0,999…

The repeating decimal 0.999 ... (also with more nines written before the ellipsis or as a 0.9 or 0, (9 ) ) called in mathematics a real number of which can be shown to be equal to one. In other words, the symbols " 0.999 ... " and "1 " the same number dar. evidence of this equation have been formulated with varying degrees of severity, depending on the preferred introduction of real numbers, background assumptions, historical context and target audience.

Also, each terminating decimal number other than 0, an alternative representation with infinitely many nines, for example, 8.31999 ... for 8.32. The terminating representation for brevity usually preferred. The same phenomenon also occurs in other bases.

While the equation in mathematics is accepted for a long time and some part of general education, they keep many students for counter-intuitive and question them or reject them. However, number of systems have been developed in which this equation actually does not apply.

3.1 Dedekind cuts
3.2 Cauchy sequences
3.3 of intervals

8.1 Hyper Real Numbers
8.2 Hackenbush
8.3 Rethinking the subtraction
8.4 p- adic numbers

Elementary proofs

The following proofs use concepts that are known from school mathematics.

Written subtraction

When 1 is written with 0.999 ... subtracted result 0.000 ...

By long division can be the quotient 1/9 in the decimal number 0.111 ... rewrite. Multiplication of 9 times 1 makes each location to 9, so 9 times 0.111 ... equal to 0.999 ..., and 9 times 1/9 is equal to 1, from which 0.999 ... = 1 follows:

The proof can also lead to other breakthroughs as 2/7 = 0.285714 ... 285 714: 285714/2 is equal to 142857, this gives 999999 times 7 They are usually marketed with the fractures 1/3 or 1/ 9 since. their periods are single digits and they only require multiplication by a single digit number.

Solving for an Unknown

The equation x = 0.999 ... can be summarized as followed to solve for x:

Average

Were 0.999 ... and 1, different numbers, the average would be ( 0.999 ... 1) / 2 = 1.999 ... / 2 again another. Indeed, 1.999 ... / 2 = 0.999 ... is thus proved that 0.999 ... = 1.

Written Division

When 1 is written divided by 1, but it started with 0, the rest 1 arises - 0 = 1 is at 10/ 1 = further made 9 results again 10-9 = 1 9 Thus, repeated periodically.

1/1 = 0.9 ... -0 - 10 - 9 - 10 . . . This procedure is often invoked against that a desired outcome will forced. So hereby leave also prove:

1/1 = 0.888 ... -0 - 10 - 8 - 20 - 8 - 120 - 8 --- 1120 . . . With an infinite number aft, however, no residual repeated, but this numbers of the form ( 10n 8) / 9 and 1n2 ( sequence A047855 in OEIS ) arise. For comparison should be 1/3 calculated using the digits before the comma is always reduced by 1:

1/3 = 0.22 ... 1 -0 - 10 3 - 6 - 40 -36 - 40 . . . At each step there is a 2, which is, however, mended with 1 to 3.

Value systems

The place value system to the base q corresponds to the number 0.999 ... the fraction 9 / (q - 1). Therefore applies to the base 10:

Discussion

The above proofs are based on assumptions whose meaning could be questioned if they are accepted as axioms. An alternative that tackles the core point in the dense order of the real numbers:

Are real numbers, are introduced by decimal, is often defined so that x is less than y when the decimal numbers are different, as seen from the left of the first different point is smaller than the corresponding point of x by y. For example, ( 0) is 43.23 less than 123.25 because the first difference is seen at 0 < 1. According to this definition it comes to actually conclude 0.999 ... < 1

At the point, however, it should be remembered that a dense order is required of the real numbers: between two real numbers is always a third. Accordingly, it is useful to define that x is less than y, if there is a number in between according to the aforementioned criteria, and because at 0.999 ... all points with 9 - the highest number - are occupied, there can be no number between 0.999 and ... give 1, so 0.999 ... = 1.

For a deeper insight worth taking a look at the analytical proof.

Analytical proof

Decimal numbers can be defined as infinite series. In general:

In the case of 0.999 ..., the convergence rate will be applied for geometric series:

As 0.999 ... is a geometric progression with a = 9 and R = 1/10, the following applies:

This proof (actually, that 10 = 9.999 ... is ) appears in Leonhard Euler's Complete Guide to Algebra.

A typical derivation of the 18th century took advantage of an algebraic proof as the above. 1811 brought John Bonnycastle in his textbook An Introduction to Algebra an argument with the geometric series. A reaction of the 19th century against such a generous summation resulted in a definition that dominates today: The number of the members of an infinite sequence is defined as the limit of the sequence of its partial sums ( sums of the first finitely many summands ).

A sequence of (a0, a1, a2, ...) has the limit x, if there are all > 0, a member of the sequence, by which all members are less than a distance of x. 0.999 ... can be understood as a limit of the sequence ( 0.9, 0.99, 0.999, ...):

The last step follows from the Archimedean property of real numbers. The limit value based behavior is also found in less precise formulations. This explains the textbook The University Arithmetic from 1846: " 999 , continued to infinity = 1, Because every annexation of a 9 brings the value closer to 1" Arithmetic for Schools (1895 ) says: "when a large number of 9s is taken, the difference in between 1 and 99999 ... Becomes inconceivably small "

The interpretation as a limit can also representations as 0.999 ... 1 is an emphasis to. 0.999 ... 1 would be as the limit of ( 0.1, 0.91, 0.991, ...) be interpreted, is thus equal to 1 but again generally places have after a period no effect.

Evidence by the construction of the real numbers

Some approaches define the real numbers as expressly structures that result from the rational numbers by axiomatic set theory. The natural numbers - 0, 1, 2, 3, and so on - start with 0 and go up continuously, so that every number has a successor. Natural numbers can be extended with their counter numbers, in order to obtain the integers, and further to the ratios between the numbers to obtain the rational numbers. These systems are accompanied by the arithmetic of addition, subtraction, multiplication and division. In addition, they have an order, so that each number can be compared with another and either smaller, larger or equal.

The step from the rational numbers to the real is a major expansion. There are at least three known ways to do it: Dedekind cuts, Cauchy sequences (both 1872 releases ) and nested intervals. Evidence of 0.999 ... = 1, the use of such structures directly are not to be found in textbooks on calculus. Even if a design is offered, it is usually used to prove the axioms of the real numbers, which then support the above proof. However, it has repeatedly expressed the opinion that it is logically reasonable to start with a design.

Dedekind cuts

A real number can be defined as Dedekind cut in, so as a complete division of rational numbers into two non- empty sets L | R, so that l < r we have for all l L and r R. The left amount of 0.999 ... contains exactly the rational numbers R, the R is less than 0.9 ... with any number of a finite number of nines, that is smaller than any number of the form:

Since each element of the left-hand quantity is less than 1 - as it is defined in the rational numbers - the section 1 is called.

The definition of the real numbers as Dedekind cuts was first published in 1872 by Richard Dedekind.

Cauchy sequences

A sequence is called Cauchy sequence if there is for every > 0 a member of the sequence, from which all elements less than apart. To all Cauchy sequences assign a specific limit, the real numbers are introduced as equivalence classes of Cauchy sequences. Two Cauchy sequences a and b are called equivalent if the sequence (an - bn ) the limit is 0, so is a null sequence. The number 1 represents the equivalence class of Cauchy sequence ( 1, 1, 1, ...), 0.999 ... stands for the equivalence class of Cauchy sequence ( 0.9, 0.99, 0.999, ...). The consequences are due to equivalent:

A possible proof of this is that all members are removed from the n-th less than 0, if = m / n. This is 0.999 ... = 1

This definition of the real numbers was first published in 1872 independently by Eduard Heine and Georg Cantor.

Of intervals

The real numbers can be defined as well as equivalence classes of rational nested intervals. A sequence of intervals [( an, bn ) ] is of intervals, when a monotonic increasing, b monotonically falls applies to b n for all n, and the sequence ( BN - to ) is a zero-sequence. Two of intervals and are equivalent if and always applies.

D0, d1d2d3 ... is the equivalence class of intervals ( [ d0, d0 1], [ D0, D1, D0, D1 0.1 ], ... ), ... Thus, 0.999 of the equivalence class of intervals ([0, 1] [ 0.9, 1] [ 0,99, 1], ...), the one of intervals of ( [1, 2 ], [1, 1,1 ], [1, 1,01 ], ...). Since the required property of equivalence is satisfied, applies 0.999 ... = 1

Generalizations

The fact that 0.999 ... = 1, can be generalized in various ways. Every terminating decimal number other than 0 has an alternative representation with infinitely many nines, for example, 0.24999 ... for 0.25. The same phenomenon also occurs in other bases. Thus, in the dual system 0.111 ... = 1, the Ternärsystem 0.222 ... = 1 and so on.

In non-integer bases, there are also different representations. With the golden ratio φ = (1 √ 5 ) / 2 as the base ( " Phinärsystem " ), there is in addition to 1 and 0.101010 ... infinitely many other ways to represent the number one. There are generally for almost all q 1-2 uncountably many base -q- representations of 1 On the other hand, there are still uncountably many q (including all natural numbers greater than 1 ) for which there is only one q is the basic representation for the trivial 1 except (1). 1998 certain Vilmos Komornik and Paola Loreti the smallest base with this property, the Komornik - Loreti constant 1.787231650 ... In this basis, 1 = 0,11010011001011010010110011010011 ...; the points arising out of the Thue - Morse sequence.

Other examples of spellings for the same are:

Harold B. Curtis points to another curiosity: 0.666 ... 0.666 ... = 1.111 ... 2

Application

1802 H. Goodwin published a discovery of the occurrence of nines in periodic decimal expansions of fractions with certain prime numbers as the denominator. Examples are:

E. Midy 1836 proved a general theorem about such breaks, which is now known as the set of Midy: Does the period of the fully truncated fraction a / p is an even number of points and p is prime, the sum of the two halves of the period a sequence of nines. The publication was obscure and it is unclear whether the evidence directly took 0.999 ..., but at least a modern proof of WG Leavitt does this.

The Cantor set, which arises when removed from the interval [0, 1] of real numbers from 0 to 1 infinitely often the open middle third of the remaining intervals can also be as a set of real numbers of [0, 1 ] describe that can be represented using only the digits 0 and 2 in Ternärsystem. The n-th decimal place here describes the position of the point after the nth step of the construction. The number 1 could be represented for example as 0.222 ... 3, which indicates that it is positioned right after each step. 1/3 = 0.13 = 0.0222 ... 3 is after the first distance, left, after every other right. 1/4 = 0.020202 ... 3 is alternately left and right.

Cantor's second diagonal argument uses a method which constructs a new every sequence of real Nachkommaanteile, and thus shows the uncountable real numbers: It is made a number whose nth decimal place is different from the nth decimal place of the nth sequence element. Is the choice of decimal arbitrary, but that's not necessarily created a new number. This can be solved by a non-terminating representation of the numbers required and the replacement of a site is prohibited by 0.

Liangpan Li put 2011 a construction of the real numbers is, in 0.999 ... and 1, and the like are defined as equivalent. A sign function is described by:

Skepticism

The equation 0.999 ... = 1 is questioned for various reasons:

Some assume any real number would have a unique decimal representation.
Some see 0.999 ... an indefinite finite or potentially infinite number of nines or -update, but no restriction to add more digits to form a number between 0.999 ... and 1. 0.999 ... 1 may be mentioned as an example.
Some interpret 0.999 ... as a direct predecessor of 1
Some see 0.999 ... as a consequence rather than limit.

These ideas do not conform to the usual definition in the real arithmetic, but may be valid in alternative payment systems that have been designed specifically for the purpose or for general mathematical benefits.

It is also conceivable, that F ( 0.999 ... ) is interpreted, so that on the one hand, 0.999 ... = 1 is accepted, but on the other hand ( 0.999 ... 2 - 1) / ( 0.999 ... - 1 ) = 2, while ( 12-1 ) / ( 1 - 1) is undefined. This write so, but is not common and misleading.

Fame

With the growth of the Internet debates over 0,999 have ... leave the classroom and are widely used in Internet forums, including those that have little to do with mathematics. The newsgroups sci.math de.sci.mathematik and the question added to the FAQ.

Lina Elbers received an award from the German Mathematical Society for the smartest question, the math professors was asked: Why ... is 0.999 not less than 1. She was sixth-grader.

The sequence of six nines in the county number from the 762nd decimal place is known as the Feynman point and was named after Richard Feynman, who once said he wanted to get the number up to this point, so he recite to the point and then "and so on " can say what suggests the number is rational.

A joke on this subject is:

Alternative Number Systems

Hyper Real Numbers

The analytical proof of 0.999 ... = 1 is based on the Archimedean property: that, for every > 0 is a natural number n such that 1 / n <. Some systems, however, do smaller numbers, so-called Infinitesimalzahlen.

For example, the dual numbers contain a new element ε, which behaves similarly to the imaginary unit i, but with the difference ε2 = 0 instead of i2 = -1. Each binary number in the form a bε with real a and b. The resulting structure is useful for the automatic differentiation. If the lexicographic order by a bε < c dε if and only if a < c or (a = c and b

A distinction can be made with the hyper- real numbers: It is an extension of the real numbers with numbers that are greater than any natural number, in which the transfer principle is satisfied: Every statement in the first order predicate logic that applies to applies also. While any real number from the interval [ 0, 1] by a sequence of digits

With natural numbers as indices can be illustrated, according to the notation of AH Lightstone each hyper- real number from the interval [0, 1] * by a hyper episode

Are presented with hyper natural numbers as indices. While Lightstone 0,999 ... did not mention directly, he showed that 1/3 with 0.333 ...; ... 333 ... is displayed. The number 1 could thus 0.999 ...; ... 999 ... shown. " 0.333 ...; ... 000 ..." and " 0.999 ...; ... 000 ... " do not correspond to hyper- real number, on the other hand we can say that 0.999 ...; ... 999000 ..., the last 9 indexed by an arbitrary hyper natural number smaller than 1.

In addition, Karin and Mikhail Katz presented an interpretation of 0.999 ... as a hyper- real number:

Ian Stewart characterizes this interpretation as a perfectly reasonable way, the intuition that in 0.999 ... "a bit " to 1 missing to justify strict.

Ultra structure of potency could 0.9 than the equivalence class of the sequence ( 0.9, 0.99, 0.999, ...) to be interpreted. This is less than 1 = (1, 1, 1, ...). In addition to Katz and Katz Robert Ely questioned the assumption that ideas about 0.999 ... <1 erroneous intuitions about real numbers are and see them rather than non-standard intuitions that could assist in the learning of calculus. José Benardete argues in his book Infinity: An essay in metaphysics that some natural vormathematische intuitions can not be expressed if a restriction on an overly restrictive number system exists.

Hackenbush

The combinatorial game theory provides alternatives. Elwyn Berlekamp in 1974 described a connection between infinite positions in the Blue -Red Hackenbush and binary numbers. For example, the Hackenbush position LRRLRLRL ... the value 0.010101 ... 2 = 1/3. The value of LRLLL ... ( 0,111 ... 2) is infinitesimally smaller than 1, the difference is the surreal number 1 / ω = 0.000 ... 2, which corresponds to the Hackenbush string ... LRRRR.

In general, two different binary numbers are always for different Hackenbush positions. Thus, in the real numbers 0.10111 ... 2 = 0.11000 ... 2 = 3/4. After Berlekamp's assignment, the first number is but the value of LRLRLLL ..., the second is the value of LRLLRRR ...

Rethinking the subtraction

The Subtraktionsbeweis can be undermined if the difference from 1 to 0.999 ... simply does not exist. Mathematical structures in which the addition, but not subtraction is complete, include commutative semigroups, commutative monoids and semirings. Fred Richman considers two such systems, which is 0.999 ... <1.

First Richman defines a non-negative decimal number as a decimal literal. It defines the lexicographic order and an addition, so 0.999 ... <1 therefore applies simply because 0 <1, however, is 0.999 ... x = 1 x for each non-terminating x. A special feature of the decimal is therefore that the addition can not always be reduced. With the addition and multiplication, the decimal form a positive totally ordered commutative semiring.

Then it defines a different system, which he calls D -section and that meets the Dedekind cuts, but with the difference that it allows for a decimal fraction d both the cut and the cut. The result is that the real numbers are " live together uneasily with decimal fractions ." There is no positive Infinitesimalzahlen in section D, but a negative type infinitesimal, 0 -, which has no decimal. He concludes that 0.999 ... = 1 0 -, while the equation 0.999 ... x = 1 has no solution.

P- adic numbers

While in the decimal 0.999 ... a first has 9 but not last, has in the 10 - adic numbers ... 999 reversed without first nine, but a last. If 1 is added, produced the number ... 000 = 0, so ... 999 = -1. Another derivation makes use of the geometric series:

While the series does not converge at the real numbers, it converges at the 10 - adic numbers. Also the possibility to apply here the proof with the multiplication by 10 is:

Finally, could be considered a theory of " Doppeldezimalzahlen " that combines the real numbers with the 10 -adic, and in the ... 999,999 ... = 0 ( because of ... 999 = -1, 0.999 ... = 1 and -1 1 = 0).