This may be more of a historical math question.

In any old, ancient, or even modern symbolism; was there a symbol or letter that represented the value $-1$?

This goes in pair with how ancient mathematicians would commonly use a symbol for $0$ before the concept of zero was well defined.

  • $\begingroup$ Any other related tags would also be useful $\endgroup$ – Graviton Jul 31 '17 at 6:44
  • $\begingroup$ For what it's worth, some formattings in excel (and probably other spreadsheet software) use $\color{red}{1}$ (a $1$ coloured red) as the negative of $1$. $\endgroup$ – Arthur Jul 31 '17 at 6:46
  • $\begingroup$ @Arthur ah, that's interesting. I wonder why that is. $\endgroup$ – Graviton Jul 31 '17 at 6:47
  • $\begingroup$ I think it aligns with standard business phrases regarding the color red as signifying deficit and black signifying surplus. I don't any more than that, though. $\endgroup$ – Arthur Jul 31 '17 at 6:48
  • $\begingroup$ @Arthur It was once standard accounting practice, to use red ink to denote a negative number (i.e. a loss). Thus the common idioms "in the red" vs "in the black". $\endgroup$ – Robert Israel Jul 31 '17 at 6:50

$-1$ has not had any special symbols in widespread use. It was "minus" that has had various symbols for its representation both to negate a quantity and/or as subtraction operation (some of them are $m^\circ$, meno, $m\tilde e$, $men$, $\tilde{m}$, $\sim$, $^{\cdot}\hspace{-6pt}{-}$, $\div$, and many others). There are lots of books on this subject.

There is just one exception in what I've just said. In 1925 the mathematician J. P. Ballantine had a similar intuition as yours. In American Mathematical Monthly Vol. 32 No. 6 (Jun.-Jul. 1925) p.302 he wrote:

Mathematical historians will tell you how many years mathematics was held back for want of a digit $0$. Though not comparing in importance with that digit, there is a certain advantage in having a digit to represent negative one.

He continues proposing the adoption for $-1$ of a symbol which is $1$ rotated by $180^\circ$ around its center (in mathjax its difficult to represent; it is something like $\downharpoonright$. In Latex it is much more simple using this syntax: \rotatebox[origin=c]{180}{1}, after having imported the graphics package with \usepackage{graphicx})

Please note that Ballantine proposes such a symbol not for the number $-1$ but for $-1$ as a digit.

In this way not only it is possible to write $${\downharpoonright}\times2=-2$$ but it is also possible to write: $${\downharpoonright}.123=-1+0.123$$ $${\downharpoonright}0=-10$$ $${\downharpoonright}2=-10+2$$ with $${\downharpoonright}2={\downharpoonright}92=...={\downharpoonright}99992=...$$

In the end Ballantine's proposal as himself says it is a way to represent numbers in their tens' complement representation.


Ballantine does not give an example of numbers where the digit $\downharpoonright$ is not the most significant digit, probably because he does not need such sofistication in his intended application (that is, to complement a given positive number $x$ w.r.t. an integer $n$-th power of ten, where $n\ge\log_{10} x$ ). Anyway following his philosophy we could also write: $${\downharpoonright}{\downharpoonright}=-11$$ $$10.{\downharpoonright}=10-0.1$$

That said, Ballantine originality is mainly about the new symbol proposed for the digit $-1$. As to the applications it is not as much as original. Think that the periodical editor recalls in a footnote (Op. cit.) that a generalization coud be one in which digits $\bar5, \bar4, \bar3, \bar2, \bar1, 0,1,2,3,4,5$ are used instead of $0,1,2,3,4,5,6,7,8,9$ for real number encoding, where $\bar5=-5,...$ Indeed this was something very well known in past centuries. As early as 1726 John Colson in "A Short Account of Negativo-Affirmative Arithmetick" Philosophical Transactions of the Royal Society of London Vol 34 pp. 161-173 introduces such negative figures, as he calls them, as a means to ease arithmetic especially when large numbers are involved. It is no more than tens' complement encoding of numbers.

  • $\begingroup$ Very interesting, thanks! One question: if ${\downharpoonright}2=-10+2$, wouldn't it be more consistent if ${\downharpoonright}.123=-1+0.123=-0.877$? Which option does Ballantine prefer? $\endgroup$ – Chris Culter Sep 1 '17 at 17:28
  • $\begingroup$ @ChrisCulter you caught me! :-) It's my error. Thanks. I'm going to edit the answer. $\endgroup$ – trying Sep 1 '17 at 17:31
  • $\begingroup$ Very interesting. It begs the question what other numbers would be such as $⇂⇂$ or $10.⇂$. $\endgroup$ – Graviton Sep 2 '17 at 23:28
  • $\begingroup$ @Graviton see edit $\endgroup$ – trying Sep 3 '17 at 13:48
  • $\begingroup$ If only I could upvote twice $\endgroup$ – Graviton Sep 8 '17 at 0:18

In balanced ternary, there is a symbol for negative one. If T is negative one, then in balanced ternary, you count 0000, 0001, 001T, 0010, 0011, 01TT, 01T0, 01T1, etc


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