0.999...

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.

Proofs fall into two main categories, depending on the level of mathematical sophistication and rigor demanded. Examples of both are given. These proofs rely on properties of the standard real numbers; there are other so called "non-standard" real numbers, for which these proofs do not hold.

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

¹⁄₃

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, in most cases 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,

¹⁄₂ = ³⁄₆.

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. (See "The proof in popular culture" below).

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

The standard method used to convert the fraction ¹⁄₃ 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 × ¹⁄₃ equals 1, so it must be the case that 0.9999… = 1.

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. To see this, consider that subtracting 0.9999... from 9.9999… 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.

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, rom 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

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

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

.999… can be defined using the infinite geometric sequence

${\displaystyle a_{n}=.9\times .1^{n-1}\,}$

By the definition of an infinite geometric series,

${\displaystyle \sum _{n=1}^{\infty }.9\times .1^{n-1}=.9\times .1^{0}+.9\times .1^{1}+.9\times .1^{2}+.9\times .1^{3}+\cdots \,}$

${\displaystyle .999...=.9\times .1^{0}+.9\times .1^{1}+.9\times .1^{2}+.9\times .1^{3}+\cdots \,}$

Therefore,

${\displaystyle .999...=\sum _{n=1}^{\infty }.9\times .1^{n-1}\,}$

By the formula for the sum of a geometric sequence,

${\displaystyle \sum _{n=1}^{\infty }.9\times .1^{n-1}={\frac {.9}{1-.1}}\,}$

${\displaystyle \sum _{n=1}^{\infty }.9\times .1^{n-1}=1\,}$

Therefore,

${\displaystyle .999...=1\,}$

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.

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, a+bε, 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.^[1]

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.

This topic provokes interest far beyond its minor status within mathematics. For example, in the newsgroup sci.math, devoted to discussion of general mathematics, statistics show over one thousand postings related to this proof; and it is one of the questions answered in its FAQ. It is also quite common in other forums of an elementary nature. One reason might be that people encounter it at a time when they are young and curious, and the usual explanations seem unconvincing. Another is that, like many such magnets, the statement of the proposition is elementary, but the proof is not. Professor David Tall has gone so far as to study characteristics of teaching and cognition that might lead to some of the misunderstandings he has encountered in his college students.

Many internet message boards contain frequent debates over this theorem since some participants reject it.

• Recurring decimal
• Geometric series
• Convergent series
• Infinite series
• Limit of a sequence
• Non-standard analysis
• Intuitionism

• Why does 0.9999... = 1 ?
• Ask A Scientist: Repeating Decimals
• Repeating Nines
• Mathematical Gazette joke
• Point nine recurring equals one
• David Tall's research on mathematics cognition