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
Essence and designation
Irrational numbers are infinite non-periodic decimal fractions. The need to introduce this concept is due to the fact that the previously existing concepts of real or real, integer, natural and rational numbers were no longer enough to solve new emerging problems. For example, in order to calculate what the square of 2 is, you need to use non-recurring 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 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 everotherwise, Indian mathematicians encountered this phenomenon in the 7th century BC, when it was discovered that the square roots of some quantities could not be indicated 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 set 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 the name
If ratio in Latin means "fraction", "ratio", then the prefix "ir"
gives this word the opposite meaning. Thus, the name of the set of these numbers indicates that they cannot be correlated with an integer or fractional, they have a separate place. This follows from their essence.
Place in the overall classification
Irrational numbers, along with rational numbers, belong to the group of real or real numbers, which in turn belong to complex numbers. There are no subsets, however, there are algebraic and transcendental varieties, which will be discussed below.
Properties
Since irrational numbers are part of the set of real numbers, all 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 (displacement law);
(ab)c=a(bc) (distributivity);
a(b+c)=ab + ac (distributive law);
a x 1=a
a x 1/a=1 (the existence of an 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 ratio) and. etc.
Of course, all irrational numbers can be converted using basic arithmetic. There are no special rules for this.
In addition, the axiom of Archimedes applies to irrational numbers. It says that for any two quantities a and b, the statement is true that by taking a as a term enough times, you can surpass b.
Use
Despite the fact that in ordinary life one does not often encounter them, irrational numbers cannot be counted. 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 essentially the base 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 part to the smaller, and vice versa, is also
belongs to this set. Lesser known "silver" - too.
On the number line, they are very dense, so between any two values related to the set of rational ones, there is always an irrational one.
There are still a lot of unsolved problems related to this set. There are such criteria as the measure of irrationality and the normality of a number. Mathematicians continue to examine the most significant examples for their belonging to one group or another. For example, it is considered that e is a normal number, that is, the probability of different digits appearing in its entry is the same. As for pi, research is still underway regarding it. A measure of irrationality is a value 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 complex numbers are hidden, which include real or real numbers.
So, an algebraic value is a value that is the root of a polynomial that is not identically zero. For example, the square root of 2 would be in this category because it is the solution to the equation x2 - 2=0.
All other real numbers that do not satisfy this condition are called transcendental. To this varietyinclude the most famous and already mentioned examples - the number 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 2,500-year controversy about the problem of squaring the circle. It is still not fully understood, so modern mathematicians have something to work on. By the way, the first sufficiently accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too approximate.
For e (the Euler or Napier numbers), proof 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 non-zero values.
