Wikipedia:Reference desk/Archives/Mathematics/2021 February 3

= February 3 =

What is the name of the number system where the only digits that exist are +1 and -1?
What is the name of the number system where the only digits that exist are +1 and -1?2804:7F2:599:7854:7DC8:A854:8F46:1667 (talk) 23:24, 3 February 2021 (UTC)


 * I think you actually are looking for either binary, in which the only digits are 0 and 1, or balanced ternary, in which the only digits are -1, 0, and +1. —SeekingAnswers (reply) 00:47, 4 February 2021 (UTC)
 * It's difficult to have a base without a digit for 0 since there's no equivalent way to add leading 0's. Not that it would be impossible, for example you might argue that a representation of n base 1 is a string of n 1's. A base with just two digits would presumably have a radix of 2. (Our article seems to be incorrect on this since the radix can be less than the number of digits, see for example Golden ratio base.) But a base 2 system with digits {+1, -1} would also have the issue that a single number can be represented an infinite number of ways, for example (+1) = (+1)(-1) = (+1)(-1)(-1) = (+1)(-1)(-1)(-1) = ... . It's normal for a single number to represented in two ways (see 0.999...) but an infinite number of ways seems a bit much. --RDBury (talk) 12:18, 4 February 2021 (UTC)
 * There would also be the issue that 2 (= (+1) + (+1)) cannot be represented; other than perhaps 0 represented by an empty digit sequence, the representable numbers are the odd numbers. --Lambiam 14:13, 4 February 2021 (UTC)
 * I suppose trailing zeros shouldn't be allowed, in which case the number of representations of 1 is still infinite:
 * 1 = .+1+1+1+1+1... = +1.+1-1-1-1-1... = +1.-1+1+1+1+1... = +1-1.+1-1-1-1-1... ....
 * You can get 2 by shifting these over one decimal, e.g. 2 = +1.+1+1+1+1... . --RDBury (talk) 11:36, 5 February 2021 (UTC)
 * As a multiplicative system it is the cyclic group of order 2, thus isomorphic to the additive group of Z/2Z. Rgdboer (talk) 01:53, 5 February 2021 (UTC)