# What do you call a number formed by a sequence of repeating digits

For example, I have a sequence

``````77777777777
``````

Is there a word in the dictionary to represent

a sequence of repetitive/recurring digits

• I'd call it repetitive
– Jim
Nov 22, 2013 at 4:29
• I bet someone at Mathematics Stack Exchange would know.
– long
Nov 22, 2013 at 4:37
• I remember using 'repeating stanza' once. As jwpat says, repetend works for the repeat in a recurring decimal. It's tempting to broaden the usage, but maths is hotter on the well-defined than English per se. Nov 22, 2013 at 9:31
• @EdwinAshworth you'd be re-broadening a narrowing. Generally repetend means anything that repeats except for numbers, because with numbers it has the more specific meaning. Nov 22, 2013 at 12:13
• Recommending migration to mathSE
– Kris
Nov 22, 2013 at 13:36

This type of number is commonly called a repdigit number, or sometimes a monodigit number. This term is used most frequently in recreational mathematics and is most often seen in investigations into prime numbers.

See wikipedia article on repdigit for details.

• Apparently I saw your answer too late, and was 13 minutes behind Jan 16, 2017 at 23:20
• @LaurentDuval I do that a lot. I post an answer before reading all of the answers only to then realise that my answer has already been given. I shall upvote your answer because I think it is correct ;-) Jan 17, 2017 at 4:20

It's called a periodic sequence. See wikipedia.

The given example is not a sequence per se, considering that the "digits" are not separated (by a comma, space).

A natural number whose digits are repeating in some positional number system is called, in recreational mathematics, a repdigit. This comes from repeated and digit. In the case it is composed of digit 1 (1, 11, 111, 11111), it is called a repunit. The latter was coined by A. H. Beiler in 1966. See Repunit and Repdigit Numbers for other details.

Otherwise, this would just be a constant sequence:

Constant sequences are sequences for which all terms are the same.

Repeating would be the usual term, as in a repeating decimal, but that could also mean a sequence such as "123123123123." If you don't mind a coinage, how about monodigital sequence?

thefreedictionary.com defines repetend as “Mathematics The digit or group of digits that repeats infinitely in a repeating decimal”.
collinsdictionary.com defines repetend as “(mathematics) the digit or series of digits in a recurring decimal that repeats itself”.
Wiktionary defines repetend as “(mathematics) A repeating decimal”, which while closer to what you've asked about – ie the sequence of repeated numbers rather than the digits that repeat – is probably an incorrect definition.

• The decimal is a significant part of the definition, not to be ignored.
– Kris
Nov 22, 2013 at 13:35
• @Kris, no it isn't. Decimal is used in those definitions to direct attention to the fractional part of a number, rather than the integer part. Repetends actually are base-agnostic; they can follow a binary point, a trinary point, etc just as well as a decimal point. Nov 22, 2013 at 17:22
• Check again. 1. It occurs only after the decimal point ('decimal', not binary, ternary [not trinary], etc.); 2. it repeats infinitely; 3. it need not be a single digit. That part of a circulating decimal which recurs continually, ad infinitum: -- sometimes indicated by a dot over the first and last figures; thus, in the circulating decimal .728328328 ..., the repetend is 283. (from the GNU version of the Collaborative International Dictionary of English)
– Kris
Nov 23, 2013 at 7:07
• @Kris, a repetend is a repeating element (digit or digits, possibly 0) in the representation of a fraction in a number system. Yes, the repetend repeats infinitely, but the infinite sequence that the question asks about is not a repetend; a repetend is merely one small but infinitely-often-repeated unit of an infinite sequence. Re trinary vs ternary, either may be used (1,2). Re point terms, see Radix Point Nov 23, 2013 at 15:48
• @tchrist, I suppose I should have said representation of a fraction's value instead of representation of a fraction. Also, reference to ℚ is correct; a real value that isn't rational (ie, that's in ℝ∖ℚ) can't have a repetend in spite of having an infinitely-long (non-terminating) decimal representation. Nov 23, 2013 at 19:17

The closest single word would be repetend, which generally means anything that is repeated (e.g. it would also be used about music and poetry).

Unfortunately, the one thing we cannot use it for is repeating numbers, because it has a specific sense when it comes to numbers where it only refers to the repeated numbers found in representing a fraction that cannot be fully represented in the base used (e.g. ⅓ in decimal ends with a repetend of 3, 1/7 in decimal has a repetend of 142857, ⅓ in binary ends with a repetend of 01, 1/10 in binary ends with a repetend of 0011, and so on).

So, while in that one case it is the perfect word to use, in any others it is a particularly bad word to use, because it will make people think of that more specific use.

In other cases I would just use "recurring sequence" or "repeated sequence".

• Interestingly, all ⁿ⁄₇ fractions have the same repetend — it just starts at a different point in the sequence. When n is 1, it’s what you said: 142857; when n is 2, it’s 285714; when n is 3, it’s 428571; when n is 4, it’s 571428; when n is 5, it’s 714285; and when n is 6, it’s 857142. So all you ever need to remember is a single repetend for all the sevenths — and how that starting point varies. １428571２8571428571４28571428５714285７142857142８571428571428571428571428571428571428571428571428571428571428571428571428571428571428571. . . .
– tchrist
Nov 23, 2013 at 17:49
• @tchrist a bit beyond what I needed to express here, but if we do move this to math.SE, we'll be well-primed (excuse the pun). Meanwhile, I see a downvoter has an objection to my answer. I look forward to their explaining it. Nov 23, 2013 at 17:51

A series or set, maybe.

Finding an exact term for this kind of repeated number is quite hard, for it will depend on where you are currently using it