# Longest equation

Im just wandering what is the longest equation in mathematics?

I could make any of those up right now.

the Taylor expansion for sin(x) for example is infinitly long.

Or perhaps &Sigma; 1/(1-r) from term 1 to infinity with r less than 1. Basically any equation involving an infinite number of terms can be called longest.

Yes there is no particular difficulty in having long equations, and there should be no prestige in long equations. For example fermats last theorem was a short theorem, but very very difficult to solve. It is easy to get infinitely many terms in an equation as demonstrated above.

thats not what i meant

An associated question might be:

What is the longest complete formula that is of true interest to the math sciences, and not just an expansion of something shorter.

thanks quarto

Clearly it is easily possible to generate an equation of any length.

Define 'interest' in this situation.

Integral
Staff Emeritus
Gold Member
Originally posted by quartodeciman
An associated question might be:

What is the longest complete formula that is of true interest to the math sciences, and not just an expansion of something shorter.

This statement is a bit misleading, acutally many "equations" are truncations of the infinite series. It would make no sense to create an infinite series of something which did not start that way. The common practise is to make a small angle, or similar, approximaition by ignoring all non linear terms of an infinite series. One simply does not arbitaraly ADD non linear terms as this question would seem to imply.

Please permit me to contest your point. First order equations are fine if you want to express a simplified case of something, but may be insufficient for a more detailed explanation. I intended for a closed formula to include terms of real interest, and to avoid arbitrary extensions of series'. Such series' terms express higher order moments and the like.

A real problem with this type of question is the fact that more and more special function names get added to the approved list, which permits shortened expression. It was common to use truncated series throughout the eighteenth century for things that were subsequently given designations and symbolic names, like Bessel, Legendre, etc. in the nineteenth century. The best case of an otherwise-longish formula is Einstein's Gik-Tik gravitational field equation. Write it out with derivatives and products of the metric tensor terms (gik), along with the determinate g, and the formula gets rather long, not to mention boringly repetitious.

Short equations are more elegant and usually have more meaning to science.

Integral
Staff Emeritus
Gold Member
Originally posted by quartodeciman
Please permit me to contest your point. First order equations are fine if you want to express a simplified case of something, but may be insufficient for a more detailed explanation. I intended for a closed formula to include terms of real interest, and to avoid arbitrary extensions of series'. Such series' terms express higher order moments and the like.

A real problem with this type of question is the fact that more and more special function names get added to the approved list, which permits shortened expression. It was common to use truncated series throughout the eighteenth century for things that were subsequently given designations and symbolic names, like Bessel, Legendre, etc. in the nineteenth century. The best case of an otherwise-longish formula is Einstein's Gik-Tik gravitational field equation. Write it out with derivatives and products of the metric tensor terms (gik), along with the determinate g, and the formula gets rather long, not to mention boringly repetitious.

What point are you contesting? The fact that many common equations are linear forms of nonliner solutions?

Integral,

What point are you contesting?
'This statement is a bit misleading...' and 'arbitaraly ADD non linear terms...this question would seem to imply'

"An associated question might be:

What is the longest complete formula that is of true interest to the math sciences, and not just an expansion of something shorter."

---

plus,

Define 'interest' in this situation.
I can't. It's of interest because it's of interest to the inventors of those formulae.

---

einsteinian77,

I haven't answered even my revised version of your question. I have just suggested that it might be interesting. We all got derailed onto discussions of series' terms, which I didn't want to bother with anyway. Tell me, please, what will you do with the world's longest formula-that-isn't-just-some-expansions-of-functions-by-series'-terms, once you figure out what it might be?

ahrkron
Staff Emeritus
Gold Member
Tell me, please, what will you do with the world's longest formula-that-isn't-just-some-expansions-of-functions-by-series'-terms, once you figure out what it might be?

I would guess einsteinian77 asked just out of sheer curiosity, which is probably the best motivation you could ask for in these forums.

In terms of answering the question (or quartodeciman's revised version of it), I seem to remember that the Wheeler-DeWitt equation is an infinite-dimensional system of functional partial differential operator equations... or something of that sort. Maybe Hurkyl, Marcus or Lethe can tell us something clearer about it.

Also, do you know about Feynman diagrams? each diagram corresponds to one term in an infinite expansion. Some theorists spend quite a bit of effort understanding the relative importance of contributions from each order. For some calculations, they need to compute the values of tens of thousands of terms in these expansions (they don't go all the way to infinity because they can't, but they try to learn as much as possible about the finite subsets of terms that correspond to third or fourth order).

Originally posted by quartodeciman

Define 'interest' in this situation.
I can't. It's of interest because it's of interest to the inventors of those formulae.

At which point we find your stumbling block.

As Akhron said: keep up the curiosity.

Ahrkron said it best, im asking out of sheer curiosity. As i said before, thats not what i meant by longest equations when Brad brought up the infinite series and expansion stuff.

da capo one says in music: go back to the head. I can't offer any examples of truly huge equations of significance, where each term tells its own part of the story. So I will quit the topic. I think I have seen some truly awesome hydrodynamics equations (with all the stops pulled out), but I don't remember the detail and have not been lucky with searches.

a parting story:

Back in the mid sixties, I met some friends for late afternoon brew consumption. One was going to the college physics colloquium beforehand, so I decided to sit in too. The guest speaker was a young postdoc. He talked about renormalization stuff. He wrote a long piece of power series terms on the blackboard. At the end of his speech one of the older physics professors asked a question.
The exchange went like this.

prof: What is the particular intuitive meaning of the first two terms?

speaker: Well, er, they really don't mean anything.

:)

Originally posted by einsteinian77
Im just wandering what is the longest equation in mathematics?

OK - Here it is :

1

+

1

=

2