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This article presents background and proofs of the fact that the recurring decimal 0.999… equals 1, not approximately but exactly. More precisely, the standard real number represented by 0.999… (where the 9s recur) is exactly equal to the standard real number 1.

In arithmetic with decimal fractions, a simple division of integers like

⁄₃

becomes a recurring decimal,

0.3333…,

in which digits repeat without end. There also exist numbers that are not quotients of integers, such as √2 = 1.41421356… and π = 3.14159265… with an endless number of digits that do not repeat. A benefit of the decimal notation is that most calculations — addition, subtraction, multiplication, division, comparison — use manipulations that are much the same as for integers. And like integers, for most numbers a different series of digits means a different number (ignoring trailing zeros as in 0.250 and 0.2500). The one notable class of exceptions is numbers with trailing repeating 9s.

It should be no surprise that a notation allows a single number to be written in different ways. For example,

⁄₂ = ⁄₆.

The 9s case does surprise, perhaps because any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1. Thus infinity, a sometimes mysterious concept, plays an important role behind the scenes.

Proofs fall into two main categories, depending on the level of mathematical sophistication and rigor demanded. Examples of both are given.

Elementary proofs

Elementary proofs assume that manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition.

Fraction proof

The standard method used to convert the fraction ⁄₃ to decimal form is long division, and the well-known result is 0.3333…, with the digit 3 repeating. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…; but 3 × ⁄₃ equals 1, so it must be the case that 0.9999… = 1.

Algebra proof

Another kind of proof adapts to any repeating decimal. When a fraction in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. Subtracting the smaller number from the larger can proceed digit by digit; the result is 9 − 9, which is 0, in each of the digits after the decimal separator. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.9999…, be called c. Then 10c − c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1.

Advanced proofs

Proofs at a more advanced level draw on the axiomatic foundations of mathematics. They use careful and sound definitions of integers, fractions, real numbers, infinity, limits, and equality. The validity of manipulations at the elementary level is a logical consequence of these foundations.

One requirement is to characterize numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a positional notation, so that the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5. Without sign the value is determined as follows:

$a_{n}\cdots a_{2}a_{1}a_{0}.b_{1}b_{2}\cdots =\,\!$	$a_{n}\times 10^{n}+\cdots +a_{2}\times 10^{2}+a_{1}\times 10^{1}+a_{0}\times 10^{0}$
	${}+b_{1}/10^{1}+b_{2}/10^{2}+\cdots .$

A minus sign negates the value. For purposes of this discussion the integer part can be summarized as b₀. To proceed further, to give any meaning to the sum of the b_k terms, requires a theoretical exploration of numbers, especially real numbers.

The natural numbers — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. Peano axioms are the usual formal definition, and these in turn draw upon axiomatic set theory. There is little difficulty, conceptual or formal, in extending natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found less than, greater than, or equal.

Order proof

The step from rationals to reals is a huge extension, and order is an essential part of any construction. In the Dedekind cut approach, each real number z is a partition of the rational numbers into two sets, (B, A), with the numbers in B being all those ordered less than (below) z and the numbers in A being the rest (above or equal). So for any non-empty set of rationals S bounded above, let U(S) be the set of all rationals that are upper bounds of S. (Thus for any x in S and y in U(S), x ≤ y.) With U(S) as A and its complement (in the rationals) as B, a definite real number is selected.

Now let the set S be {⁄_{1, ⁄_{10, ⁄_{100, ⁄_{1000, …}, the rational numbers obtained as truncations of 0.9999… to 0, 1, 2, 3, or any number of decimal places. In this way, every number in decimal notation determines a Dedekind cut, which is taken to define its meaning as a real number. The task is thus to show that U({1}) is the same set (and thus gives the same Dedekind cut) as U(S), or equivalently, to show that 1 is the least rational greater than or equal to every member of S.}}}}

If an upper bound less than 1 exists, it can be written as 1−x for some positive rational x. To bound ⁄_{10, which is ⁄_{10 less than 1, x can be at most ⁄_{10. Continuing in this fashion through each decimal place in turn, induction shows that x must be less than ⁄_{10 for every positive integer n. But the rationals have the Archimedean property (they contain no infinitesimals), so it must be the case that x = 0. Therefore U(S) = U({1}), and 0.9999… = 1.}}}}

Limit proof

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where |z| is the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that are Cauchy using this distance. That is, in the sequence (x₀, x₁, x₂, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |x_m − x_n| ≤ δ for all m, n > N. (The distance between terms becomes arbitrarily small.)

A sequence (x₀, x₁, x₂, ...) has a limit x if the distance |x − x_n| becomes arbitrarily small as n increases. Now if (x_n) and (y_n) are two Cauchy sequences, taken to be real numbers, then they are defined to be equal as real numbers if the sequence (x_n − y_n) 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

\left(1-0,1-{9 \over 10},1-{99 \over 100},\dots \right)=\left(1,{1 \over 10},{1 \over 100},\dots \right)

has the limit 0. But this is clear by inspection, and so again it must be the case that 0.9999… = 1.

Generalizations

These proofs immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0's) has a doppelgänger with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. Second, a comparable theorem applies in each radix or base. For example, in the radix 3 version 0.222… equals 1.

Alternative algebras and expansions

These proofs rely, explicitly or implicitly, on properties of the standard real numbers, including the Archimedean property that there are no nonzero infinitesimals. There are mathematically coherent ordered algebras, including various alternatives to standard reals, which are non-Archimedean; but it is difficult to discuss decimal expansions in them, because:

They may have multiple elements with the same decimal expansion to an infinite number of places.
Dividing through by an infinitesimal, when defined, would result in elements larger than every integer, which therefore cannot be expressed by decimals in the usual fashion at all.

The non-standard properties make these systems unsuitable for ordinary calculations, though they are of theoretical interest. For example, the p-adic numbers are constructed from rationals in the same way as the reals, but using different orderings (one for each prime p). Their equivalent of "decimal expansions" is of interest in number theory.

Standard reals can also be extended to become dual numbers, by including a new element ε defined to combine with other reals in the usual way, but such that its product with itself is zero. Every dual number then consists of a standard real component and an "infinitesimal" component, either of which may be zero. However, the infinitesimals are displaced off the real line, rather than ordered between standard reals.

Another way to construct alternatives to standard reals is to use topos theory and alternative logics rather than set theory and classical logic (which is a special case). For example, smooth infinitesimal analysis has infinitesimals with no reciprocals, as explained by Bell .

Game theory provides alternative reals as well, with Hackenstrings as one particularly relevant example.

The existence of such alternatives is one reason why we must insist on standard reals, and why the advanced proofs require more care than might be supposed.

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