# Why no interest in negative triangular numbers? **URL:** \ **Category:** Factual Questions\ **Created:** [June 1, 2018, 3:00pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262 "2018-06-01T15:00:05Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Crane](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crane/32/3495_2.png) [@Crane](https://boards.straightdope.com/u/Crane)\ **Post date:** [June 1, 2018, 3:00pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/1 "2018-06-01T15:00:05Z")
Googling about for negative triangular numbers doesn’t yield any responses. Yet it is easy to generate the series by summing negative integers just as you do the positive ones: 0 -1 -3 -6 -10 etc. There is an interesting square root algorithm for perfect squares: S^.5 = 2\*((S/2)-T) Where T is the nearest triangular number less than S/2 Applied to negative triangular numbers: -1^.5 = 2((-1/2)-0) = -1 This plots as a square in quadrant 3 with sides of -1 and an area of -1. -25^.5 = 2\*((-25/2)-10) = -5 This plots as a square in quadrant 3 with sides of -5 and an area of -25. Seems like the kind of thing that interests mathematicians, but I don’t find any references. ---
**Author:** ![Kimstu](https://avatars.discourse-cdn.com/v4/letter/k/ecd19e/32.png) [@Kimstu](https://boards.straightdope.com/u/Kimstu)\ **Post date:** [June 1, 2018, 3:35pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/2 "2018-06-01T15:35:35Z")
The absence of negative numbers in most areas of classical number theory is largely a historical artifact of its development in ancient Greek mathematics: number-theoretic results on the natural numbers were studied long before negative numbers came into standard use. And since most elementary number-theoretic results can be trivially extended to the negative integers, as in your example, mathematicians don’t bother with them much. If you found out something about the properties of negative integers that is **not** already covered by a simple reformulation of corresponding descriptions of the natural numbers, that would definitely get mathematicians’ attention. But don’t hold your breath. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [June 1, 2018, 3:39pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/3 "2018-06-01T15:39:59Z")
The length of sides and areas of ordinary shapes like triangles are always positive. No real point in noting the negative of a very well known sequence. Even the amazing The On-Line Encyclopedia of Integer Sequences doesn’t carry it. [Plugging](https://oeis.org/search?q=0%2C+-1%2C+-3%2C+-6%2C+-10+&sort=&language=english&go=Search) in the first few terms gives a sequence that starts off right but goes off in a different direction and then the regular triangle numbers. If OEIS doesn’t specifically ID this, it isn’t worth studying on its own. Just make the numbers positive and go to town on generating identities. ---
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [June 1, 2018, 5:05pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/4 "2018-06-01T17:05:53Z")
> [@](#): > > -1^.5 = 2((-1/2)-0) = -1 Huh? -1^.5 = i where i^2 = -1 ---
**Author:** ![Crane](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crane/32/3495_2.png) [@Crane](https://boards.straightdope.com/u/Crane)\ **Post date:** [June 1, 2018, 7:30pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/5 "2018-06-01T19:30:00Z")
The algorithm doesn’t require ‘i’ ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [June 1, 2018, 7:43pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/6 "2018-06-01T19:43:16Z")
I gotta say “Huh?” as well. You show a formula that produces the wrong answer for sqrt(-1) and then you say it “doesn’t require” the correct answer. I have no idea what you mean by that. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 1, 2018, 8:59pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/7 "2018-06-01T20:59:48Z")
One could also argue that if T(0) = 0, T(1) = 1, T(2) = 3, T(n) = 0 + 1 + … + n, then when you sum _from_ zero _to_ a negative index you pick up a minus sign, so T(-1) = 0, T(-2) = 1, etc. (T(n) = (n^2 + n)/2 for all n in this case), and the “negative triangular numbers” are the same as the positive triangular numbers after a change of index. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [June 1, 2018, 10:48pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/8 "2018-06-01T22:48:39Z")
> [@Crane](#): > > This plots as a square in quadrant 3 with sides of -5 and an area of -25. The signed area of a square with sides -5 is 25, not -25. ---
**Author:** ![Crane](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crane/32/3495_2.png) [@Crane](https://boards.straightdope.com/u/Crane)\ **Post date:** [June 2, 2018, 3:25am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/9 "2018-06-02T03:25:31Z")
Why not? In quadrant 3 there is a square with an altitude of -5 and a base of -5. Is the nature of it’s area identical to a 5 by 5 square in quadrant 1? Using the algorithm above: -25^.5 = 2\*((-25/2)-(-10) = -5 (I am aware that this contradicts elementary math) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 2, 2018, 7:17am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/10 "2018-06-02T07:17:13Z")
False arithmetic is… false. Note that the square of an integer number is never negative. Other than that, your identity seems to amount to n = 2(n^2/2 - (n^2 - n)/2) or n = 2(-n^2/2 + (n^2 + n)/2). ---
**Author:** ![DavidwithanR](https://avatars.discourse-cdn.com/v4/letter/d/3bc359/32.png) [@DavidwithanR](https://boards.straightdope.com/u/DavidwithanR)\ **Post date:** [June 2, 2018, 7:41am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/11 "2018-06-02T07:41:04Z")
I don’t know much math, but I’m getting the impression that what happened here is a demonstration of the difference between ‘i’ (imaginary numbers) and what I’ll call ‘f’ (fake numbers). 🙂 ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 2, 2018, 7:52am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/12 "2018-06-02T07:52:24Z")
One thing we do see in all this is that under some circumstances it makes sense to define the sum of f(k) as k runs _from_ 0 _to_ n, where n \< 0, as -f(-1) - f(-2) - … -f(-n+1). ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [June 2, 2018, 7:52am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/13 "2018-06-02T07:52:31Z")
> [@md2000](#): > > Huh? -1^.5 = i where i^2 = -1 If you are expecting square roots with what you have in the first post that is why you are having problems finding information. square sqrt(-5) == √5\*ⅈ The reason you aren’t finding anything is because you are trying to avoid _imaginary_ numbers. Just replace the term _imaginary_ with _complex_ in your mind and you will find a lot. There is nothing _imaginary_ about i, and it is used everywhere. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [June 2, 2018, 8:15am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/14 "2018-06-02T08:15:27Z")
> [@Crane](#): > > Why not? In quadrant 3 there is a square with an altitude of -5 and a base of -5. Is the nature of it’s area identical to a 5 by 5 square in quadrant 1? _Nature_ of it’s area? That’s a philosophical discussion of some kind. Starting with a positive-area square in quadrant 1, we can flip it across the X axis to put it in quadrant 4. It now has a negative height and positive width, and so a negative area. Or, we can just consider that we’ve taken the mirror-image of the shape and that it has negative area for that reason. We can then flip across the Y axis to give it a negative width. Negative width times negative height makes positive area. Alternatively, the negative-area shape mirrored goes back to being positive-area. Both ways of looking at it are consistent. ---
**Author:** ![Isilder](https://avatars.discourse-cdn.com/v4/letter/i/cc9497/32.png) [@Isilder](https://boards.straightdope.com/u/Isilder)\ **Post date:** [June 2, 2018, 8:26am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/15 "2018-06-02T08:26:28Z")
> [@Crane](#): > > Why not? In quadrant 3 there is a square with an altitude of -5 and a base of -5. Is the nature of it’s area identical to a 5 by 5 square in quadrant 1? > > Using the algorithm above: > > -25^.5 = 2\*((-25/2)-(-10) = -5 > > (I am aware that this contradicts elementary math) No it doesn’t . It just trivially just restores the negative to the front after taking the square root of the absolute. What if I square root your bank balance ? If its in the +ve thats fine. If its negative, what , do you suddenly turn a debt (you owing money) into +ive balance, you being owed money ? Its already been said, the -ve version of your triangular series is trivial. And its trivial to have "if ( x \< 0 ) sqrt\_x= -1 \* sqrt(-1 \* x ) else sqrt\_x = sqrt(x) " Its overcomplicating it to say he needs to use complex numbers. Its still going to require an arbitrary correction for the negative case. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [June 2, 2018, 8:44am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/16 "2018-06-02T08:44:32Z")
Also I should point out, be careful about floating point on computers. Floating point numbers in computers are finite. Being finite numbers a large number of postulates like they are not associative nor distributive. These rules do not necessarily hold in computer floating point arithmetic. Associative: A\*(B_C) = (A_B)\*C Distributive: (A+B)_C = A_C + B_C -25\*\*(1/2) == -5 only in floating point. But that .5 in the -25^.5 you have above is really significand \* base ^ exponant or 5_10\*\*-1 You really need to use either complex numbers, or a symbolic math function to mess around with these types of numbers on a computer. -25^.5 == sqrt(-25) ==5⋅ⅈ -1^.5 == sqrt(-1) == ⅈ That last one is the identity of ⅈ and this all just works in the complex plane. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [June 2, 2018, 8:58am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/17 "2018-06-02T08:58:23Z")
> [@Isilder](#): > > No it doesn’t . It just trivially just restores the negative to the front after taking the square root of the absolute. > What if I square root your bank balance ? If its in the +ve thats fine. > If its negative, what , do you suddenly turn a debt (you owing money) into +ive balance, you being owed money ? > > Its already been said, the -ve version of your triangular series is trivial. And its trivial to have "if ( x \< 0 ) sqrt\_x= -1 \* sqrt(-1 \* x ) else sqrt\_x = sqrt(x) " > Its overcomplicating it to say he needs to use complex numbers. Its still going to require an arbitrary correction for the negative case. Your bank account balance is hopefully only one dimensional as far as numbers go. But lets say banks do use square roots and we take the absolute. -25^1/2 - 25^1/2 == sqrt(-25) - sqrt(25) == -5 + 5⋅ⅈ |-25|^1/2 - |25|^2 == sqrt(|-25|)-sqrt(|25|) == 0 That is a big change in [displacement](https://www.mathsisfun.com/definitions/displacement-distance-.html) which is the primary concern for both banks and consumers when it comes to money! ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [June 2, 2018, 9:07am UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/18 "2018-06-02T09:07:51Z")
Here is a quick plot of -5 + 5⋅ⅈ to show just how far from zero it is. [https://s33.postimg.cc/py90iwszj/complex.png](https://s33.postimg.cc/py90iwszj/complex.png) ---
**Author:** ![Johanna](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johanna/32/8318_2.png) [@Johanna](https://boards.straightdope.com/u/Johanna)\ **Post date:** [June 2, 2018, 12:08pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/19 "2018-06-02T12:08:17Z")
> [@md2000](#): > > Huh? -1^.5 = i where i^2 = -1 > [@rat\_avatar](#): > > If you are expecting square roots with what you have in the first post that is why you are having problems finding information. > > square sqrt(-5) == √5\*ⅈ > The reason you aren’t finding anything is because you are trying to avoid _imaginary_ numbers. Just replace the term _imaginary_ with _complex_ in your mind and you will find a lot. > > There is nothing _imaginary_ about i, and it is used everywhere. These two give the Straight Dope, listen to them. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [June 2, 2018, 1:34pm UTC](https://boards.straightdope.com/t/why-no-interest-in-negative-triangular-numbers/815262/20 "2018-06-02T13:34:07Z")
> [@rat\_avatar](#): > > -25\*\*(1/2) == -5 only in floating point. Floating point or not, many calculators and computers are going to evaluate -25\*\*(1/2) or -25^(1/2) or -25^.5 as -5, simply because the order of operations they’re programmed to follow specifies that exponents are evaluated before the -.