The impersonal irrational numbers ring out to be signified by the great Latin letter I (\displaystyle \mathbb (I) ) at the top of the fat cross without filling. In this manner: I = R ∖ Q (\displaystyle \mathbb(I) =\mathbb(R) \backslash \mathbb(Q) ), then impersonal irrational numbers є difference of multiplicity of speech and rational numbers.

About the basis of irrational numbers, more precisely, innumerable numbers, innumerable in a single singularity, already old mathematicians knew: it was known, for example, the innumerability of the diagonal of that side of the square, which is equal to the irrationality of the number.

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  Irrational є:

  Apply proof of irrationality

  Korin z 2

  Let's not accept: 2 (\displaystyle (\sqrt (2))) rational, so it seems to be a fraction m n (\displaystyle (\frac (m)(n))), de m (\displaystyle m)- whole number, and n (\displaystyle n) is a natural number.

  Zvedomo perebachuvanu equanimity at the square:

  2 = m n ⇒ 2 = m 2 n 2 ⇒ m 2 = 2 n 2 (\displaystyle (\sqrt (2))=(\frac (m)(n))\Rightarrow 2=(\frac (m^(2 ) ))(n^(2)))\Rightarrow m^(2)=2n^(2)).

  History

  antiquity

  The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, if Manava (bl. 750 BC - bl. 690 BC) z'yasuvav, which is the square root of some natural numbers , such as 2 and 61, cannot be clearly expressed [ ] .

  The first proof of the foundation of irrational numbers is attributed to Hippasus of Metapontus (bl. 500 BC), a Pythagorean. For the hours of the Pythagoreans, it was important that there was only one loneliness, it was small and unbearable, as a whole number of times to enter the be-any kind of vіdrіzok [ ] .

  There are no exact data about those, the irrationality of such a number was confirmed by Hippasus. Zgіdno z legend, vіn znayshov yogo vvchayuchi dozhini sides with pentagrams. Therefore, it is wise to let go, what was the cost of gold peretin [ ] .

  Greek mathematicians named the goal of non-obvious quantities alogos(Nevimovnim), prote zgіdno with legends did not see Hippas’s burden. There is a legend that Hippasus zdіysniv vіdkrittya, perebuvayuchi in a sea voyage, and was taken overboard by the other Pythagoreans "for the creation of the element of the omniscience, which will refute the doctrine that all the uraniums of the omniscience can be added to the cycle of hundred numbers". The discovery of Hippas posed a serious problem for Pythagorean mathematics, destroying the assumptions that lay at the basis of all theories, that numbers and geometric objects are united and inseparable.

  Tsya vlastivist vіdіgraє important role vіrishenny differential rivnyan. So, for example, to the only decisions of differential equalization

  

  є function

  

  de c- Pretty constant.

  • 1. Number e irrationally and navit transcendentally. Yogo transcendence was brought only in 1873 to Charles Hermit. Tell me what e- The number is normal, so that the possibility of the appearance of different digits in the same record is the same.
  • 2. Number eє countable (also, і arithmetic) number.

  Euler formula, zocrema

  

  5. t.z. "Poisson integral" or "Gauss integral"

  

  8. Giving Catalana:

  

  9. Submission via tvir:

  

  10. Through Bell numbers:

  

  11. World of irrationality of number e dorіvnyuє 2 (which is the least possible value for irrational numbers).

  Proof of irrationality

  Let's assume that

  de a and b are natural numbers. Vrakhovuuchi tsyu equanimity and looking at the layout in a row:

  

  we will take away the jealousy:

  

  Let's imagine that the sum is given to the sum of two dodankiv, one of which is the sum of the members of the series according to n id 0 to a, and the other - the sum of the members of the series:

  

  Now let's transfer the sum to the left part of the equanimity:

  

  Let's multiply the offending parts of the gained zeal. Take away

  Now we can easily take away viraz:

  

  Let's take a look at the left part of the taken away equivalence. Obviously, it's a whole number. Tsіlim є і number, oskіlki (zvіdsi viplivaє, scho tse numbers mean the number). Tim himself, the lion's share of the gained equivalence - the whole number.

  Let's move on to the right part. Tsya suma maє vyglyad

  
  

  For the sign of Leibniz, the whole series converge, that yogo sum Sє speech number, put between the first dodank and the sum of the first two dodankiv (with signs), tobto.

  Offended qi numbers lie between 0 and 1. Otzhe, tobto. - the rights of a part of equality - can be a whole number. They took away the rub: the whole number cannot be equal to the number, as it is not the number. Tse protirichchya to bring, scho number e not rational, but also irrational.

  1. Proof with the butts of deductive mirkuvannya and inductive chi empirical arguments. The proof is guilty of demonstrating that hardness, what to bring, invariably, along the way, to resurrect all possible moods and show that hardness is victorious in the skin of them. The proof can be based on the obvious and glaringly accepted phenomena and the fall, as axioms. As a result, the irrationality of the “square root of two” is brought up.
  2. The introduction of topology here is explained by the very nature of speeches, which means that there is no purely algebraic way of proving irrationality, zocrema, arising from rational numbers. + 1/8 …≠ 2 ???
  If you accept 1+1/2 + 1/4 + 1/8 +…= 2, which is important by the “algebraic” approach, then it’s not important to show that n/m ∈ ℚ, as if on an inexhaustible sequence, irrational and final number. Tse suggest that irrational numbersє zamikannyam field ℚ, but tse stosuetsya topological singularities.
  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
  It is better to show that there is an uninterrupted homomorphism ℚ → I, and you can show suvoro that the basis of such an isomorphism is not a logical legacy of algebraic axioms.

  The very understanding of the irrational number is so powerful that it is signified through the list of authorities "be rational", to which the proof of the opposite is the most natural here. You can, prote proponuvati axis yak mirkuvannya.

  What are the principally rational numbers in the irrational ones? Like tі, so і і інші, it is possible to approximate by rational numbers with any given accuracy, but for rational numbers it can be approximated with "zero" accuracy (by the number itself), but for irrational numbers it is not so. Let's try it for whomever you want.

  Nasampered, it is significant such a simple fact. Let $%\alpha$%, $%\beta$% - two positive numbers, so that they approximate one to another exactly $%\varepsilon$%, so $%|\alpha-\beta|=\varepsilon$%. What will happen if we replace the numbers on the return? How to change the accuracy? It's easy to bachiti that $$\left|\frac1\alpha-\frac1\beta\right|=\frac(|alpha-beta|)(\alpha\beta)=\frac(\varepsilon)(\alpha\ beta) ,$$ which will be strictly less than $%\varepsilon$% for $%\alpha\beta>1$%. Tse firmness can be seen as an independent lemma.

  Now let's put $%x=\sqrt(2)$%, and let $%q\in(\mathbb Q)$% be a rational approximation of $%x$% with $%\varepsilon$% accuracy. We know that $%x>1$%, and if the proximity of $%q$% leads to $%q\ge1$% unevenness. For all numbers less than $%1$%, the approximation accuracy will be higher, lower for $%1$%, and we will not consider them.

  Up to skin numbers $%x$%, $%q$% add $%1$%. Obviously, the accuracy of the proximity will be lost. Now we have є numbers $% alfa=x+1$% and $%beta=q+1$%. Moving on to turning numbers and stagnating "lemma", we will come to the point where the accuracy of our approximation has improved, becoming strictly less than $%\varepsilon$%. $% alpha beta>1$% is required, we have a lot of reserve: we really know that $% alpha>2$% and $% betage2$%, the stars can be used to generate whiskers, so the accuracy is improved at least at $%4 $% once, so it doesn't move $%\varepsilon/4$%.

  І here is the main point: for the mind, $%x^2=2$%, then $%x^2-1=1$%, but it means that $%(x+1)(x- 1)= 1$%, so the numbers $%x+1$% and $%x-1$% are wrapped one to one. And it means that $%\alpha^(-1)=x-1$% will be close to the (rational) number $%\beta^(-1)=1/(q+1)$% with strictly less accuracy $%\varepsilon$%. No more adding $%1$% to x numbers, and it appears that $%x$%, then $%\sqrt(2)$%, has a new rational approach, which is better $%\beta^( - 1)+1$%, then $%(q+2)/(q+1)$%, with "reduced" accuracy. To complete the proof, the shards of rational numbers, as we meant more, it is "absolutely more accurate" rational approximation with accuracy $% \ varepsilon = 0 $ %, it is not possible to accurately move the principle. And we were wondering what to say about the irrationality of our number.

  In fact, this mirroring shows how there will be specific rational approximations for $% \ sqrt (2) $ % with the accuracy that everything is going wrong. Required to take an approximation $%q=1$%, and then put one and the same replacement formula: $%q\mapsto(q+2)/(q+1)$%. At the beginning of the process, the next step is: $$1,\frac32,\frac75,\frac(17)(12),\frac(41)(29),\frac(99)(70)$$ and so on.

  What numbers are irrational? Ір rational number - rational speech number, tobto. it cannot be represented as drіb (like the introduction of two whole numbers), de m- whole number, n is a natural number. Irrational number it is possible to reveal as an inexhaustible non-periodic dozen dribs.

  Irrational number the mother of the exact meaning is impossible. Only in format 3.333333. For example, the square root of two is an irrational number.

  What number is irrational? irrational number(on the vіdmіnu vіd rational) are called innumerable dozens of non-periodic drіb.

  Anonymity of irrational numbers most often denoted by the great Latin letter in bold baptism without flooding. That.:

  Tobto. impersonal irrational numbers ce difference of multiplicity of speech and rational numbers.

  Power of irrational numbers.

  • The sum of 2 non-negative irrational numbers is, perhaps, a rational number.
  • Irrational numbers signify Dedekind's revisions in impersonal rational numbers, for the lower class some have the largest number, and for the upper class there is no less.
  • Be like speech transcendent number- A rational number.
  • All irrational numbers are either algebraic or transcendental.
  • Anonymous irrational numbers cross over on a numerical straight line: between a skin pair of numbers is an irrational number.
  • The order on impersonal irrational numbers is isomorphic to the order on impersonal speech transcendental numbers.
  • Anonymity of irrational numbers is infinitely insignificant of the 2nd category.
  • The result of a skin arithmetic operation with rational numbers (crim subdivided by 0) is rational numbers. The result of arithmetic operations on irrational numbers can become a rational, and irrational number.
  • The sum of rational and irrational numbers will always be an irrational number.
  • The sum of irrational numbers can be a rational number. For example, come on x irrational, also y=x*(-1) also irrational; x+y=0, and the number 0 rational (like, for example, add up the root of any degree from 7 and minus the root of the same degree from seven, then we take the rational number 0).

  Irrational numbers, applications.

  γ — ζ (3) — ρ — √2 — √3 — √5 — φ — δs — α — e — π — δ