0.999 ...

Artist's impression of the decimal number

The recurring decimal 0.999 ... (with more or less nines before the ellipsis or written as 0, 9 , or 0, (9) ) refers to the real number 1 . The symbols “0.999…” and “1” therefore represent the same number (see place value notation ). Evidence of this equation has been formulated with varying degrees of rigor depending on the preferred introduction of the real numbers, background assumptions, historical context, and target audience.

Furthermore, every decimal number that is not equal to 0 has an alternative representation with an infinite number of nines, for example 8.31999 ... for 8.32. The interrupted representation is usually preferred because of its brevity, which favors the misconception that it is the only one. The same phenomenon occurs in other bases as well. However, systems have been developed in which the equation does not hold.

This article is intended to give an overview under which mathematical rules of the symbol sequence 0.999 ... which mathematical meaning can be given.

Elementary evidence

The following evidence uses concepts known from school math. It is assumed that periodic decimal fractions always represent a rational number. For example, 1/3 = 0.333 ... is taken as given, since the handwritten division produces this result. In the following considerations, it is not a question of whether 0.999 ... represents something meaningful, but only about the question of which number is meant by it. The handwritten procedures for performing the four basic arithmetic operations are also carried over to non-terminating decimal fractions and are assumed to remain valid.

Written subtraction

If 0.999 ... is subtracted from 1 in writing from left to right , the result is 0.000 ...

Fractions

The quotient 1/9 can be converted into the decimal number 0.111 ... by division in writing . A multiplication of 9 times 1 makes every digit a 9, so 9 times 0.111 ... equals 0.999 ..., and 9 times 1/9 equals 1, which means 0.999 ... = 1:

{\ displaystyle {\ begin {aligned} 0 {,} 333 \ ldots & {} = {\ frac {1} {3}} \\ 3 \ cdot 0 {,} 333 \ ldots & {} = 3 \ cdot { \ frac {1} {3}} \\ 0 {,} 999 \ ldots & {} = 1 \ end {aligned}}}

{\ displaystyle {\ begin {aligned} 0 {,} 111 \ ldots & {} = {\ frac {1} {9}} \\ 9 \ cdot 0 {,} 111 \ ldots & {} = 9 \ cdot { \ frac {1} {9}} \\ 0 {,} 999 \ ldots & {} = 1 \ end {aligned}}}

The proof can also be carried out with other fractions such as 2/7 = 0.285714 285714…: 285714/2 is equal to 142857, this times 7 results in 999999. However, they are usually given with the fractions 1/3 or 1/9, there their periods are single-digit and they only require multiplication by a single-digit number.

Transformation of an indeterminate

${\ displaystyle x = 0 {,} 999 \ ldots}$ can be reworked as follows:

{\ displaystyle {\ begin {aligned} x & = 0 {,} 999 \ ldots \\ 10x & = 9 {,} 999 \ ldots \\ 10x-x & = 9 {,} 999 \ ldots -0 {,} 999 \ ldots \\ 9x & = 9 \\ x & = 1 \ end {aligned}}}

average

If 0.999… and 1 were different numbers, the average (0.999… + 1) / 2 = 1.999… / 2 would be different again. In fact, 1.999… / 2 = 0.999…, which proves that 0.999… = 1.

Place value systems

In the place value system based on q , the number 0.999 (written in standard notation) corresponds to the fraction 9 / ( q - 1). So for the base q = 10:

{\ displaystyle 0 {,} 999 \ ldots = {\ frac {9} {10-1}} = {\ frac {9} {9}} = 1}

discussion

The above proofs are based on assumptions the meaning of which could be questioned if accepted as axioms . An alternative that starts with the dense ordering of the real numbers:

If real numbers are to be introduced through decimal representations, it is often defined that x is smaller than y if the decimal representations of the numbers are different and the first different digit of x seen from the left is smaller than the corresponding digit of y . For example, (0) 43.23 is less than 123.25 because the first difference can be seen at 0 <1. According to this definition, it actually comes to the conclusion 0.999 ... <1.

At this point, however, it should be borne in mind that a dense order is required of the real numbers: Between two different real numbers there is always a third, different from the two. Accordingly, it makes sense to define that x is smaller than y if there is a number in between according to the criterion already mentioned, and because at 0.999 ... all digits are occupied with 9 - the highest digit - there can be no number between 0.999 ... and give 1, which means 0.999 ... = 1.

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

Analytical evidence

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

{\ displaystyle d_ {0} {,} d_ {1} d_ {2} d_ {3} d_ {4} \ ldots = d_ {0} + d_ {1} \ left ({\ tfrac {1} {10} } \ right) + d_ {2} \ left ({\ tfrac {1} {10}} \ right) ^ {2} + d_ {3} \ left ({\ tfrac {1} {10}} \ right) ^ {3} + d_ {4} \ left ({\ tfrac {1} {10}} \ right) ^ {4} + \ cdots}

For the case 0.999 ... the convergence theorem for geometric series can be applied:

If then .

{\ displaystyle | r | <1 \, \!}

{\ displaystyle ar + ar ^ {2} + ar ^ {3} + \ cdots = {\ frac {ar} {1-r}}}

Since 0.999 ... is a geometric series with a = 9 and r = 1/10, the following applies:

{\ displaystyle 0 {,} 999 \ ldots = 9 \ left ({\ tfrac {1} {10}} \ right) +9 \ left ({\ tfrac {1} {10}} \ right) ^ {2} +9 \ left ({\ tfrac {1} {10}} \ right) ^ {3} + \ cdots = {\ frac {9 \ left ({\ tfrac {1} {10}} \ right)} {1 - {\ tfrac {1} {10}}}} = 1}

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

A typical derivation from the 18th century used an algebraic proof similar to the above. In 1811, John Bonnycastle made an argument with the geometric series in his textbook An Introduction to Algebra . A reaction of the 19th century to such a generous summation resulted in a definition that still dominates today: the series of terms of an infinite sequence is defined as the limit value of the sequence of its partial sums (sums from the first finitely many summands).

A sequence ( a ₀ , a ₁ , a ₂ , ...) has the limit value x if there is one term in the sequence for all > 0, from which all terms are less than distant from x . 0.999 ... can be understood as the limit value of the sequence (0.9, 0.99, 0.999, ...): ${\ displaystyle \ varepsilon}$ ${\ displaystyle \ varepsilon}$

{\ displaystyle 0 {,} 999 \ ldots = \ lim _ {n \ to \ infty} 0 {,} \ underbrace {99 \ ldots 9} _ {n} = \ lim _ {n \ to \ infty} \ sum _ {k = 1} ^ {n} {\ frac {9} {10 ^ {k}}} = \ lim _ {n \ to \ infty} \ left (1 - {\ frac {1} {10 ^ { n}}} \ right) = 1- \ lim _ {n \ to \ infty} {\ frac {1} {10 ^ {n}}} = 1}

The last step follows from the Archimedean property of real numbers. The limit-based attitude can also be found in less precise formulations. The textbook The University Arithmetic from 1846 explains : ".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 between 1 and .99999… becomes inconceivably small "

By interpreting it as a limit value, representations such as 0.999… 1 can also be assigned a meaning. 0.999… 1 would then be understood as a limit value of (0.1, 0.91, 0.991,…), but it is then equal to 1. In general, digits after a period have no effect .

Prove by constructing the real numbers

Some approaches explicitly define the real numbers as structures that result from the rational numbers through axiomatic set theory . The natural numbers - 0, 1, 2, 3, and so on - start with 0 and go up, so that each number has a successor. The natural numbers can be expanded with their opposites to get the whole numbers , and further with the ratios between the numbers to get 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 to another and is either less, greater, or equal.

The step from the rational numbers to the real ones is a significant extension. There are at least three known ways to accomplish this: Dedekindian cuts, Cauchy sequences (both published in 1872), and interval nesting. Evidence for 0.999 ... = 1 using such constructions directly cannot be found in textbooks on calculus. Even when a construction is offered, it is usually used to prove the axioms of the real numbers, which then support the above proof. However, the opinion has been expressed several times that it is logically more appropriate to start with a construction.

Dedekindian cuts

A real number can be defined as a Dedekind cut in , i.e. as a complete subdivision of the rational numbers into two non- empty sets L | R , so l < r for all l L and r R applies. The left set of 0.999 ... contains exactly the rational numbers r for which r is less than 0.9 ... with an arbitrary number of finitely many nines, i.e. smaller than any number of the form: ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ in}$ ${\ displaystyle \ in}$

{\ displaystyle 1- \ left ({\ frac {1} {10}} \ right) ^ {n}}

Since every element of the set on the left is less than 1 - as it is defined for the rational numbers - the cut is called 1.

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

Cauchy episodes

A sequence is called a Cauchy sequence if, for all > 0, there is a term in the sequence from which all terms are less than apart from one another. In order to be able to assign a specific limit value to all Cauchy sequences, the real numbers are introduced as equivalence classes of Cauchy sequences. Two Cauchy sequences a and b are called equivalent if the sequence ( a _n - b _n ) has the limit value 0, i.e. is a zero sequence . The number 1 stands for the equivalence class of the Cauchy sequence (1, 1, 1,…), 0.999… stands for the equivalence class of the Cauchy sequence (0.9, 0.99, 0.999,…). The consequences are equivalent because of: ${\ displaystyle \ varepsilon}$ ${\ displaystyle \ varepsilon}$

{\ displaystyle \ lim _ {n \ to \ infty} \ left (1- \ sum _ {k = 1} ^ {n} {\ frac {9} {10 ^ {k}}} \ right) = \ lim _ {n \ to \ infty} {\ frac {1} {10 ^ {n}}} = 0}

A possible proof for this is that all terms from the nth onwards are less than 0 when = m / n . So 0.999 ... = 1. ${\ displaystyle \ varepsilon}$ ${\ displaystyle \ varepsilon}$

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

Interval nesting

Illustration of the equation 1 = 0.222… ₃ with nesting intervals

The real numbers can also be defined as equivalence classes of rational nesting of intervals. A sequence of intervals ([ a _n , b _n ]) is of intervals, when a monotonic increasing, b monotonically falls, a _n b _n for all n is valid and the sequence ( b _n - a _n ) is a zero-sequence. Two nesting intervals and are equivalent if and is always true. ${\ displaystyle \ leq}$ ${\ displaystyle ([a_ {n}, b_ {n}])}$ ${\ displaystyle ([a '_ {n}, b' _ {n}])}$ ${\ displaystyle a_ {n} \ leq b '_ {n}}$ ${\ displaystyle a '_ {n} \ leq b_ {n}}$

d ₀ , d ₁ d ₂ d ₃ … stands for the equivalence class of the interval nesting ([ d ₀ , d ₀ + 1], [ d ₀ , d ₁ , d ₀ , d ₁ + 0.1],…), consequently 0.999 ... the equivalence class of the interval nesting ([0, 1], [0.9, 1], [0.99, 1], ...), 1 that of the interval nesting ([1, 2], [1, 1.1] , [1, 1,01], ...). Since the required property of equivalence is fulfilled, 0.999… = 1 applies.

Generalizations

The fact that 0.999… = 1 can be generalized in different ways. Every terminating decimal number not equal to 0 has an alternative representation with an infinite number of nines, for example 0.24999 ... for 0.25. An analogous phenomenon occurs in every base for the digits with value : In the dual system 0.111 ... = 1, in the ternary system 0.222 ... = 1 and so on. ${\ displaystyle q}$ ${\ displaystyle q-1}$

There are also different representations in non-integer bases. With the golden ratio φ = (1 + √ 5 ) / 2 as the basis ("Phinärsystem") there are next to 1 and 0.101010 ... infinitely many more possibilities to represent the number one. In general, for almost all q between 1 and 2 there are uncountably infinitely many base q representations of 1. On the other hand, there are still uncountably infinitely many q (including all natural numbers greater than 1) for which there is only one Base q representation for 1 except for the trivial (1) there. In 1998 Vilmos Komornik and Paola Loreti determined the smallest basis with this property, the Komornik-Loreti constant 1.787231650… In this basis 1 = 0.11010011001011010010110011010011…; the positions result from the Thue-Morse sequence .

Further examples of different representations of the same value are:

1/2 = 0.111… = 1, 1 1 1 … in the balanced ternary system .
1 = 0.1234… in the reversed faculty-based number system with the bases 2 !, 3 !, 4 !,… for decimal places .

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

application

Positions of 1/4, 2/3 and 1 in the Cantor set

In 1802 H. Goodwin published a discovery about the occurrence of nines in periodic decimal representations of fractions with certain prime numbers as denominators. Examples are:

1/7 = 0.142857142857 ... and 142 + 857 = 999.
1/73 = 0.0136986301369863 ... and 0136 + 9863 = 9999.

E. Midy proved a general theorem about such fractions in 1836, now known as Midy's Theorem: If the period of the fully abbreviated fraction a / p has an even number of digits and p is prime, then the sum of the two halves of the period a series of nines. The publication was obscure and it is unclear whether the evidence directly used 0.999 ... but at least modern evidence from WG Leavitt does.

The Cantor set, which arises when the open middle third of the remaining intervals is removed infinitely from the interval [0, 1] of real numbers from 0 to 1, can also be expressed as the set of real numbers from [0, 1 ], which can only be represented with the digits 0 and 2 in the ternary system. The nth decimal place describes the position of the point after the nth step of the construction. The number 1 could be represented as 0.222 ... ₃ , indicating that it is positioned to the right after each step. 1/3 = 0.1 ₃ = 0.0222… ₃ is on the left after the first distance, on the right after every further distance. 1/4 = 0.020202… ₃ lies alternately on the left and right.

Cantor's diagonal argument used a method of real to each sequence Nachkommaanteile a new constructed, and thus shows the uncountable of real numbers: There is formed a number whose n -th decimal place other than the n th decimal place of the n th Sequence member. If the choice of decimal representation is arbitrary, it does not necessarily result in a new number. This can be remedied by requiring a non-terminating display of the numbers and forbidding a digit to be replaced by 0.

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

{\ displaystyle {\ text {sign}} (x): = {\ begin {cases} 0 & \; {\ text {if}} \; x \ geq 0 {,} 000 \ ldots \\ 1 & \; {\ text {if}} \; x \ leq (-1) {,} 999 \ ldots \\\ end {cases}}}

skepticism

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

Some assume that every real number has a unique decimal representation.
Some see 0.999 ... as an indefinite finite or potentially or actually infinite number of nines, but no restriction on adding more decimal places to form a number between 0.999 ... and 1. 0.999… 1 could be given as an example.
Some interpret 0.999 ... as a direct predecessor of 1.
Some see 0.999 ... as a consequence instead of a limit value.

These ideas do not correspond to the usual decimal notation in (real) arithmetic, but may be valid in alternative systems designed specifically for the purpose or for general mathematical use.

It is also conceivable that f (0.999 ...) is interpreted as, so that on the one hand 0.999 ... = 1 is accepted, but on the other hand also (0.999 ... ² - 1) / (0.999 ... - 1) = 2, while (1 ² - 1 ) / (1 - 1) is undefined. To write it this way, however, is uncommon and misleading. ${\ displaystyle \ lim _ {x \ to 1} f (x)}$

Notoriety

With the growth of the Internet, debates of .999 ... have left the classroom and are rampant on Internet forums, including those that have little to do with math. The newsgroups de.sci.mathematikand sci.mathhave included the question in the FAQ.

Lina Elbers received a prize from the German Association of Mathematicians for the smartest question that math professors were asked: Why 0.999 ... is not less than 1. She was a sixth grader then.

The sequence of the six nines in the circle number from the 762nd decimal place is known as the Feynman point and was named after Richard Feynman , who once said he wanted to learn the number up to that point, so that he could recite it up to that point and then "And so on", suggesting that the number is rational. ${\ displaystyle \ pi}$

A joke on the subject goes:

Question: How many mathematicians does it take to change a lightbulb?

Answer: 0, 9

Other structures

Hyperreal numbers

The analytical proof for 0.999… = 1 is based on the Archimedean property. This means that for every > 0 there is a natural number n such that 1 / n < . However, some systems offer even smaller numbers, so-called infinitesimal numbers. ${\ displaystyle \ varepsilon}$ ${\ displaystyle \ varepsilon}$

For example, the dual numbers contain a new element ε, which behaves analogously to the imaginary unit i, but with the difference ε ² = 0 instead of i ² = −1. Every dual number has the form a + b ε with real a and b . The resulting structure is useful for automatic differentiation . If the lexicographical order is defined by a + b ε < c + d ε if and only if a <c or both a = c and b <d, the multiples of ε are infinitesimal. The same conventions apply to the decimal representation, so 0.999… = 1 still applies.

A difference can be made with the hyper-real numbers : It is an extension of the real numbers with numbers that are larger than any natural number for which the transfer principle is fulfilled: Every statement in first-order predicate logic that is true for also for . While every real number from the interval [0, 1] by a sequence of digits ${\ displaystyle {} ^ {*} \ mathbb {R}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle {} ^ {*} \ mathbb {R}}$

0, d ₁d ₂d ₃ …

can be represented with natural numbers as indices, according to the notation of AH Lightstone, every hyperreal number from the interval [0, 1] * can be replaced by a hypersequence

0, d ₁d ₂d ₃ …;… d _{ω - 1}d _ωd _{ω + 1} …

represented with hypernatural numbers as indices. While Lightstone did not directly mention 0.999…, he showed that 1/3 is represented as 0.333…;… 333…. The number 1 could thus be represented with 0.999…;… 999…. "0.333 ...; ... 000 ..." and "0.999 ...; ... 000 ..." do not correspond to a hyper-real number, on the other hand it can be said that 0.999 ...; ... 999000 ..., the last 9 of which is indexed by any hypernatural number, is smaller than 1 is.

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

{\ displaystyle {\ underset {\ omega} {0 {,} \ underbrace {999 \ ldots}}} \; = 1 \; - \; {\ frac {1} {10 ^ {\ omega}}}}

Ian Stewart characterizes this interpretation as a perfectly adequate way to strictly justify the intuition that in 0.999 ... "a little" is missing to 1.

In the Ultra potency construction might 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 questions the assumption that ideas about 0.999 ... <1 are faulty intuitions about real numbers and sees them more as non-standard intuitions that could be helpful in learning analysis. José Benardete argues in his book Infinity: An essay in metaphysics that some natural pre-mathematical intuitions cannot be expressed when constrained to an overly restrictive system.

Hackenbush

The combinatorial game theory provides alternatives. In 1974 Elwyn Berlekamp described a connection between infinite positions in the blue-red Hackenbush and binary numbers. For example, the Hackenbush position LRRLRLRL… has 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 always represent different Hackenbush positions. So for the real numbers 0.10111… ₂ = 0.11000… ₂ = 3/4. According to Berlekamp's assignment, however, the first number is the value of LRLRLLL ..., the second the value of LRLLRRR ...

Rethinking subtraction

The proof of subtraction can be undermined if the difference 1 - 0.999 ... simply does not exist. Mathematical structures in which addition but not subtraction is complete include some commutative semigroups , commutative monoids, and half rings , among others . Fred Richman considers two such systems where 0.999 ... <1.

First, Richman defines a non-negative decimal number as a literal decimal representation. It defines the lexicographical order and an addition, which means that 0.999… <1 simply applies because 0 <1, but 0.999… + x = 1 + x for every non-terminating x . A special feature of the decimal numbers is that the addition cannot always be reduced. With the addition and multiplication, the decimal numbers form a positive, totally ordered commutative half-ring .

Then he defines another system, which he calls section D , and which corresponds to Dedekind's cuts, with the difference that for a decimal fraction d, he allows both the cut and the cut . The result is that the real numbers "live uncomfortably with the decimal fractions". There are no positive infinitesimal numbers in the section D , but a kind of negative infinitesimal number, 0 ^- , which has no decimal representation. He concludes that 0.999… = 1 + 0 ^- , while the equation 0.999… + x = 1 has no solution. ${\ displaystyle (- \ infty, d)}$ ${\ displaystyle (- \ infty, d]}$

p -adic numbers

While 0.999… in the decimal system has a first 9, but not a last, with the 10-adic numbers… 999, conversely, has no first 9, but has a last one. Adding 1 results in a number… 000 = 0, so that… 999 = −1 (at least if we are in an additive group with a 0 and a generating 1). Another "derivation" uses the geometric series:

{\ displaystyle \ ldots 999 = 9 + 9 (10) +9 (10) ^ {2} +9 (10) ^ {3} + \ cdots = {\ frac {9} {1-10}} = - 1 .}

While the series does not converge for the real numbers (i.e. it is not a real number), it converges for the 10-adic numbers. There is also the possibility to apply the "proof" with the multiplication by 10:

{\ displaystyle {\ begin {aligned} x & = \ ldots 999 \\ 10x & = \ ldots 990 \\ 10x & = x-9 \\ x & = - 1 \ end {aligned}}}

These 10-adic numbers form a non-zero divisor -free ring in which the zero divisors have non-terminating representations (see Pro-finite number # 10-adic numbers ).

Finally, a “theory” of “double decimal numbers” could be considered, which combines the real numbers with the 10-adic, and in which… 999.999… = 0 (based on… 999 = −1, 0.999… = 1 and −1 + 1 = 0).

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James Stewart: Calculus: Early transcendentals . 4th edition. Brooks / Cole, 1999, ISBN 0-534-36298-2 .

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David O. Tall: Cognitive Development In Advanced Mathematics Using Technology . In: Mathematics Education Research Journal . tape 12 , no. 3 , 2000, pp. 210-230 ( warwick.ac.uk [PDF]).

Dr. Hans von Mangoldt: Introduction to higher mathematics . 1st edition. Publishing house by S. Hirzel, Leipzig 1911, serial numbers.

David F. Wallace: Everything and more: a compact history of infinity . Norton, 2003, ISBN 0-393-00338-8 .

SE Burkov: One-dimensional model of the quasicrystalline alloy . In: Journal of Statistical Physics . tape 47 , no. 3/4 , 1987, pp. 409 , doi : 10.1007 / BF01007518 .

Bob Burn: 81.15 A Case of Conflict . In: The Mathematical Gazette . tape 81 , no. 490 , March 1997, p. 109-112 , doi : 10.2307 / 3618786 .

JB Calvert, ER Tuttle, Michael S. Martin, Peter Warren: The Age of Newton: An Intensive Interdisciplinary Course . In: The History Teacher . tape 14 , no. 2 , February 1981, p. 167-190 , doi : 10.2307 / 493261 .

Younggi Choi, Jonghoon Do: Equality Involved in 0.999… and (-8) ⅓ . In: For the Learning of Mathematics . tape 25 , no. 3 , November 2005, pp. 13-15 .

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Petr Eisenmann: Why is it not true that 0.999… <1? In: The Teaching of Mathematics . tape 11 , no. 1 , 2008, p. 35-40 .

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Web links

Why 0.999 period is not less than 1
Right ?: Is 0.99999999999999999 ... equal to 1? at Zeit Online
.999999 ... = 1? at cut-the-knot (English)
Why does 0.9999 ... = 1? in the Dr. Math FAQ
Repeating Decimals at Ask A Scientist (English)
1 = 0.999 ... in Math Central (English)
Repeating nines (English)
Point nine recurring equals one at Things Of Interest (English)
0.999 ... in the Metamath Proof Explorer (English)
A Friendly Chat About Whether 0.999 ... = 1 at BetterExplained
9,999 ... reasons why 0.999 ... = 1 by Vi Hart (English)

Individual evidence

↑ William Byers argues that if you accept 0.999 ... = 1 on the basis of such evidence, but have not resolved the ambiguity, you have not really understood the equation (Byers pp. 39–41).
↑ In the article rank system # Lexicographical order it is shown that the orderhomoMorphism , which assigns a real number to the character strings, for example using an alphabet {0,1, ..., 9} and which arranges the character strings lexicographically and the real numbers as usual, never an orderisomorphism can be.
↑ Rudin p. 61, Theorem 3.26; J. Stewart p. 706.
↑ Euler p. 170.
↑ Grattan-Guinness p. 69; Bonnycastle p. 177.
↑ The limit value follows, for example, from Rudin p. 57, Theorem 3.20e.
↑ Davies p. 175; Smith and Harrington p. 115.
^ Griffiths and Hilton p. Xiv and Pugh p. 10 prefer Dedekind cuts to axioms. For the use of the cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For positions on logic, see Pugh p. 10, Rudin p. Ix or Munkres p. 30.
↑ Enderton, p. 113 uses a similar definition that corresponds to what is referred to here as the left set.
↑ Rudin p. 17–20, Richman p. 399 and Enderton p. 119 name this cut 1 *, 1 ^- and 1 _R and identify it with the traditional real number 1. What Rudin and Enderton call a Dedekind cut, Richman calls nonprincipal Dedekind cut .
^ ^A ^b J. J. O'Connor, EF Robertson: History topic: The real numbers: Stevin to Hilbert. (No longer available online.) Archived from the original on September 29, 2007 ; accessed on January 23, 2014 .
^ Komornik and Loreti p. 636.
↑ Kempner p. 611; Petkovšek p. 409.
↑ Pugh p. 97; Alligood, Sauer and Yorke pp. 150-152. Protter and Morrey p. 507 and Pedrick p. 29 assign this description as a task.
↑ Prize for arithmetic ace: student asks smartest math question. In: Spiegel Online . April 24, 2008, accessed January 17, 2015 .
^ Lightstone pp. 245-247.
↑ Katz and Katz 2010.
↑ Stewart 2009, p. 175; the full discussion of 0.999… can be found in 172–175.
↑ Katz and Katz 2010b
^ R. Ely 2010.
↑ Richman pp. 397-399.
↑ Richman pp. 398-400. Rudin p. 23 prescribes this alternative construction (though using the rational numbers) as the last task of Chapter 1.
↑ ^a ^b Fjelstad p. 11.
↑ That there is actually a suitable amount function for these is shown in the article Pro-finite number # 10-adic numbers .
↑ DeSua pp. 901-903.

[1] William Byers argues that if you accept 0.999 ... = 1 on the basis of such evidence, but have not resolved the ambiguity, you have not really understood the equation (Byers pp. 39–41).

[2] In the article rank system # Lexicographical order it is shown that the orderhomoMorphism , which assigns a real number to the character strings, for example using an alphabet {0,1, ..., 9} and which arranges the character strings lexicographically and the real numbers as usual, never an orderisomorphism can be.

[3] Rudin p. 61, Theorem 3.26; J. Stewart p. 706.

[4] Euler p. 170.

[5] Grattan-Guinness p. 69; Bonnycastle p. 177.

[6] The limit value follows, for example, from Rudin p. 57, Theorem 3.20e.

[7] Davies p. 175; Smith and Harrington p. 115.

[8] Griffiths and Hilton p. Xiv and Pugh p. 10 prefer Dedekind cuts to axioms. For the use of the cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For positions on logic, see Pugh p. 10, Rudin p. Ix or Munkres p. 30.

[9] Enderton, p. 113 uses a similar definition that corresponds to what is referred to here as the left set.

[10] Rudin p. 17–20, Richman p. 399 and Enderton p. 119 name this cut 1 *, 1 ^- and 1 _R and identify it with the traditional real number 1. What Rudin and Enderton call a Dedekind cut, Richman calls nonprincipal Dedekind cut .

[MacTutor-11] A ^b J. J. O'Connor, EF Robertson: History topic: The real numbers: Stevin to Hilbert. (No longer available online.) Archived from the original on September 29, 2007 ; accessed on January 23, 2014 .

[12] Komornik and Loreti p. 636.

[13] Kempner p. 611; Petkovšek p. 409.

[14] Pugh p. 97; Alligood, Sauer and Yorke pp. 150-152. Protter and Morrey p. 507 and Pedrick p. 29 assign this description as a task.

[SPON-549422-15] Prize for arithmetic ace: student asks smartest math question. In: Spiegel Online . April 24, 2008, accessed January 17, 2015 .

[16] Lightstone pp. 245-247.

[17] Katz and Katz 2010.

[18] Stewart 2009, p. 175; the full discussion of 0.999… can be found in 172–175.

[19] Katz and Katz 2010b

[20] R. Ely 2010.

[21] Richman pp. 397-399.

[22] Richman pp. 398-400. Rudin p. 23 prescribes this alternative construction (though using the rational numbers) as the last task of Chapter 1.

[Fjelstad-23] Fjelstad p. 11.

[24] That there is actually a suitable amount function for these is shown in the article Pro-finite number # 10-adic numbers .

[25] DeSua pp. 901-903.