What are irrational numbers? Why are they called that? Where are they used and what are they? Few people can answer these questions without thinking. 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 numbers. The need to introduce this concept is due to the fact that to solve new problems that arise, the previously existing concepts of real or real, integer, natural and rational numbers were no longer sufficient. For example, in order to calculate which quantity is the square of 2, you need to use non-periodic infinite decimals. 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, the numerator of which will be an integer, and the denominator will be
For the first time, one way or another, Indian mathematicians encountered this phenomenon in the 7th century when it was discovered that the square roots of some quantities cannot 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 the isosceles right triangle. Some other scientists who lived before our era made a serious contribution to the study of this set. The introduction of the concept of irrational numbers entailed a revision of the existing mathematical system, that's why they are so important.
Origin of the name
If ratio translated from Latin is “fraction”, “ratio”, then the prefix “ir”
gives this word opposite meaning. Thus, the name of the set of these numbers indicates that they cannot be correlated with an integer or fraction and have a separate place. This follows from their essence.
Place in the general 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, but 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 + (-a) = 0 (existence of the opposite number);
ab = ba (commutative law);
(ab)c = a(bc) (distributivity);
a(b+c) = ab + ac (distribution law);
a x 1/a = 1 (existence of a reciprocal number);
The 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 for this.
In addition, the Archimedes axiom applies to irrational numbers. It states that for any two quantities a and b, it is true that if you take a as a term enough times, you can beat b.
Usage
Despite the fact that you don’t encounter them very often in everyday life, irrational numbers cannot be counted. There are a huge number of them, but they are almost invisible. Irrational numbers are all around us. Examples that are familiar to everyone are pi, which is 3.1415926..., or e, which is essentially the base natural logarithm, 2.718281828... In algebra, trigonometry and geometry they have to be used constantly. By the way, the famous meaning of the “golden ratio”, that is, the ratio of both the larger part to the smaller part, and vice versa, also
belongs to this set. The lesser known “silver” one too.
On the number line they are located very densely, so that between any two quantities classified as rational, an irrational one is sure to occur.
There are still a lot unresolved problems associated with this set. There are criteria such as the measure of irrationality and the normality of a number. Mathematicians continue to study the most significant examples to determine whether they belong to one group or another. For example, it is believed that e is a normal number, i.e. the probability of appearing in its notation different numbers is the same. As for pi, research is still underway regarding it. The measure of irrationality is a value that shows how well a given number can be approximated by rational numbers.
Algebraic and transcendental
As already mentioned, irrational numbers are conventionally divided into algebraic and transcendental. Conditionally, since, strictly speaking, this classification is used to divide the set C.
Hidden under this designation are complex numbers, which include real or material ones.
So, an algebraic value is a value that is the root of a polynomial that is not identically equal to zero. For example, square root of 2 would 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 - the number pi and the base of the natural logarithm e.
Interestingly, neither one nor the other were originally developed by mathematicians in this capacity; their irrationality and transcendence were proven many years after their discovery. For pi, the proof was given in 1882 and simplified in 1894, ending a 2,500-year debate about the problem of squaring the circle. It has not yet been fully studied, so modern mathematicians there is something to work on. By the way, the first fairly accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too approximate.
For e (Euler's or Napier's number), proof of its transcendence was found in 1873. It is used in solving logarithmic equations.
Other examples include the values of sine, cosine, and tangent for any algebraic nonzero value.
1.Proofs are examples of deductive reasoning and are different from inductive or empirical arguments. A proof must demonstrate that the statement being proven is always true, sometimes by listing all possible cases and showing that the statement holds in each of them. A proof may rely on obvious or generally accepted phenomena or cases known as axioms. Contrary to this, the irrationality of the “square root of two” is proven.
2. The intervention of topology here is explained by the very nature of things, which means that there is no purely algebraic way to prove irrationality, in particular based on rational numbers. Here is an example, the choice is yours: 1 + 1/2 + 1/4 + 1/8 ….= 2 or 1+1/2 + 1/4 + 1/8 …≠ 2 ???
If you accept 1+1/2 + 1/4 + 1/8 +…= 2, which is considered the “algebraic” approach, then it is not at all difficult to show that there exists n/m ∈ ℚ, which on an infinite sequence is irrational and finite number. This suggests that the irrational numbers are the closure of the field ℚ, but this refers to a topological singularity.
So for Fibonacci numbers, F(k): 1,1,2,3,5,8,13,21,34,55,89,144,233,377, … lim(F(k+1)/F(k)) = φ
This only shows that there is a continuous homomorphism ℚ → I, and it can be shown rigorously that the existence of such an isomorphism is not a logical consequence of the algebraic axioms.
Definition of an irrational number
Irrational numbers are those numbers that in decimal notation represent endless non-periodic decimal fractions.
So, for example, numbers obtained by taking the square root of natural numbers are irrational and are not squares of natural numbers. But not all irrational numbers are obtained by extraction square roots, because the number “pi” obtained by division is also irrational, and you are unlikely to get it when trying to extract the square root of a natural number.
Properties of irrational numbers
Unlike numbers written as infinite decimals, only irrational numbers are written as non-periodic infinite decimals.
The sum of two non-negative irrational numbers can end up being a rational number.
Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class which do not have the large number, and in the upper there is no less.
Any real transcendental number is irrational.
All irrational numbers are either algebraic or transcendental.
The set of irrational numbers on a line is densely located, and between any two of its numbers there is sure to be an irrational number. rational number.
The set of irrational numbers is infinite, uncountable and is a set of the 2nd category.
When performing any arithmetic operation on rational numbers, except division by 0, the result will be a rational number.
When adding a rational number to an irrational number, the result is always an irrational number.
When adding irrational numbers, we can end up with a rational number.
The set of irrational numbers is not even.
Numbers are not irrational
Sometimes it is quite difficult to answer the question whether a number is irrational, especially in cases where the number has the form decimal or in the form numerical expression, root or logarithm.
Therefore, it will not be superfluous to know which numbers are not irrational. If we follow the definition of irrational numbers, then we already know that rational numbers cannot be irrational.
Irrational numbers are not:
First, all natural numbers;
Secondly, integers;
Thirdly, common fractions;
Fourthly, different mixed numbers;
Fifthly, these are infinite periodic decimal fractions.
In addition to all of the above, an irrational number cannot be any combination of rational numbers that is performed by the signs of arithmetic operations, such as +, -, , :, since in this case the result of two rational numbers will also be a rational number.
Now let's see which numbers are irrational:
Do you know about the existence of a fan club, where fans of this mysterious mathematical phenomenon are looking for more and more information about Pi, trying to unravel its mystery? Any person who knows by heart a certain number of Pi numbers after the decimal point can become a member of this club;
Did you know that in Germany, under the protection of UNESCO, there is the Castadel Monte palace, thanks to the proportions of which you can calculate Pi. King Frederick II dedicated the entire palace to this number.
It turns out that they tried to use the number Pi in the construction of the Tower of Babel. But unfortunately, this led to the collapse of the project, since at that time the exact calculation of the value of Pi was not sufficiently studied.
Singer Kate Bush in her new disc recorded a song called “Pi”, in which one hundred and twenty-four numbers from the famous number series 3, 141… were heard.
Fraction m/n we will consider it irreducible (after all, a reducible fraction can always be reduced to an irreducible form). By squaring both sides of the equality, we get m^2=2n^2. From here we conclude that m^2, and after this the number m- even. those. m = 2k. That's why m^2 = 4k^2 and therefore 4 k^2 =2n^2, or 2 k^2 = n^2. But then it turns out that n Also even number, but this cannot be, since the fraction m/n irreducible. A contradiction arises. It remains to conclude: our assumption is incorrect and the rational number m/n, equal to √2, does not exist.”
That's all their proof.
A critical assessment of the evidence of the ancient Greeks
But…. Let's look at this proof of the ancient Greeks somewhat critically. And if you are more careful in simple mathematics, then you can see the following in it:
1) In the rational number adopted by the Greeks m/n numbers m And n- whole, but unknown(whether they even, whether they odd). And so it is! And in order to somehow establish any dependence between them, it is necessary to accurately determine their purpose;
2) When the ancients decided that the number m– even, then in the equality they accepted m = 2k they (intentionally or out of ignorance!) did not quite “correctly” characterize the number “ k " But here is the number k- This whole(WHOLE!) and quite famous a number that quite clearly defines what was found even number m. And don't be this way found numbers " k"the ancients could not in the future" use" and number m ;
3) And when from equality 2 k^2 = n^2 the ancients received the number n^2 is even, and at the same time n– even, then they would have to don't rush with the conclusion about " the contradiction that has arisen", but it is better to make sure of the maximum accuracy accepted by them " choice» numbers « n ».
How could they do this? Yes, simple!
Look: from the equality they obtained 2 k^2 = n^2 one could easily obtain the following equality k√2 = n. And there is nothing reprehensible here - after all, they got from equality m/n=√2 is another equality adequate to it m^2=2n^2 ! And no one contradicted them!
But in the new equality k√2 = n for obvious INTEGERS k And n it is clear that from it Always get the number √2 - rational . Always! Because it contains numbers k And n- famous WHOLE ones!
But so that from their equality 2 k^2 = n^2 and, as a consequence, from k√2 = n get the number √2 – irrational (like that " wished"the ancient Greeks!), then it is necessary to have in them, at least , number " k» in the form not whole (!!!) numbers. And this is precisely what the ancient Greeks did NOT have!
Hence the CONCLUSION: the above proof of the irrationality of the number √2, made by the ancient Greeks 2400 years ago, is frankly incorrect and mathematically incorrect, not to say rudely - it is simply fake .
In the small brochure F-6 shown above (see photo above), released in Krasnodar (Russia) in 2015 with a total circulation of 15,000 copies. (obviously with sponsorship investment) a new, extremely correct from the point of view of mathematics and extremely correct ] proof of the irrationality of the number √2 is given, which could have happened long ago if there were no hard " teacher n" to the study of the antiquities of History.
What numbers are irrational? Irrational number is not a rational real number, i.e. it cannot be represented as a fraction (as a ratio of two integers), where m- integer, n- natural number. Irrational number can be represented as an infinite non-periodic decimal fraction.
Irrational number may not have an exact meaning. Only in format 3.333333…. For example, the square root of two is an irrational number.
Which number is irrational? Irrational number(as opposed to rational) is called an infinite decimal non-periodic fraction.
Set of irrational numbers often denoted by a capital Latin letter in bold style without shading. That.:
Those. The set of irrational numbers is the difference between the sets of real and rational numbers.
Properties of irrational numbers.
- The sum of 2 non-negative irrational numbers can be a rational number.
- Irrational numbers define Dedekind cuts in the set of rational numbers, in the lower class of which there is no largest number, and in the upper class there is no smaller one.
- Every real transcendental number is an irrational number.
- All irrational numbers are either algebraic or transcendental.
- The set of irrational numbers is dense everywhere on the number line: between every pair of numbers there is an irrational number.
- The order on the set of irrational numbers is isomorphic to the order on the set of real transcendental numbers.
- The set of irrational numbers is infinite and is a set of the 2nd category.
- The result of every arithmetic operation with rational numbers (except division by 0) is a rational number. The result of arithmetic operations on irrational numbers can be either a rational or an irrational number.
- The sum of a rational and an irrational number will always be an irrational number.
- The sum of irrational numbers can be a rational number. For example, let x irrational then y=x*(-1) also irrational; x+y=0, and the number 0 rational (if, for example, we add the root of any degree of 7 and minus the root of the same degree of seven, we get the rational number 0).
Irrational numbers, examples.
γ — ζ (3) — ρ — √2 — √3 — √5 — φ — δs — α — e — π — δ
