Algebraic and transcendental numbers. transcendental numbers transcendental numbers

ie for a = 1 served us the purpose of the sum of the geometric progression. Assuming that Gauss theorem has been proved, it is assumed that a = a 1 is equal root (17),

Considering the virase s f(x) and regrouping terms, we take into account the sameness

f(x) = f(x) − f(a1) = (xn − a n 1 ) + an−1 (xn−1 − a n 1 −1 ) + . . . + a1 (x - a1).

(21) Now scouring the formula (20), we can see the multiplier x − a 1 from the skin member and then blame Yogo for the bow, moreover, the feet of the rich member, which is left in the bows, become one less. Regrouping new members, we take away the sameness

f(x) = (x − a1 )g(x),

where g(x) is a rich term of step n − 1:

g(x) = xn−1 + bn−2 xn−2 + . . . + b1x + b0.

(The calculation of the coefficients, which are known through b, we are here to be called.) It is necessary to distance the very same calculation from the polynomial g (x). According to the Gauss theorem, the square root a2 equals g(x) = 0, so that

g(x) = (x − a2 )h(x),

where h(x) is a new polynomial of step n − 2. Repeating n − 1 times

From the sameness (22) not only those that are complex numbers a1, a2,

An is the essence of the root of equal (17), and those that have no other roots of equal (17). True, yakbi number y was the root of equal (17), then s (22) slid bi

f(y) = (y - a1) (y - a2). . . (y - an) = 0.

Ale mi bachili (p. 115) that the addition of complex numbers to zero in that and less that way, as one of the multipliers to zero. Also, one of the multipliers y−ar is equal to 0, so y = ar, which is necessary to set.

§ 6.

1. The purpose is that nutritional reason. Any number x is called an algebraic number;

130 MATHEMATICAL NUMBER SYSTEM ch. II

de numbers ai numbers. So, for example, the number 2 is algebraic, to that which is pleased with equal

x2 − 2 = 0.

In the same rank of algebraic number, whether there is a root, whether it be equal, with the whole coefficients of the third, fourth, fifth, whether it be the world, and independently, in addition, it can be expressed or not expressed by the radicals. The concept of an algebraic number is a natural comprehension of the concept of a rational number, in a way that confirms the okremy fall n = 1.

Not every real number is algebraic. Tse vipliva z offensive, with Kantor, theorems: the impersonality of all numbers of the algebra of rachunkiv. Bo bezlich usikh day numbers is indistinguishable, then obov'yazkovo is due to use the actual numbers, as they are not algebraic.

Let us point out one of the methods for resolving impersonal algebraic numbers. Skin equal to the appearance (1) equal to the target number

h = | an | + | an-1 | +. . . + | a1 | + | a0 | +n,

for the sake of style, we call it “high” equal. Up to the skin fixed value n is only the last number equal to the form (1) with the height h. Skin from such equals can be more than n roots. For this, it is possible to use only the last number of numbers of algebra, which are generated by equals with height h; father, everything algebraic numbers you can roztashuvati at the sight of the sequence, overshooting the head of them, as they are born by the equals of height 1 then - height 2 and so on.

This proof of the identity of impersonal algebraic numbers establishes the basis of real numbers, as they are not algebraic. Such numbers are called transcendental (from the Latin transcendere - go over, turn over); Euler gave such a name to him, that stinks "to overturn the tightness of the methods of algebra."

Cantor's proof of the foundation of transcendental numbers does not lie before constructive ones. Theoretically speaking, it would be possible to induce a transcendental number for an additional diagonal procedure, which is carried out over an explicit list of tens of expansions of all numbers of algebra; But such a procedure was spared any practical significance and would not lead to a number that could be written in the tens (or any other) drib. Most of the problems associated with transcendental numbers are related to proving that peevn, specific numbers (here are the numbers p and e, about div. 319-322) are transcendental.

**2. Liouville's theorem and the construction of transcendental numbers. The proof of the foundation of transcendental numbers was given before Cantor by J. Liouville (1809–1862). It allows us to actually construct examples of such numbers. Lіouvil's proof is more important, lower than Cantor's proof, and it's not surprising, shards to construct a butt, inflamed seeming, folded, lower to bring the foundation. Leading lower is Liouville's proof, perhaps it looks less like a trained reader, wanting to understand the proof with sufficient knowledge of elementary mathematics.

As having shown Lіouville, irrational algebraic numbers have that power that they cannot be approximated by rational numbers with an already great level of accuracy, just do not take the banners of fractions that they approximate, they are superbly great.

Assume that the number z satisfies the equation of algebra with integer coefficients

but you are not satisfied with such a leveling of the lower step. Todi

it seems that x itself is the number of the algebra of degree n. So, for example,

the number z \u003d 2 is the number of the algebra of level 2, so that the level x2 − 2 = 0 √ is satisfied with level 2, but not the level of the first level is not satisfied; number z = 3 2 - level 3, which is satisfied with x3 - 2 = 0, but not satisfied (as we show in section III) with the level of the lower level. Algebraic number of step n > 1

cannot be rational, because the rational number z = p q

satisfies the level qx − p = 0 step 1. Skin irrational number z can be, with some degree of accuracy, approximated by an additional rational number; does not mean that you can always indicate the sequence of rational numbers

p1, p2,. . .

q 1 q 2

s not surrounded by growing banners, that Volodya Tim-

what, what

p r → z. qr

Liouville's theorem is staggering: if there were no number of algebra z of step n > 1, it could not be closer to an additional rational

to finish the great bannermen obov'yazkovo vykonuetsya nerіvnіst

We choose to prove the theorem, and earlier it will be shown how transcendental numbers can be obtained for additional help. Let's look at the number

z = a1 10-1! + a2 10-2! + a3 10-3! +. . . + am · 10−m! +. . . = = 0,a1 a2 000a3 00000000000000000a4 000. . . ,

de ai mean certain numbers from 1 to 9 (it would be easier to put all ai equal to 1), and the symbol n! . . n. The characteristic power of the tenth expansion of such a number is those who are groups, who quickly grow up behind their dozhina, zeros are drawn into the new one with okremi digits, which look like zero. Significantly through zm, the end of the tenth drіb, which is settled, if all the members are taken in the layout up to am · 10−m! inclusive. Todi take away the nervousness

Assume that z is the number of the algebra of step n. Todi, respecting the nervousness of Lіouville (3) pq = zm = 10pm! , we are guilty mothers

|z - zm | > 10(n+1)m!

at high values ​​of m. Comparison of the remaining unevenness with the nervousness (4) yes

stars follow (n + 1) m! > (m + 1)! − 1 for great m. Alece is wrong for values ​​of m greater than n (let the reader try to give a detailed proof of this assertion). We didshli super-sharpness. Also, the number z is transcendental.

It remains to finish Liouville's theorem. Assume that z is the number of the algebra of degree n > 1, which satisfies the equation (1), so that

f(zm ) = f(zm ) − f(z) = a1 (zm − z) + a2 (zm 2 − z2 ) + . . . + an (zm n − zn).

Dealing with insulting parts on zm − z and coring with an algebraic formula

u n − v n = un−1 + un−2 v + un−3 v2 + . . . + uvn−2 + vn−1 , u − v

we accept:

Since zm is the right z, then when you reach the great m, it is rational the number zm to be taken into account z less lower by one. Therefore, for dosing great m, you can earn such a rough estimate:

N|an|(|z|+1)n−1 = M, (7)

moreover, to be right-handed, the number M is constant, the shards z do not change during the proof process. Vibero now m flooring great, shob

fraction z m = p m standard q m higher, lower M; also qm

so that it was also possible to see the multiplier (x − zm ) from the polynomial f(x), i, also, z was satisfied with the level of the lower lower n. Otzhe, f(zm) 6= 0. Ale numeral at the right part of equality (9) In such a manner, zіzstavlennya sіvvіdnіshen (8) and (9) vyplyaє nerіvnіst

still warehouse zmіst zaznachenї theorem.

With a stretch of a few remaining decades, the possibility of approximating algebraic numbers by rational ones poked their way far into the distance. For example, the Norwegian mathematician A. Tue (1863–1922) found that the Liouville unevenness (3) could have an exponent n + 1 replaced by a smaller exponent n 2 + 1.

Siegel showing that you can take even smaller (smaller

with larger n) indicator 2 n.

Transcendental numbers have always been a topic, as they have riveted the respect of mathematicians to themselves. Ale, until the recent hour of the middle of the day, like tsіkavі by powerful forces, there were not many such, the transcendental nature of such bulo was installed. (Because of the transcendence of the number p, as it happens in section III, there is the impossibility of squaring the stake with the help of a ruler and a compass.) At his speech at the Paris International Mathematical Congress 1900 r. David Hilbert chanting thirty mathematical

problems that allow simple formulary, deyakі - navіt zovsіm elementary and more popular, z аnу аlѕt аlѕt bіlka vyrіshenа, аlаnіt аnd didn't hаvіd zdatnoy but allowed by the mathematicians of the tієї epoch. Qi "Hilbert's problems" gave a strong wake-up call to the development of mathematics in the coming period. Mayzhe all the stinks were allowed step by step, and in rich vipadkas their virishennia was due to clearly manifested successes in the sense of more outrageous and glib methods. One of the problems that the hopeless one dared to deal with

proof that the number

є transcendental (chi wanta b irrational). For three decades it was not possible to put pressure on such a pidhіd to feed from someone else's side, which spurred hope for success. Zreshtoyu, Zigel and, independently, young Russian mathematician A. Gelfond discovered new methods for proving the transcendence of riches

numbers, which can mean the meaning of mathematics. Zokrema, Bulo inserted

transcendence like a Hilbert number 2 2 , and th integer to a great class of numbers of the form ab , where a is an algebraic number, a rule is 0 and 1, and b is an irrational algebraic number.

ADDITION TO RAZDILU II

Algebra of multiples

1. Hot theory. The concept of class, sukupnostі, chi impersonal objects - one of the most fundamental in mathematics. The impersonal signifies a deaco power (“attribute”) A, which is the fault of either the mother, or not the mother of the skin analyses of the object; those objects, like the power of A, make up the impersonality of A. So, as we see the purpose of the number that power of A is in the fact that we forgive, then the impersonality of A is added up from the usual prime numbers 2, 3, 5, 7 , . . .

Mathematical theory multiplies come out of the fact that it is possible to establish new multipliers for additional operations (similar to the fact that new numbers appear from numbers for an additional operation of folding that multiplier). Vyvchennya operations on multiplies to become the subject of "multiple algebra", as it can be richly coherent with a great numerical algebra, wanting to see why and in it. The fact that the methods of algebra can be staggered to the point of including non-numerical objects, such as impersonal, ilu-

a stream of great convergence of ideas of modern mathematics. In the rest of the hour, it was clear that the algebra of multiplies was throwing a new light on the rich magic of mathematics, for example, the theory of the world and the theory of imaginary things; vona korisna is also a pіd hour of systematization math understand that z'yasuvannі їх logical zv'yazkіv.

Nadal I mean the deak of postiynu impersonal objects, the nature of such baiduzh, and as we can call it universal impersonality (or the universe of mirkuvannya), and

A, B, C, . . . If I is the plurality of all natural numbers, then A, let's say, can mean the absence of all paired numbers, B - the absence of all unpaired numbers, C - the absence of all prime numbers, and so on. on the flat, then A can be a pointless point in the middle of this stake, B - a pointless point in the middle of another stake, etc. Before the "subsets" we can manually turn on I itself, and also the "empty" pointless, so as not to avenge any elements. Meta, as if following such a piece of expansion, poking at the saving of that position, that the skin authority A shows a lot of elements from I, which will lead to the power of authority. In times, as A є universally vykonuvan authority, the butt of which you can serve (as you can find about numbers) authority satisfies the trivial equivalence x = x, then in the case of a multiplier I will be itself I, the skin element may have such authority; from the other side, as A є as an internally super-powerful power (on kshtalt x 6 = x), then it’s not possible to avenge the elements, it’s “empty” and is denoted by a symbol.

It seems that the multiplier A is the submultiplier of the multiplier B, in short, “A enters at B”, or “B avenges A”, because the multiplier A does not have such an element, which is not the same as the multiplier B.

A B or B A.

For example, the impersonal A of all integer numbers, which is divisible by 10, is the submultiple of the impersonal B of all integer numbers, which is divisible by 5, so the skin number, which is divisible by 10, is also divisible by 5. A B does not include the B A. maє mіsce i te y іnshe, then

Tse means that the skin element A є at the same time the element B, і back, so multiply A and B to replace the same elements.

Spivvіdnoshennia A B mizhiny rich in what guess spіvіdnoshennia a 6 b mizh numbers. Zokrema, obviously traced

blowing the power of this spіvvіdnoshennia:

1) A A.

2) If AB and BA, then A = B.

3) Like A B and B C, then A C.

For reasons of spіvvіdnoshennia AB are sometimes called "to order". Golovna Vidmіnniy Analized SPIVVISHENYNYA VID SPIVVISHENYNYA A 6 b mines in the numbers of Polega in the one, the cousin of the cow of the number of numbers a і b is not a reserv analogous assertion is wrong. For example, that A is impersonal, which is composed of the numbers 1, 2, 3,

and B is a multiplier, which is added up from the numbers 2, 3, 4,

then there is no time for A B, or B A. There are no reasons to say that A, B, C, . . . multipliers I є “partially ordered”, the same as the effective numbers a, b, c, . . .

establish a “fully ordered” order.

Respectfully, among others, that there was no difference between A and B, that, if there were not a multiplier of A, a multiplier of I,

Power 4) may be somewhat paradoxical, but, if you think about it, it is logically subservient to the exact change of the appointed sign. True, spіvvіdnoshnya A was broken only

in to that vipadka, as if empty, many elements misplaced the element, which did not avenge b A; but so, like an empty impersonal, do not take revenge on the elements, then you can’t be, if it weren’t for A.

We now signify two operations on multiplies, which formally allow you to be rich with algebraic powers to add that multiplicity of numbers, wanting for your internal zmіsto zovsіm vіdminnі vіd tsikh arithmetic diy. Let A and B be two multipliers. Under the terms, or "logical sum", A and B understand the impersonal, which is composed of quiet elements, which are located in A or

in B (including and those elements that can be found in A and B). This multiplier is denoted by A + B. 1 Under the "peretina", or "logical creation", A and B are understood impersonally, which are composed of quiet elements, which can be found in A and in B. This multiplier is indicated by AB.2

Among the important powers of the algebra of operations A + B and AB, the offensive is overwhelmed. The reader can reverse the fairness, depending on the purpose of the operations themselves:

The re-verification of all these laws is the simplest logic on the right. For example, rule 10) states that the elements are impersonal, that either A, or A, or the impersonal A; rule 12) stating that the impersonal elements, if they are in A and at the same time are either B or C, are impersonal elements, if they are either in A and B, or in A and C vykoristovuyutsya in proving a similar kind of rules, hand-illustrated, as if we were able to imagine the impersonal A, B, C, . . . at the sight of such figures on the square, we will be more respectful in that respect, so as not to miss the logical possibilities, if there is about the presence of the main elements of two sets, or, on the contrary, the presence of one set of elements, if not to be found in the other.

A reader, no doubt, having lost respect for those who laws 6), 7), 8), 9) and 12) are called the same with the well-known commutative, associative and distributive laws of sonic algebra. Zvіdsi viplivaє, scho tse rules zvichaynoї algebra, yakі z tsikh laws, effective in the algebra of sets. Navpaki, laws 10), 11) and 13) there are no analogues of the original algebra, and they give the algebra many simple structure. For example, the binomial formula in the algebra of multiplies can be reduced to the simplest equality

(A + B) n = (A + B) · (A + B). . . (A + B) = A + B,

as a matter of law 11). Laws 14), 15) and 17) to speak about those that the power of the plurals I in terms of the number before the operation of adding that number is similar to the power of the numbers 0 and 1 in terms of the power of the numbers 0 and 1 in terms of the number before the operation. Ale law 16) has no analogue in numerical algebra.

One more operation in the algebra of sets remains to be given. Let A be the submultiplier of the universal multiplier I. So, under the additive A in I, the impersonal of all elements of I can be understood, if not in A. For the multiplier, we introduce the value A0. So, if I is impersonal of all natural numbers, and A is impersonal of all prime numbers, then A0 is impersonal, which is added up from all warehouse numbers and the number 1. authority:

23) A00 = A.

24) Spivvіdnenja A B 0A0.

25) (A + B) 0 = A0 B0. 26) (AB)0 = A0 + B0.

Re-verification of these powers I re-nadaemo chitachev.

Laws 1)-26) underlie the algebra of sets. The stench of the miraculous power of "duality" in the offensive sensation:

Like in one of the laws 1)–26) replace one for one

(for the dermal input), then as a result, one of these laws reappears. For example, law 6) transforms into law 7), 12) - in 13), 17) - in 16) just. bud. , "Dvіyna" їth theorem, which comes out of the first for additional meanings of permutations of symbols. True, shards of proof

Goal. II ALGEBRA MNOZHIN 139

the first theorem is composed of successive stagnation (at different stages of the reconciliation to be carried out) of the laws 1–26), then the stagnation at the final stages of the “two” laws in the warehouse is the proof of the “double” theorem. (Because of the drive of such a “doubleness” in the geometry of the div. Section IV.)

2. Zastosuvannya mathematical logic. The re-verification of the laws of the algebra of multiplies was based on the analysis of the logical sense of spiving A B and operations A + B, AB and A0. We can now reverse this process and consider the laws 1)–26) as the basis for the "algebra of logic". To put it more precisely: that part of the logic, as there are many, or, in fact, the very same, the powers of the objects that are looked at, can be reduced to a formal algebraic system based on the laws 1)–26). The logical "smart omniscience" signifies the impersonal I; dermal power A signifies impersonal A, which is composed of quiet objects I, like it can be power. Rules for translating the most logical terminology into language

In terms of algebra, there is a syllogism “Barbara”, which means that “if every A є B and every B є C, then every A є C”, it looks simple:

3) If AB and BC, then AC.

Similarly, the “law of resistance”, which affirms that “an object cannot simultaneously lead and cannot lead such power”, is recorded by the viewer:

20) AA 0 = ,

a “the law of the included third”, which is to say that “the object is to blame for the mother, but not the mother for the deacon of power”, is written:

19) A+A0=I.

In this way, that part of the logic, as seen in terms of symbols, +, · і 0, can be interpreted as a formal system of algebra, according to the laws 1)–26). On the basis of a logical analysis of mathematics and mathematical analysis of logic, a new discipline has been created - mathematical logic, like none of them is rebuking the process of turbulent development.

From the axiomatic point of view, due to the respect of that miraculous fact, which is confirmed by 1)-26), together with other theorems of the algebra of sets, can be logically seen from the coming three equalities:

27) A + B = B + A,

(A + B) + C = A + (B + C),

(A0 + B0) 0 + (A0 + B) 0 = A.

It is evident that the algebra of multiplies can be motivated as a deductive theory, on the basis of Euclidean geometry, on the basis of these three positions, which are accepted as axioms. As axiomatically accepted, then the operation AB and the proposition A B are defined in terms of A + B and A0 :

We call another example of a mathematical system, in which all the formal laws of the algebra of multipliers are encoded, is given by a system of eight numbers 1, 2, 3, 5, 6, 10, 15, 30: here a + b means,

the highest, lowest multiple of a і b, ab - the highest dіlnik a і b, a b - hardness "b is subdivided by a" and a0 - the number 30 a. Su-

The basis of such applications has caused the development of outrageous algebraic systems, which satisfies the laws 27). Such systems are called "Boolean algebras" - in honor of George Boole (1815-1864), an English mathematician and logician, whose book "An investigation of the laws of thought" appeared in 1854.

3. One of the stops before the theory of immovability. Algebra can be much closer to the theory of immovability and allows you to look at it in a new light. Let's take a look at the simplest example: let's make our own experiment from the last number of possible nasledkiv, yakі all think like "equally able". An experiment can, for example, lie in the fact that we can draw a card from a new deck, which is well shuffled. If the multiplier of all the results of the experiment is significant through I, and A means that it is a submultiplier of I, then the possibility that the result of the experiment will lie up to the submultiplier of A is signified as an extension

p(A) = number of elements in A. number of elements in I

If we think of the number of elements in any multiplier A as n(A), then the rest of the equality can be given by looking at

Ideas of algebra of plurals appear when counting the possibilities, if it is possible, knowing the imovirness of some plurals, to count the imovirness of others. For example, knowing the dynamics of p(A), p(B) and p(AB), we can calculate the dynamics of p(A + B):

It doesn't matter to bring it. Mi maєmo

n(A + B) = n(A) + n(B) − n(AB),

shards of elements that can be occupied at the same time in A and B, then elements of AB are taken into account when counting sums n(A) + n(B), and, therefore, it is necessary to see n(AB) from the sum of sums, so n(A + B) the letter of division is correct. Let's keep the offenders offended by part of the equivalence on n(I), we will take away the spontaneity (2).

Cіkavіsha formula to go out, so there are about three multipliers A, B, C z I.

p(A + B + C) = p[(A + B) + C] = p(A + B) + p(C) − p[(A + B)C].

Law (12) from the previous paragraph gives us (A + B) C = AC + BC. Sounds are screaming:

p[(A + B)C)] = p(AC + BC) = p(AC) + p(BC) − p(ABC).

Substituting in the previous order the value of p[(A + B)C] and the value of p(A + B), taken from (2), we arrive at the required formula:

p(A + B + C) = p(A) + p(B) + p(C) − p(AB) − p(AC) − p(BC) + p(ABC). (3)

Like a butt, we can look at an offensive experiment. Three numbers 1, 2, 3 are written in any order. What is the meaning of the fact that one of the digits is accepted to be based on the overhead (in the sensi numbering) space? Let A be an impersonal permutation, for which the number 1 should cost the first place, B - an impersonal permutation, for which the number 2 should cost another place, C - an impersonal permutation, for which the number 3 should cost the third place. We need to calculate p(A+B+C). I realized that

p(A) = p(B) = p(C) = 2 6 = 1 3;

effectively, as if the figure is standing on the right place, then there are two possibilities to rearrange the solution of two digits from the main number 3 2 1 = 6 possible permutations of three digits. Dali,

Right. Enter a valid formula for p(A + B + C + D) and wait until the experiment, which involves 4 digits. Vidpovidna umovirnіst dorіvnyuє 58 = 0.6250.

A common formula for combining n multiplies may look

combinations to avenge one, two, three, . . . , (n − 1) letter from the number A1 , A2 , . . .

an. This formula can be inserted after additional mathematical induction - just like formula (3) was introduced from formula (2).

From the formula (4) it is possible to add wisps, so that there are n digits 1, 2, 3, . . . n written in any order, then the ability to accept one of the digits to lean on a proper place is more

moreover, before the remaining member, there is a sign + or −, calling for those that are paired and unpaired. Zocrema, for n = 5

p5 = 1 − 2! + 3! − 4! +5! = 30 = 0.6333. . .

In the VIII division, we would like to know that if there is no incompatibility, viraz

1 1 1 1 Sn = 2! − 3! +4! − . . . ±n!

pragne between 1 e, the meaning of which, with five signs after Komi,

one 0.36788. From the formula (5) it is clear that pn = 1 − Sn, then the star is clear, that for n → ∞

pn → 1 − e ≈ 0.63212.

The word "transcendental" is associated with transcendental meditation and various esotericism. But in order to live yoga correctly, it is necessary as a minimum to resurrect yoga from the term "transcendental", and as a maximum - to guess the role of yoga in Kant's robots and other philosophers.

Tse understandable to resemble the Latin transcendens - "to cross", "to cross", "to go beyond". In general, wines mean those that are importantly inaccessible to empirical knowledge, or based on evidence. Rethink the term viniklische philosophy of neoplatonism - the founder directly Plotin having made a vchennya about the One - the all-good pershopochka, as it is impossible to recognize the thoughts with the help of the mind, without the aid of a sensitive mind. "One does not exist, but father Yogo" - explains the philosopher.

The most recent term “transcendental” was developed in the philosophy of Immanuel Kant, de vin vikoristovuvsya to characterize, clearly indispensable to the knowledge and how to feel our bodies are sensitive, being left in principle unrecognizable, like in practice, and in theory. Proliferation of transcendence - : it means either invisibility, internal link, be it as the object is with the object itself, or the recognition of the object on special certificate. For example, let's assume that the All-World of creations, behind a great idea, thought itself transcendent for us - we can only make hypotheses about the new. And yet, as I conceived it, it’s true, and the consequences for us are immanent, influencing the physical laws and conditions, which we can consume. Therefore, in some theological concepts, God is transcendent and perebuvaet posture created by him butts.

Actual speeches are still accessible to a priori knowledge: for example, space and time, ideas of God, goodness and beauty, logical categories. Tobto transcendental objects - tse, figuratively seeming, "behind the line set" in our mind

The statement about transcendental nature in mathematics: a transcendental number is a number that cannot be calculated using additional algebra or algebraically (that is, it cannot be the root of a rich term with multiple coefficients that is not the same as zero). Before them enter, for example, the numbers π і e.

Understanding, close to "transcendental", and even beyond the meanings - "transcendental". On the back it meant simply the area of ​​abstract rozum categories, and by the end of the year, having raised Kant, having drank pasta from the vlasnu: it was impossible to induce the philosophical system only on empirical data, but it was impossible to recognize other people’s old ones, the crime of empirics, without knowing wine. In order to turn around, philosophers had a chance to admit that some speeches are still accessible to a priori knowledge: for example, space and time, ideas of God, goodness and beauty, logical categories. That transcendental objects - tse, figuratively seeming, "before put behind the mind" in our minds - with which information about them is self-evident and does not vyplyvaet from our knowledge.

There is one more controversial understanding - transcendence. In a broad sense, the word “vono” means the transition to the cordon between two different regions, especially the transition from the sphere of this world to the sphere of the future, the transcendent. For simplicity, let's take an example from science fiction: a parallel world for great people- transcendental manifestation. But if the hero drank at his parallel light, it seems that the rank is manifested by the building yoga spriymati, tse transcendence. A more foldable example of existential philosophy: Jean-Paul Sartre, having realized that a person is transcendent, the shards will not go beyond the limits of any possible wet truth: we can navkolishniy svit from different sides, but in any case we can’t get close to full recognition of ourselves. Ale, at once, a person can build up to transcendence: he transcends whether it’s a river, giving it a meaning. Transcendence is an important element in religion: it helps people to grow in their material nature and reach something foreign.

From philosophy, the concept of transcendentality has migrated to psychology: the Swiss psychologist Carl Jung has developed the concept of “transcendental function” - the same function that goes along with that incomprehensibility. Zocrema, the transcendental function can be overcome by a psychoanalyst - help the patient to analyze the images of the unseen (for example, dreaming) and show them at once from their own psychic processes.

Yak talk

Incorrect "I signed up for a class in transcendental meditation." That's right - "transcendental".

That's right, "When I go to the temple, I watch something transcendent."

Correctly, “The art of transcendence knows us objects from the material world, reminiscent of them with the greatest light.”

Illya Shchurov

Mathematician Illya Shchurov about tens of fractions, transcendence and irrationality of the number Pi.

How did “loneliness” help to inspire the first place and that great empire? How did you blow the minds of the people? What role did she play in the appearance of pennies? Yak "one" united with zero, to rule modern world? The history of singleness is inextricably linked with the history of European civilization. Terry Jones is virushaya in a humorous way more expensively with the method of taking together the marvelous history of our simplest number. For the help of computer graphics in this program, one comes alive in different forms. From the history of the loneliness, it became clear, the stars appeared today, and like the faults of zero, vryatuvav in the light of the need to win the Roman numerals.

Jacques Cesiano

We know little about Diophantus. Well, Vin is alive at Oleksandriya's. None of the Greek mathematicians figured it out until the 4th century, for that, ymovirno, is alive in the middle of the 3rd century. The head of the robot of Diophantus, "Arithmetic" (Ἀριθμητικά), was taken on the cob of 13 "books" (βιβλία), to be divided. Today we may have 10 of them, and in itself: 6 for the Greek text and 4 others for the middle Arabic translation, and a few for the middle of the Greek books: books I-III in Greek, IV-VII in Arabic, VIII-X in Greek . "Arithmetic" of Diophantus is ahead of schedule, only close 260. Theories, seemingly true, nothing; There are no more general instructions at the beginning of the book, and more private respect for other directors, if necessary. "Arithmetic" already looks like an algebraic treatise. Diophantus on the cob different signs, schob vyslovlyuvati nevidome that yogo step, also deakі calculus; like all algebraic symbols of the middle, its symbolism resembles mathematical words. Then, Diophantus explains how to solve the problem using the algebra method. But the task of Diophantus is not algebraic in the primary meaning, so that everything can be reduced to the height of an undefined equal, or systems of such equals.

George Shabat

Course program: History. First ratings. The problem of the consistency of a stake with a її diameter. Neskіchennі rows, create that іnshі vrazi for π. Zbіzhnist and її yakіst. Virazi, what to revenge π. Sequences that converge quickly up to π. Modern methods calculation of π, number of computers. About the irrationality and transcendence of π and other numbers. Forward knowledge is not necessary for the course.

Officials from Oxford University said that the early introductions of the number 0 to indicate the number of days in a row (as in the number 101) should include the text of the Indian manuscript of Bakhshali.

Vasil Pispanen

Who is not engraved by children in the group "name the greatest number"? Milyoni, trillioni and other "-they" can be seen in the thoughts already smoothly, but we will try to solve the "mastodon" in mathematics - Graham's number.

Viktor Kleptsin

The right number can be approximated exactly by rational ones. And if we kindly do it, can we get close to each other - is it aligned with yoga folding? For example, breaking tenth entry numbers x on k-th digit after that, we take away the proximity x≈a/10^k with a pardon of the order of 1/10^k. I vzagali, having fixed the banner q in the fraction that is approaching, we can definitely take the approach with a pardon of the order 1/q. And what can you do better? Knowing to everyone, the proximity π≈22/7 gives a pardon of the order of 1/1000 - that is clearly better, lower could be corrected. Why? We were spared, why is π so close to є? It appears that for any irrational number є impersonal fractions p / q, which is closer to it, lower 1 / q ^ 2. Tseverzhuє Dirichlet's theorem - and mi pochnemo course іz її troha non-standard proof.

In 1980, the Guinness Book of Records repeated Gardner's assertions, further boosting public interest until that number. Graham's number in the name of the number of times more, lower otherwise good in the house great numbers, so, like googol, googolplex and navit more, lower Skewes number and Moser number. In truth, the whole world is too small for someone to take in his own tenth record of Graham's number.

Dmitro Anosov

Lectures read Anosov Dmitro Viktorovich, Doctor of Physical and Mathematical Sciences, Professor, Academician of the Russian Academy of Sciences. Summer School "Modern Mathematics", Dubna. April 16-18, 2002

It’s not possible to correctly respond to the food chain, shards number series do not maє upper boundary. So, until a certain number, it’s enough to add one more, in order to take the number even more. Although the numbers themselves are not limited, their names are not so rich and rich, so that most of them are satisfied with names that are added up from smaller numbers. I realized that in the final set of numbers, which people have heaped up for their powerful names, they can be the most. But how is it called and why is it equal? Come on, let's try to figure it out in some way and recognize the infection, mathematicians have come up with some great numbers.

The number is called algebraic yakscho it’s the root of a deaky rich term with a lot of coefficients

a n x n +a n-1 x n-1 +... +a 1 x+a 0(i.e. the root of equal a n x n +a n-1 x n-1 +... +a 1 x+a 0 =0, de a n, a n-1, ..., a 1, a 0--- numbers, n 1, a 0).

An impersonal algebraic number is meaningfully a letter .

It is easy to see that whether a rational number is algebraic. True, - the root of the river qx-p=0 with many coefficients a 1 =qі a 0 =-p. Otzhe, .

However, not all algebraic numbers are rational: for example, the number is the root of equality x 2 -2 = 0, otzhe, --- algebraic number.

The old hour was left untouched, important for mathematics nutrition: ? Less than 1844, the fate of Lіouville first navіv an example of a transcendental (tobto. non-algebraic) number.

On the first day of the month, the proof of its transcendence is even more foldable. It is possible to bring the theorem on the basis of transcendental numbers in a significantly simpler way, by pointing out the equivalence and non-equivalence of numerical multiplies.

And itself, we can bring, that impersonal algebraic numbers are Rakhunkov. However, the shards of all the real numbers are not equal, we can set the base of non-algebraic numbers.

Let's mutually unambiguously distinguish between and with a dozen . Tse is meaningful, sho - It's good chi rakhunkovo. Ale oskilki , then neskіchenno, otzhe, rakhunkovo.

Come on - deyake number of algebra. Let's look at all the rich terms with the number of coefficients, the root of which is є, and choose the middle of the rich terms P the minimum step (so that it will not be the root of the same rich term with the whole coefficients of the lesser step).

For example, for a rational number, such a polynomial can have step 1, and numbers - step 2.

Let's divide all the coefficients of a rich member P to their biggest sleeper. We take away the polynomial, the coefficient of which is mutually simple at once (their largest sleeper is 1). Zreshtoyu, as a senior coeficient a n vіd'єmniy, we multiply all the coefficients of the polynomial by -1 .

The subtraction of the rich term (that is, the rich term with large coefficients, the root of which is the number, which can be the least possible step, mutually simple coefficient and the positive senior coefficient) is called the minimum rich term of the number.

It can be proved that such a polynomial is uniquely assigned: the skin number of an algebra can be exactly one minimal polynomial.

The number of real roots of a polynomial is no more than the lower step. Also, you can number (for example, for growth) the roots of such a rich term.

Now, be it the number of algebra, it is now recognized by its minimal rich term (that is, by the set of its coefficients) and the number, which is different from the other roots of the polynomial: (a 0 ,a 1 ,...,a n-1 ,a n ,k).

Later, for the dermal algebraic number, we set the distinction of the final set of whole numbers, moreover, it is uniquely followed by this set (so different sets are given to different numbers).

All prime numbers are numbered in order of growth (it doesn’t matter to show that they are too rich). We take away the inexcusable sequence (pk): p1=2,p2=3, p3=5, p4=7, ... Now a set of integers (a 0 ,a 1 ,...,a n-1 ,a n ,k) you can put u vіdpovidnіst tvіr

(This number is more positive and rational, but do not be natural, even the middle of numbers a 0, a 1, ..., a n-1, may be negative). Respectfully, that the number is not short-lived, the shards are simple multipliers, to enter before the laying out of the number book and banner, difference. It is also worth respecting that two non-short fractions with positive numerals and stanzas are equal, even if they are numerals equal, those їх are equal stanzas.

Now let's look at it with a grain of salt:

(a 0 ,a 1 ,...,a n-1 ,a n ,k) =

Oskіlki different numbers of algebra have set different sets of integer numbers, and different sets --- different rational numbers, then we, in this order, established mutually unambiguous validity between a multiplicity and with a dozen . Therefore, the impersonal algebraic numbers are significant.

Shards of impersonal real numbers are indistinguishable, we have brought the basis of non-algebraic numbers.

However, the reasoning theorem does not show how to determine what whole number algebraic. And nutrition is sometimes important for mathematics.

transcendent number

a number (dіysne abo yavne), which is not satisfied with any equalization of algebra (Div. Algebraic equalization) with many coefficients. In this rank, T. h. are assigned to algebraic numbers. Іsnuvannya T. H. first having established J. Liouville (1844). The right point for Liouville was the th theorem, which states that any order of approximation of a rational fraction with a given standard to the th irrational algebraic number cannot be sufficiently high. The most algebraic number a satisfies the unreduced equal of algebra n with many coefficients, then for any rational number to deposit only α ). Therefore, for a given irrational number α, it is possible to show impersonal rational approximations that do not satisfy the induction of unevenness for any hі n(some and quiet for all close), then α є T. h. The butt of such a number is yes:

R. Kantor (1874), having mentioned that the impersonality of all algebraic numbers is distinguishable (so that all algebraic numbers can be renumbered; div. Multiplicity theory), then impersonality of all real numbers is immutable. It sounded like the impersonal T. h.

The most important task of the theory of T. h. - tse z'yasuvannya that chi є T. h. the value of analytical functions, which may have those other arithmetic arithmetic powers with algebraic values ​​of the argument. The task of which family lies before the most important task of modern mathematics. U 1873 Sh.

In 1882, the German mathematician F. Lindemann took a more significant result: since α is the number of algebra, then eα - T. h. Lipdeman's result was significantly aggravated by the German mathematician K. Siegel (1930), who proved, for example, the transcendence of the value of a wide class of cylindrical functions with the values ​​of the algebra argument. In 1900, at the Mathematical Congress in Paris, D. Hilbert, among 23 inviolable problems of mathematics, pointing out the offensive: chi є transcendental number α β , de α і β - algebraic numbers, moreover β - irrational number, i, zokrema, chi є transcendental number e π α β bula first in private form was put by L. Euler, 1744). The outer version of the problem (in a solid sense) was more or less taken into account in 1934 by A. O. Gelfond. From the statement of Gelfond, zokrema, it is clear that all tens of logarithms of natural numbers (that is, “tabular logarithms”) are T. h. Methods of theory T. h.

Lit.: Gelfond A. O., Transcendental and algebraic numbers, M., 1952.

Great Radianska Encyclopedia. - M: Radianska Encyclopedia. 1969-1978 .

Marvel at such a "Transcendent number" in other dictionaries:

A number that is not satisfied with any equal of algebra with any number of coefficients. Transcendental numbers є: number? 3.14159...; the tenth logarithm of any whole number, which is not represented by one with zeros; number e = 2.71828 ... ta in ... Great Encyclopedic dictionary

- (Latin transcendere go over, turn over) tse recheve abo complex number, which is not algebraic in other words, a number that cannot be a root of a rich term with many coefficients. Zmist 1 Power 2 ... ... Wikipedia

A number that is not satisfied with any equal of algebra with any number of coefficients. Transcendental numbers є number π = 3.14159...; the tenth logarithm of any whole number, which is not represented by one with zeros; number e = 2.71828... ta in. Encyclopedic dictionary

A number that does not satisfy the same algebra. ur nіu with qіlimi coefficients. T. year. є: number ПІ = 3.14159...; the tenth logarithm of any whole number, which is not represented by one with zeros; number e = 2.71828... ta in. Natural science. Encyclopedic dictionary

The number, which is not the root of the same rich term with the same coefficients. The scope of such numbers is the zero of real, complex and radial numbers. Іnuvannya that obviously prompted the action of T. h. obguruntuvav J. Liouville ... Mathematical Encyclopedia

Equal, like not є algebraic. Call the price alignment, which can be shown, logarithmic, trigonometric, reversible trigonometric functions, for example: Suvorishe of the designation such: Transcendental alignment of the goal ... Wikipedia

The number, approximately 2.718, is often used in mathematics and natural sciences. For example, when radioactive speech breaks down after the end of the hour t, in the end of the speech period, a part is lost that is more expensive e kt, de k number, ... Collier Encyclopedia

E is a mathematical constant, the basis of the natural logarithm, an irrational and transcendental number. In other words, the number e is called the Euler number (do not confuse with the so-called Euler numbers of the first kind) or the Napier number. It is signified by the small Latin letter "e".

E is a mathematical constant, the basis of the natural logarithm, an irrational and transcendental number. In other words, the number e is called the Euler number (do not confuse with the so-called Euler numbers of the first kind) or the Napier number. It is signified by the small Latin letter "e".