In mathematics, the repeating decimal 0.999..., or ) denotes a real number
that can be shown to be the number one. In other words, the symbols "0.999..."
and "1" represent the same number. Proofs of this equality have been
formulated with varying degrees of mathematical rigor, taking into account
preferred development of the real numbers, background assumptions,
historical context, and target audience. Every nonzero, terminating decimal has
an equal twin representation with infinitely many trailing 9s. The
terminating decimal representation is usually preferred, contributing to the
misconception that it is the only representation. The same phenomenon
occurs in all other bases or in any similar representation of the real
numbers. The equality of 0.999... and 1 is
closely related to the absence of nonzero infinitesimals in the real
number system, the most commonly used system in mathematical analysis. Some
alternative number systems, such as the hyperreals, do contain nonzero
infinitesimals. In most such number systems, the standard interpretation of
the expression 0.999... makes it equal to 1, but in some of these number
systems, the symbol "0.999..." admits other interpretations that contain
infinitely many 9s while falling infinitesimally short of 1.
The equality 0.999... = 1 has long been accepted by mathematicians and is part
of general mathematical education. Nonetheless, some students find it
sufficiently counterintuitive that they question or reject it. Such skepticism
is common enough that the difficulty of convincing them of the validity of this
identity has been the subject of several studies in mathematics education.
Algebraic proofs Algebraic proofs showing that 0.999...
represents the number 1 use concepts such as fractions, long division, and
digit manipulation to build transformations preserving equality from
0.999... to 1. However, these proofs are not rigorous as they do not include a
careful analytic definition of 0.999... = Fractions and long division =
One reason that infinite decimals are a necessary extension of finite decimals
is to represent fractions. Using long division, a simple division of integers
like 1⁄9 becomes a recurring decimal, 0.111..., in which the digits repeat
without end. This decimal yields a quick proof for 0.999... = 1. Multiplication
of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 ×
1⁄9 equals 1, so 0.999... = 1: This result is consistent with other
ninth fractions, all of which have repeating decimals, such as 3/9 and 8/9.
If 0.999... is to be consistent, it must equal 9/9 = 1.
= Digit manipulation = When a number in decimal notation is
multiplied by 10, the digits do not change but each digit moves one place to
the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the
original number. To see this, consider that in subtracting 0.999... from
9.999..., each of the digits after the decimal separator cancels, i.e. the
result is 9 − 9 = 0 for each such digit: = Discussion =
Although these proofs demonstrate that 0.999... = 1, the extent to which they
explain the equation depends on the audience. In introductory arithmetic,
such proofs help explain why 0.999... = 1 but 0.333... N.
If and are two Cauchy sequences, then they are defined to be equal as real
numbers if the sequence has the limit 0. Truncations of the decimal number
b0.b1b2b3... generate a sequence of rationals which is Cauchy; this is taken
to define the real value of the number. Thus in this formalism the task is to
show that the sequence of rational numbers
has the limit 0. Considering the nth term of the sequence, for , it must
therefore be shown that This limit is plain if one understands
the definition of limit. So again 0.999... = 1.
The definition of real numbers as Cauchy sequences was first published separately
by Eduard Heine and Georg Cantor, also in 1872. The above approach to decimal
expansions, including the proof that 0.999... = 1, closely follows Griffiths
& Hilton's 1970 work A comprehensive textbook of classical mathematics: A
contemporary interpretation. The book is written specifically to offer a second
look at familiar concepts in a contemporary light.
= Infinite decimal representation = Commonly in secondary schools'
mathematics education, the real numbers are constructed by defining a number
using an integer followed by a radix point and an infinite sequence written
out as a string to represent the fractional part of any given real
number. In this construction, the set of any combination of integers and digits
after the decimal point is the set of real numbers. This construction can too
be rigorously shown to satisfy all of the real axioms after defining an
equivalence relation over the set that defines 1 =eq 0.999... as well as for
any other nonzero decimals with only finitely many nonzero terms in the
decimal string with its trailing 9s version. With this construction of the
reals, all proofs of the statement 1 = .999... can be viewed as implicitly
assuming the equality when any operations are performed on the real
numbers. Generalizations
The result that 0.999... = 1 generalizes readily in two ways. First, every
nonzero number with a finite decimal notation has a counterpart with trailing
9s. For example, 0.24999... equals 0.25, exactly as in the special case
considered. These numbers are exactly the decimal fractions, and they are
dense. Second, a comparable theorem applies in
each radix or base. For example, in base 2 0.111... equals 1, and in base 3
0.222... equals 1. In general, any terminating base b expression has a
counterpart with repeated trailing digits equal to b − 1. Textbooks of real
analysis are likely to skip the example of 0.999... and present one or both of
these generalizations from the start. Alternative representations of 1 also
occur in non-integer bases. For example, in the golden ratio base, the two
standard representations are 1.000... and 0.101010..., and there are
infinitely many more representations that include adjacent 1s. Generally, for
almost all q between 1 and 2, there are uncountably many base-q expansions of 1.
On the other hand, there are still uncountably many q for which there is
only one base-q expansion of 1, other than the trivial 1.000.... This result
was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In
1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the
Komornik–Loreti constant q = 1.787231650.... In this base, 1 =
0.11010011001011010010110011010011...; the digits are given by the Thue–Morse
sequence, which does not repeat. A more far-reaching generalization
addresses the most general positional numeral systems. They too have multiple
representations, and in some sense the difficulties are even worse. For
example: In the balanced ternary system, 1/2 =
0.111... = 1.111.... In the reverse factorial number system,
1 = 1.000... = 0.1234.... = Impossibility of unique representation
That all these different number systems suffer from multiple representations for
some real numbers can be attributed to a fundamental difference between the real
numbers as an ordered set and collections of infinite strings of
symbols, ordered lexicographically. Indeed, the following two properties
account for the difficulty: If an interval of the real numbers is
partitioned into two non-empty parts L, R, such that every element of L is less
than every element of R, then either L contains a largest element or R contains
a smallest element, but not both. The collection of infinite strings of
symbols taken from any finite "alphabet", lexicographically ordered,
can be partitioned into two non-empty parts L, R, such that every element of L
is less than every element of R, while L contains a largest element and R
contains a smallest element. Indeed, it suffices to take two finite prefixes p1,
p2 of elements from the collection such that they differ only in their final
symbol, for which symbol they have successive values, and take for L the
set of all strings in the collection whose corresponding prefix is at most
p1, and for R the remainder, the strings in the collection whose corresponding
prefix is at least p2. Then L has a largest element, starting with p1 and
choosing the largest available symbol in all following positions, while R has a
smallest element obtained by following p2 by the smallest symbol in all
positions. The first point follows from basic
properties of the real numbers: L has a supremum and R has an infimum, which are
easily seen to be equal; being a real number it either lies in R or in L, but
not both since L and R are supposed to be disjoint. The second point
generalizes the 0.999.../1.000... pair obtained for p1 = "0", p2 = "1". In fact
one need not use the same alphabet for all positions or consider the full
collection of possible strings; the only important points are that at each
position a finite set of symbols can be chosen from, and that making a valid
choice for any position should result in a valid infinite string. Under these
assumptions, the above argument shows that an order preserving map from the
collection of strings to an interval of the real numbers cannot be a bijection:
either some numbers do not correspond to any string, or some of them correspond
to more than one string. Marko Petkovšek has proven that for any
positional system that names all the real numbers, the set of reals with
multiple representations is always dense. He calls the proof "an
instructive exercise in elementary point-set topology"; it involves viewing
sets of positional values as Stone spaces and noticing that their real
representations are given by continuous functions.
Applications One application of 0.999... as a
representation of 1 occurs in elementary number theory. In 1802, H. Goodwin
published an observation on the appearance of 9s in the
repeating-decimal representations of fractions whose denominators are certain
prime numbers. Examples include: 1/7 = 0.142857142857... and 142 + 857 =
999. 1/73 = 0.0136986301369863... and 0136 +
9863 = 9999. 1/77 = 0.012987012987... and 0129 + 870
= 999. E. Midy proved a general result about
such fractions, now called Midy's theorem, in 1836. The publication was
obscure, and it is unclear if his proof directly involved 0.999..., but at least
one modern proof by W. G. Leavitt does. If it can be proved that a decimal of
the form 0.b1b2b3... is a positive integer, then it must be 0.999..., which
is then the source of the 9s in the theorem. Investigations in this
direction can motivate such concepts as greatest common divisors, modular
arithmetic, Fermat primes, order of group elements, and quadratic
reciprocity. Returning to real analysis, the base-3
analogue 0.222... = 1 plays a key role in a characterization of one of the
simplest fractals, the middle-thirds Cantor set:
A point in the unit interval lies in the Cantor set if and only if it can be
represented in ternary using only the digits 0 and 2.
The nth digit of the representation reflects the position of the point in
the nth stage of the construction. For example, the point 2⁄3 is given the
usual representation of 0.2 or 0.2000..., since it lies to the right of
the first deletion and to the left of every deletion thereafter. The point 1⁄3
is represented not as 0.1 but as 0.0222..., since it lies to the left of
the first deletion and to the right of every deletion thereafter.
Repeating nines also turn up in yet another of Georg Cantor's works. They
must be taken into account to construct a valid proof, applying his 1891
diagonal argument to decimal expansions, of the uncountability of the unit
interval. Such a proof needs to be able to declare certain pairs of real numbers
to be different based on their decimal expansions, so one needs to avoid pairs
like 0.2 and 0.1999... A simple method represents all numbers with
nonterminating expansions; the opposite method rules out repeating nines. A
variant that may be closer to Cantor's original argument actually uses base 2,
and by turning base-3 expansions into base-2 expansions, one can prove the
uncountability of the Cantor set as well.
Skepticism in education Students of mathematics often reject the
equality of 0.999... and 1, for reasons ranging from their disparate appearance
to deep misgivings over the limit concept and disagreements over the
nature of infinitesimals. There are many common contributing factors to the
confusion: Students are often "mentally committed
to the notion that a number can be represented in one and only one way by a
decimal." Seeing two manifestly different decimals representing the same
number appears to be a paradox, which is amplified by the appearance of the
seemingly well-understood number 1. Some students interpret "0.999..." as a
large but finite string of 9s, possibly with a variable, unspecified length. If
they accept an infinite string of nines, they may still expect a last 9 "at
infinity". Intuition and ambiguous teaching lead
students to think of the limit of a sequence as a kind of infinite process
rather than a fixed value, since a sequence need not reach its limit. Where
students accept the difference between a sequence of numbers and its limit, they
might read "0.999..." as meaning the sequence rather than its limit.
These ideas are mistaken in the context of the standard real numbers, although
some may be valid in other number systems, either invented for their
general mathematical utility or as instructive counterexamples to better
understand 0.999... Many of these explanations were found by
David Tall, who has studied characteristics of teaching and
cognition that lead to some of the misunderstandings he has encountered in
his college students. Interviewing his students to determine why the vast
majority initially rejected the equality, he found that "students
continued to conceive of 0.999... as a sequence of numbers getting closer and
closer to 1 and not a fixed value, because 'you haven't specified how many
places there are' or 'it is the nearest possible decimal below 1'".
Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a
successful strategy for convincing reluctant students that 0.999... = 1.
Still, when confronted with the conflict between their belief of the first
equation and their disbelief of the second, some students either begin to
disbelieve the first equation or simply become frustrated. Nor are more
sophisticated methods foolproof: students who are fully capable of
applying rigorous definitions may still fall back on intuitive images when they
are surprised by a result in advanced mathematics, including 0.999.... For
example, one real analysis student was able to prove that 0.333... = 1⁄3 using
a supremum definition, but then insisted that 0.999...