nLab radix notation

Idea

At least in classical mathematics, every real number may be written in base $n$ for any natural number $n \geq 2$ (binary for $n = 2$, decimal for $n = 10$, etc), which can be generalized to fractional $n \gt 1$ (especially $n = \mathrm{e} \approx 2.718$) and beyond.

For integers, this idea goes back to the Old Babylonians using base $60$ (around 2000 BCE), perfected in the Gupta Empire using base $10$ (around 400), and popularized in Europe by Fibonacci? (in 1202). For arbitrary real numbers, these were first used by Abu'l-Hasan al-Uqlidisi? (around 952) and popularized by Simon Stevin? (in 1585). This notation (in base $10$) is now ubiquitous, and (despite the technical difficulties of doing so rigorously) serves as a de facto definition of the real numbers in elementary mathematics.

Radix notation in base $2$ is the basis of floating point arithmetic?, the fast but imprecise method of calculation with real numbers usually used in modern computing.

Definition

Given a natural number $n \geq 2$, let $[n]$ be the set of natural numbers (including zero) strictly less than $n$: $[n] = \{0, 1, 2, \ldots, n - 2, n - 1\}$; in this context, an element of $[n]$ is called a digit (in base $n$). Let $\mathbb{Z}$ be the set of integers: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$; in this context, an element of $\mathbb{Z}$ is called a place. Then a function from $\mathbb{Z}$ to $[n]$ is a doubly-infinite sequence of digits, one digit for each place. Such a sequence $a = (a_i)_{i\colon{\mathbb{Z}}}$ defines a doubly-infinite series of rational numbers:

$$\sum_{-\infty \lt i \lt \infty} a_i n^{-i} .$$

(The term $a_i n^{-i}$ is the place value of the digit $a_i$ at the place $i$.) This series converges (to a real number) if and only if there is a place $N$ such that $a_i = 0$ for all $i \lt N$. That is, we can write the series as

$$\sum_{i = N}^\infty a_i n^{-i} .$$

In this way, a sequence of digits (finite on one end, infinite on the other) represents a nonnegative real number. (Sometimes $n^i$ is used instead of $n^{-i}$, so that integers have nonzero digits only at positive places instead of only at negative ones. Ultimately this is a matter of convention, but this article uses $n^{-i}$ so that the places are written in order.)

As a representation of real numbers, this is almost unique. Specifically, two distinct sequences $a$ and $b$ of digits represent the same real number (meaning that the sums of the series are equal) if and only if there is some place $m$ such that

• $a_i = b_i$ for every place $i \lt m$,
• $a_m = b_m + 1$ (or the reverse), and
• $a_i = n - 1$ for every $i \gt m$ while $b_i = 0$ for every $i \gt m$.

In this case, the sequence with $0$s is generally considered standard. The real numbers arising in this way are precisely the positive $n$-adic rational? numbers, that is the rational numbers that are positive integer multiples of $n^{-m}$ for some integer $m$ (the same $m$ as above).

In classical mathematics, every nonnegative real number may be represented in this way. Given such a number $x$ and a place $i$, let $a_i$ be the remainder? modulo $n$ of the floor of $x n^i$: that is, $a_i \coloneqq \lfloor{x n^i}\rfloor \mod n$; then $x = \sum_i a_i n^{-i}$. (For $n$-adic rational $x$, this will produce the representation with $0$s; to automatically produce the other representation, use $a_i = (\lceil{x n^i} - 1\rceil) \mod n$.) The number $0$ is represented only by a sequence of all $0$s.

Given $n$ unique symbols for the $n$ digits, the real number represented by a sequence of digits may be written by, beginning at the smallest place whose digit is nonzero, writing the symbols in order, with a dot (or comma), called the radix point, between place $0$ and place $1$ (and padding zeroes after the radix point if the first nonzero digit has not yet been reached). Of course, the sequence is still infinite and cannot be written down, but we may write any finite portion. The $n$-adic rationals can be represented exactly by leaving out the infinitely repeating digit $0$; arbitrary rational numbers may be represented exactly by a bar over a list of digits that repeats infinitely. (It is a theorem that every non-$n$-adic rational number can be uniquely represented in this way; in fact, only rational numbers can be so represented.) A subscript may be used to indicate the base $n$ (with a default base $10$ in practice). Finally, a negative real number is written by writing its absolute value after a minus sign.

Generalizations

Given any number $n$ (not necessarily a natural number), still called the base, and any set $D$ of numbers, whose elements are called digits, we may consider finite or infinite sequences of digits, indexed by integers (still called places). Such a sequence should include a smallest place $N$ as part of its data (since $0$ may not be one of the digits), so the places are all (or perhaps only some) of the integers $i \geq N$. Then the infinite series

$$\sum_{i=N}^\infty a_i n^{-i}$$

still converges to a real number.

If $n \geq 2$ is a natural number and $D$ is $[n]$, then we recover the representation above of all nonnegative real numbers, unique except for the positive $n$-adic rationals. More generally, if $n \gt 1$ is any real number and $D$ is $[\lceil{n}\rceil]$, then we get a representation of all nonnegative real numbers, which (for non-integer $n$) is never unique (although in practice one uses the last sequence in lexicographic ordering).

If $n \geq 2$ is a natural number and $D$ is $[n] + 1 = \{1, 2, 3, \ldots, n - 1, n\}$, then we get a representation of all positive real numbers (but not $0$ unless we allow the empty list of digits). If $n = 1$ and $D$ is again $[n] + 1 = \{1\}$, then every infinite sequence gives a divergent series, but the finite sequences represent all positive integers, uniquely up to a shift of place; this is called tally notation. In the other direction, if $n \geq 3$ is a natural number and $D$ is a set of $n$ consecutive integers that includes both $-1$ and $1$, then we get a representation of all real numbers (positive, negative, and zero), which is unique except for a few rational numbers (in the case of $n = 3$ and so $D = \{-1, 0, 1\}$, those which are half of a power of $1/3$); this is called balanced radix notation.

Examples

Here are some representations of well-known real numbers in well-known radixes:

Number	Decimal ($n = 10$, $D = [10]$	Binary ($n = 2$, $D = \{0, 1\}$)	Positive binary ($n = 2$, $D = \{1, 2\}$)	Natural ($n = \mathrm{e}$, $D = \{0, 1, 2\}$)	Ternary ($n = 3$, $D = \{0, 1, 2\}$)	Balanced ternary ($n = 3$, $D = \{-1, 0, 1\}$, writing $-$ for $-1$)	Tally ($n = 1$, $D = \{1\}$)
$0$	$0$ (or $-0$)	$0$ (or $-0$)	$0$ (nonce symbol, or empty list)	$0$ (or $-0$)	$0$ (or $-0$)	$0$	$0$ (nonce symbol, or empty list)
$1$	$1$ (or $0.\overline{9}$)	$1$ (or $0.\overline{1}$)	$1$ (as a finite list, or $0.\overline{1}$ with padding $0$, or $0.0\overline{2}$)	$1$ (or $0.2121111212001\ldots$, etc)	$1$ (or $0.\overline{2}$)	$1$	$1$
$2$	$2$ (or $1.\overline{9}$)	$10$ (or $1.\overline{1}$)	$2$ (as a finite list, or $1.\overline{1}$, or $0.\overline{2}$ with padding $0$)	$2$ (or $1.2121111212001\ldots$, etc)	$2$ (or $1.\overline{2}$)	$2$	$11$
$\mathrm{e}$	$2.7182818284590\ldots$	$10.10110111111\ldots$	$1.21221222222\ldots$	$10$ (or $2.121111212001\ldots$, etc)	$2.2011011212211\ldots$	$10.{-}0111{-}{-}0{-}0{-}\ldots$	(not possible)
$3$	$3$ (or $2.\overline{9}$)	$11$ (or $10.\overline{1}$)	$11$ (as a finite list, or $2.\overline{1}$, or $1.\overline{2}$)	$10.020011200001\ldots$ (etc)	$10$ (or $2.\overline{2}$)	$10$	$111$
$\pi$	$3.14159265358979\ldots$	$11.001001000011\ldots$	$2.112112111122\ldots$	$10.101002020002$ (etc)	$10.010211012222\ldots$	$10.011{-}111{-}000{-}\ldots$	(not possible)
$1/2$	$0.5$ (or $0.4\overline{9}$)	$0.1$ (or $0.0\overline{1}$)	$0.1$ (as a finite list with padding $0$, or $0.0\overline{1}$ with padding $0$, or $0.00\overline{2}$)	$0.1021200201202\ldots$ (etc)	$0.\overline{1}$	$0.\overline{1}$ (or $1.\overline{-}$)	(not possible)
$1/3$	$0.\overline{3}$	$0.0\overline{10}$	$0.00\overline{21}$ (with padding $0$)	$0.021012102010201\ldots$ (etc)	$0.1$ (or $0.0\overline{2}$)	$0.1$	(not possible)
$-1$	$-1$ (or $-0.\overline{9}$)	$-1$ (or $-0.\overline{1}$)	$-1$ (as a finite list, or $-0.\overline{1}$ with padding $0$, or $-0.0\overline{2}$)	$-1$ (or $-0.2121111212001\ldots$, etc)	$-1$ (or $-0.\overline{2}$)	$-$	$-1$

Foundational issues

In constructive mathematics, it is not generally true that every real number has a radix expansion. However, one does have the following results:

• In the presence of weak countable choice, there exists a radix expansion in any base (e.g., a decimal expansion or binary expansion) for every Cauchy real number iff $\mathrm{LLPO}_\mathbb{N}$ holds; there exists a radix expansion for every Dedekind real number has iff the analytic $\mathrm{LLPO}$ holds. Without weak countable choice, Lifschitz realizability gives a model in which $\mathrm{LLPO}_\mathbb{N}$ holds but it is not true that there exists a radix expansion in any base for every Cauchy real number, which implies that there are models in which the analytic LLPO holds but it is not true that there exists a radix expansion in any base for every Dedekind real number. See Andrew Swan‘s answer to Birchfield (2024). In addition, the analytic $\mathrm{LLPO}$ holds for the Dedekind real numbers in condensed sets, but every function from the Dedekind real numbers to the boolean domain $\mathbb{2}$ is a constant function, which implies that it is not true that there exists a radix expansion in any base for every Dedekind real number.

• In the presence of countable choice, $\mathrm{LLPO}_\mathbb{N}$ is equivalent to the claim that the rings of radix expansions in any two bases are isomorphic. See Daniel Mehkeri's answer to Feldman (2010).

• That every Cauchy real number has a choice of radix expansion in any base implies that the weak limited principle of omniscience (WLPO) for the natural numbers holds; that every Dedekind real number has a choice of radix expansion implies that the analytic WLPO holds.

Fred Richman considered a number system (a noncancellable rig) of nonnegative decimal sequences in which $0.\overline{9} \lt 1$; the usual rig of nonnegative real numbers is a subrig; see Richman 1999. In this rig, $0.\overline{9}$ is not cancellable, and there is no value of $1 - 0.\overline{9}$. Although Richman was a prominent constructivist, the development is not (and probably cannot be made) constructive.

In nonstandard analysis, every hyperreal number has a radix expansion (by the transfer principle), but these now have digits with nonstandard natural places. The expansion of a nonstandard hyperreal will have digits at nonstandard places that cannot be predicted by the digits at standard places. A well-known example is that (in base $10$) $0.\overline{9} \lt 1$ if it is understood that the bar applies a hyperfinite but nonstandard number of places (although $0.\overline{9} = 1$, as usual, if the bar applies to all places). One wants to write $1 - 0.\overline{9} = 0.\overline{0}1$, but this is misleading, because the bar on the left-hand side includes one more digit than the bar on the right-hand side; $1 - 0.\overline{9}9 = 0.\overline{0}1$ would be more accurate. Of course, one could write this same equation in standard mathematics if the bar is understood to apply to a large but finite number of places. (Note that it’s not possible to have the bar apply to exactly the standard places, since the set of standard natural numbers is not definable. Indeed, in that case the $1$ would have to be at the least nonstandard place, and there is no such number.)

Mathematician and Usenet legend Alexander Abian?, before branching into speculative physics, advocated polemically that the real numbers should be defined as finite or infinite sequences of decimal digits; see Abian 1981. While not favoured by most mathematicians due to the inelegance of the definitions and proofs, this is the form in which the real numbers are often presented to elementary students.

References

• Alexander Abian?. 1981. Calculus must consist of the study of real numbers in their decimal representation and not of the study of an abstract complete ordered field or nonstandard real numbers. International Journal of Mathematical Education in Science and Technology 12(4). Doi:10.1080/0020739810120417.

• Fred Richman, Is 0.999… = 1?. Mathematics Magazine, Volume 72, Issue 5, 1999, Pages 396-400, [doi:10.1080/0025570X.1999.11996777]

• David Feldman (2010) on Math.Overflow, Radix notation and toposes

• Madeleine Birchfield (2024) on Category Theory Zulip, Radix expansions in constructive mathematics