Talk:Triangular number

Easier proof

Is there a reason why there isn't an easier proof on the page? I mean, it can be solved by anyone who saw $\sum _{x=1}^{n}x=n(n+1)/2$ in school; by induction.

Extended content

Obviously 1³ = 1². if applies for $n$, then it applies for $(n+1)$ as well; so $(\sum _{x=1}^{n+1}x)^{2}=\sum _{x=1}^{n+1}x^{3}$ seperate $n+1$ $((\sum _{x=1}^{n}x)+(n+1))^{2}=(\sum _{x=1}^{n}x^{3})+(n+1)^{3}$ considering we assume it applies for $n$, we can substitute $\sum _{x=1}^{n}x^{3}=(\sum _{x=1}^{n}x)^{2}$ $((\sum _{x=1}^{n}x)+(n+1))^{2}=(\sum _{x=1}^{n}x)^{2}+(n+1)^{3}$ solving the quadratic equation $(\sum _{x=1}^{n}x)^{2}+2(\sum _{x=1}^{n}x)(n+1)+(n+1)^{2}=(\sum _{x=1}^{n}x)^{2}+(n+1)^{3}$ remove $(\sum _{x=1}^{n}x)^{2}$ from both sides $2(\sum _{x=1}^{n}x)(n+1)+(n+1)^{2}=(n+1)^{3}$ considering $(\sum _{x=1}^{n}x)=n(n+1)/2$ $2n(n+1)(n+1)/2+(n+1)^{2}=(n+1)^{3}$ as $2*a*b*b/2=a*b^{2}$ $n(n+1)^{2}+(n+1)^{2}=(n+1)^{3}$ next, we single out $(n+1)^{2}$ as $a*b^{2}+b^{2}=b^{2}(a+1)$ $(n+1)^{2}(n+1)=(n+1)^{3}$ which proves the theorem

Qube0 (talk) 17:18, 5 January 2020 (UTC)

@Qube0: Easier than what? Other than a graphical illustration, there's no proof in this article at all, which is perfectly reasonable. Inductive proofs of identities like this aren't generally very enlightening and just tend to clutter up the article. On a side note, your proof is full of errors – most seriously, starting by writing what you're trying to ultimately prove, rather than winding up with that as the final result. –Deacon Vorbis (carbon • videos) 17:44, 5 January 2020 (UTC)

I did find the existing proof by induction here rather hard to follow, so I tried to make it easier to understand, based on a short proof in Spivak's lovely Calculus. He presents his proof as the first "real" proof by induction in the book, and in the context of introducing the reader to induction in general, so I think this particular article is actually a great place to give a proof by induction. I wrote it out in an especially explicit style, such that someone who's never used induction could hopefully follow it, since I do think it makes a great introduction to induction as a proof strategy. Mesocarp (talk) 00:53, 27 October 2022 (UTC)

non-integer triangular numbers

Since the ${\frac {n(n+1)}{2}}$ formula can be evaluated for all real numbers, both positive and negative, integer or non-integer, is there any special term for the numbers that lie along the resulting graph? x^2 evaluates to the same value regardless of sign, so the resulting curve is symmetrical across the axis of x=-1/2. But Im more interested in the fractional terms and if there is anything special about them. e.g. the π^th "triangular number" is approximately 6.50559852734. If there's anything to this, it certainly would be worth mentioning, but I couldnt find anything about "generalized triangular numbers" anywhere. Thank you, —Soap— 03:42, 7 April 2020 (UTC)

Proposed merge from Termial

Termials and triangular numbers are the same thing. Since Wikipedia articles are about concepts not names (WP:NOTDICT), they should have together only one article. Therefore, I am proposing a merge. I think that termial is also a rather obscure term for what is much more widely known as a triangular number, so the merge should go from termial to triangular number. Any other opinions? —David Eppstein (talk) 19:26, 20 August 2020 (UTC)

Support I wouldn't say they are the same thing. One is a sequence while the other a series, and termial can apparently apply to non-integers. However, they are certainly intertwined, and Termial might be better as a section of the Triangular Number article. The term "termial" doesn't seem to appear in much mathematical literature though, so I question its notability in the first place. Kstern (talk) 20:54, 3 September 2020 (UTC)

Support They're the same thing! If someone wants to know more about termials, and they read the article about triangular numbers, that will serve their purpose just fine. — Preceding unsigned comment added by 73.241.189.0 (talk) 17:00, 24 September 2020 (UTC)

• Support it doesn't make much difference whether the concept is presented as a sequence or a series, the two are clearly functionally equivalent and this article does already note that triangular numbers can be represented as a series. Triangular number is definitely the primary usage and I suggest we just note somewhere that they are sometimes known as termials (The Art of Computer Programming is very influential). Termial doesn't actually specify how it can be calculated for non-integers. Hut 8.5 19:11, 24 September 2020 (UTC)
• This MUST be merged with triangular numbers. It has no independent conceptual difference, and much less content, and it's only used by Knuth, while triangular number has been used since the pythagoreans at least

Termial means exactly the same as triangular number. The idea is to simplify knowledge, not to create for each synonim a different new article. And this is obviously a term only used by no one.

Santropedro (talk) 03:13, 4 January 2021 (UTC)

Seeing support and no opposition, I have gone ahead and done the merge. —David Eppstein (talk) 05:08, 4 January 2021 (UTC)

Rotations in n-dimensional geometry

The amount of rotations in any n-dimensional geometry is the triangular number of n-1. Unfortunately, I haven't found anything on this on the internet so it would count as OR. Is there any way to get around this?

You can play around with it yourself if you want: https://www.mathgoodies.com/calculators/triangular-numbers

75.129.231.4 (talk) 23:19, 28 October 2020 (UTC)

What exactly do you mean by "amount of rotations?" Do you mean possible axes of rotation? In three dimensions an infinite number of rotational axes exist.—Anita5192 (talk) 00:00, 29 October 2020 (UTC)

Edward Waring

While looking at Edward Waring's Meditationes Algebraicae (3rd ed, 1782) for another purpose, I noticed that he states (in Latin, of course), without proof, that every whole number is the sum of 1, 2 or 3 triangular numbers. (This doesn't exclude the possibility that it is also the sum of 4, 5, 6... etc such numbers). (op. cit. p 349 prop. 3.) Waring goes on to state the corresponding propositions for pentagonal and hexagonal numbers (prop. 4) and squares (prop. 5). (This is on the same page as the proposition often known as Waring's Theorem.) I was wondering if this was already known before Waring. Looking at the article here, I see that it was later rediscovered by Gauss, who was excited enough to call it his Eureka theorem. However, I see that it is a special case of a conjecture of Fermat, known as his Polygonal Number Theorem, but characteristically stated without proof. Quite possibly Waring found it in Fermat. As a conjecture, without proof, it is therefore certainly earlier than Gauss, though Gauss may have been the first to prove it for the triangular case.2A00:23C8:7907:4B01:6415:B77:5411:4DE (talk) 14:06, 17 June 2022 (UTC)