, the name of a plurality of these numbers indicates that they can not be correlated to an integer or fractional, are separate place.This follows from their essence.
place in the general classification
Irrational numbers along with rational refers to a group of real or virtual, which in turn are integrated.There is a subset, but distinguish algebraic and transcendental species, which will be discussed below.
Properties
Since irrational numbers - it's part of the set of real, that are applicable to them all their properties, which are studied in arithmetic (also called basic algebraic laws).
a + b = b + a (commutative);
(a + b) + c = a + (b + c) (associativity);
a + 0 = a;
a + (-a) = 0 (the existence of additive inverse);
ab = ba (commutative law);
(ab) c = a (bc) (Distributivity);
a (b + c) = ab + ac (distributive law);
ax 1 = a
ax 1 / a = 1 (the existence of return);
Comparison is also made in accordance with the general laws and principles:
If a & gt;b, and b & gt;c, then a & gt;c (transitive relation) and.t. e.
course, all the irrational numbers can be converted using the basic arithmetic operations.No special rules for this.
In addition, the irrational numbers covered by the axiom of Archimedes.It states that for any two values of a and b is true that, by taking a as term enough times, it is possible to beat b.
use
Despite the fact that in real life is not so often have to deal with them, irrational numbers do not give account.They are a great many, but they are practically invisible.We are surrounded by irrational numbers.Examples familiar to everyone - the number pi, equal to 3.1415926 ..., or e, is in fact a base of natural logarithms, 2.718281828 ... In algebra, trigonometry and geometry have to use them constantly.By the way, the well-known importance of the "golden section", ie the ratio of how much of a lower, and vice versa, also applies to this set.Less well-known "silver" - too.
on the number line, they are very close, so that between any two values, covered by a set of rational, irrational necessarily occur.
Until now, there are a lot of unresolved issues related to this set.There are criteria such as the measure of irrationality and the normal number.Mathematicians continue to investigate the most significant examples for their belonging to this or that group.For example, it assumes that E - normal number, t. E. The probability of his record different figures the same.As wee, you respect it is under investigation.The measure also called irrationality value indicates how well a particular number can be approximated by rational numbers.
algebraic and transcendental
As already mentioned, irrational numbers conditionally divided into algebraic and transcendental.Conventionally, since, strictly speaking, this classification is used to divide the set C.
Under this designation hiding complex numbers, which include the actual or real.
So algebraic called a value, which is the root of the polynomial is not identically zero.For example, the square root of 2 will fall into this category, since it is a solution of the equation x2 - 2 = 0.
All other real numbers that do not satisfy this condition are called transcendental.This species and are the most well-known and already mentioned examples - pi and the base of the natural logarithm e.
Interestingly, none, nor the second were originally bred by mathematicians as such, their irrationality and transcendence has been proven through many years after their discovery.PI evidence 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 over 2500 years.It is still not fully understood, so that modern mathematics has work to do.By the way, the first reasonably accurate calculation of this value had Archimedes.Before him all the calculations were too approximate.
for e (Euler's number, or Napier), proof of his transcendence was found in 1873.It is used in solving the equations logarithmic.
Among other examples - the values of sine, cosine and tangent for any non-zero algebraic values.
