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I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.

My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.

I came with some ideas like the Turing machine but I think I'm not grasping the right examples.

Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'?

Edit: As some people pointed out that I've published this on MSE I present a code here to find some candidates as Lychrel numbers.

def reverseNum(n):
    st = str(n)
    return int("".join([st[i] for i in xrange(len(st)-1,-1,-1)]))

def isPalindrome (n):
    st = str(n)
    rev = str(reverseNum(st))
    return st==rev

def isLychrel (n, num_interations):
    p = n
    for i in xrange(num_interations):
            if isPalindrome(p):
                    return i
            p = p + reverseNum(p)
    return -1

for i in xrange(1000):
    p = isLychrel(i,100)
    if (p < 0):
            print i,p
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What about math that was once useful but now useless? Like all of the tricks engineers had to use to multiply using slide rules... – Brian Rushton Dec 17 '12 at 19:12
This sort of appendix seems contrary to the nature of mathematics. The argument isn't countered by providing a list of other ideas that people might have said were useless. Instead, why not focus on the education aspects? According to the Wikipedia article, the search has led a few computer programmers into what is ostensibly number theory, and may have introduced many young people to a fundamental idea behind proofs - even if you haven't found a palindrome by $10^9, there might still be one. Sounds a lot like Skewes' number, also probably called useless. – Zack Wolske Dec 17 '12 at 19:13
I think that "usefulness" is probably not the correct measure for mathematics. Other properties, such as beautiful results, the occurrence of complicated structure, or the use of unexpected techniques are also good ways of judging math. – André Henriques Dec 17 '12 at 21:39
It is interesting to look for things that turned out to be much more useful than initially thought, but I think you ought to look for reasons that you know for studying Lychrel numbers instead of hoping that more will come in the future. It seems to me like the primary motivation is that this is a simple question that seems like it should be easy to answer, but apparently isn't, so by searching for the answer, we may come to understand the integers better. – Miles Dec 18 '12 at 7:05
I agree with Zack. If you don't see a way of arguing directly for the usefulness of Lychrel numbers research (arguing for the usefulness of other allegedly useless results in mathematics is really no argument at all for the case in question), then don't go into it. Focus on other types of payoff. – Todd Trimble Jan 15 '13 at 13:18

13 Answers 13

up vote 27 down vote accepted

Number theory, in particular investigations related to prime numbers, was famously considered as useless (cf Hardy), for practical matters. Now, since "everybody" needs some cryptography it is quite useful to know how to generate primes (eg, for an RSA key) and alike, sometimes involving prior 'useless' number theory results.

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This answer being given, let me add that I am not convinced your idea regarding the article is a good one. – user9072 Dec 17 '12 at 18:20
Number theory may have been regarded as "useless" in this sense, but was it ever considered to be as useless as Lychrel numbers? – Franz Lemmermeyer Dec 17 '12 at 18:23
It is also true that in a perfect world --say, a world ruled by mathematicians, cryptography itself would be useless: why to hide things? – Pietro Majer Dec 17 '12 at 19:08
Pet peeve: "cf" stands for "conferre", which means "to compare"; you are using it as reference or a "see for example". Though an extremely common usage, it is incorrect. "cf" should be used for "compare with", and you don't want to compare the writings of Hardy with the statement that Number Theory was considered useless; rather, you want to use Hardy's writings as a reference to the assertion that Number Theory was considered useless... – Arturo Magidin Dec 17 '12 at 22:20
@Arturo Magidin: yes, the usage is a bit odd here; also but perhaps not only as I changed the phrase after the reference was typed. I might sometimes use it strangely or even in a wrong way, but I can assure you I do know the meaning. Incidentally, since we are discussing such matters, it seems to me it actually does not stand for (the infinitive) 'conferre.' :) – user9072 Dec 17 '12 at 23:54

The Radon transform, when introduced by Johann Radon in 1917, was useless, until Cormack and Hounsfield developed Tomography in the 60's (Nobel prize for medicine 1979).

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The most famous example is conic sections. Conic sections were of great interest to Greek mathematicians, and their theory was highly developed in the 2-nd century BC.

However I don't know of any application until Kepler's discovery that the celestial bodies move on conic sections. Thus 18 centuries passed between math research and the first application!

EDIT. There is a conjecture discussed in the paper The Astronomical Origin of the Theory of Conic Sections by O. Neugebauer, Proc. Amer. Phil. Soc., Vol. 92, No. 3

that conic sections appeared for the first time in the theory of sundials. But this is only a conjecture, and Apollonius does not mention sundials. Thanks to user Miles who brought this fact to my attention.

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This actually seems to be a non-example. Conic sections were apparently first studied by Menaechmus in the 4th centure BCE. We're not sure what his motivation was, but he definitely used them in his method of doubling the cube. Some speculate that this problem led him to discover conics; others suggest that he was prompted by the fact that the tip of a sundial traces a hyperbola on any given day (outside the Arctic circles, anyway). In any case, it looks like conic sections had applications as soon as people knew about them. – Miles Dec 18 '12 at 6:57
2 (cites Boyer's and Cooke's history of math texts) W.W. Dolan: Early Sundials and the Discovery of the Conic Sections, Mathematics Magazine, 1972. 45(1): p. 8-12. – Miles Dec 18 '12 at 16:02
Sorry about the formatting there---I tried to make it nice . . . – Miles Dec 18 '12 at 16:08
Conic sections were apparently used by the Greeks (possibly Archimedes) in real life: – YangMills Jan 16 '13 at 0:53
@YangMills: if you mean the story of Archimedes burning the ships, this is a later legend. Both theory and experiments show that this is impossible. – Alexandre Eremenko Sep 14 '14 at 21:47

Divergent series, anyone?

It was devil's work, just a curiosity, unorthodox idea for Euler and a strange concept for Abel, Ramanujan (Abel claiming that it can't and mustn't be used for serious calculations)... but today, we use it for "real" things.

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Useless concept for Euler? Is my sarcasm detector broken? – Franz Lemmermeyer Dec 17 '12 at 18:21
Done. Interesting enough, our both answers are related to Hardy, champion of "mathematics without practical use" (or at least what he hoped to be without practical use). – Harun Šiljak Dec 17 '12 at 18:22
Previous comment goes for quid's (now deleted) comment. @Franz: strange goes for Euler, useless goes for Abel and Ramanujan, I should have phrased it better. – Harun Šiljak Dec 17 '12 at 18:23
@Harun: that's bettet. Even if the idea was very orthodox for Euler, who defended using divergent series at each and every opportunity. It wasn't orthodox for the Bernoullis, however. – Franz Lemmermeyer Dec 17 '12 at 18:29
The more I think about it, the more I think that divergent series are rather useful math that became useless than the other way round. Nevertheless I like your answer better than the question. – Franz Lemmermeyer Dec 17 '12 at 21:17

Fast Fourier transform: Originally developed by Gauss in early 19th century. Gauss thought it is unworthy of publication, because there were better computational techniques. It only appeared in his collected works after his death, where nobody noticed it. Rediscovered by Cooley and Tukey, and was instantly recognized as important. See e.g.

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Negative numbers and complex numbers were regarded as absurd and useless by many mathematicians prior to $15^{th}$ century. For instance, Chuquet referred negative numbers as "absurd numbers." Michael Stifel has a chapter on negative numbers in his book "Arithmetica integra" titled "numeri absurdi". And so too were complex/imaginary numbers. Gerolamo Cardano in his book "Ars Magna" calls the square root of negative numbers as a completely useless object.

I guess the same attitude towards Quaternions and Octonions would have been prevalent, when they were initially discovered.

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Quaternions were designed as a way of representing positions in space (like complex numbers do in the plane). They were superseded by vectors, only to find use recently for efficient computation in graphics. – vonbrand Feb 19 '14 at 11:18
Indeed, also Maxwell used the language of quaternions for electrodynamics. I think, that quaternions were deemed quite an essential topic until the more general vector calculus appeared. – Lennart Meier Apr 20 at 12:15

Perhaps homology and the simplicial methods of algebraic topology (and algebraic geometry) which are just now finding applications in topological data analysis. I think algebraic geometry and topology have often been thought to be the pinnacle of math for math's sake, but their machinery is finding uses in very modern technology.

Also, the theory of graphs and its applications to networks and network and systems analysis.

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Investigations on the independence of Euclid's 5th axiom: about 2000 years of fruitless research, until Bolyai and Lobachevski resolved the issue (cf e.g. and Gauss also raised his hand, but still not useful for practical problems, until Einstein developed a non-euclidean explanation of the universe.

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The answer to this question depends on what one means by "useful". We have to distinguish between "useful" in other branches of mathematics or useful in Theoretical Physics. Being useful in actually describing the laws of the Universe can be understood as being several orders of magnitude more important than just being useful in other areas of mathematics. For example, Generalized Complex Geometry has been extensively used in String Theory, and as a result Gualtieri's PhD. Thesis is close to have 900 citations in ten years. I think this would have never happened if Generalized Complex Geometry was only interesting in Mathematics. By the way, is this the more cited paper in mathematics?

Dealing with applications of Mathematics to Physics, there are many examples of mathematical theories that in the beginning seemed useless to Physics and years later found its place in it. To name a few:

  • Lie groups: I once heard that when Sophus Lie introduced Lie groups, he said that finally mathematicians had created something that would never be used by physicists. Modern Physics uses Lie groups at so many levels that I cannot even begin to mention. Let me just say that the Standard Model that describes all the known the fundamental interactionx and particles is based on the Lie group $SU(3)\times SU(2)\times U(1)$.

  • Holonomy theory of the Levi-Civita of a Riemannian manifold. In principle (in the 50's), this was unrelated to physics. However, in the 80's it found its realization in physics through String Theory: the simplest compactification manifolds are six, seven, and eight-dimensional Riemannian manifolds of special holonomy. In addition, the scalar manifold associated to the non-linear sigma model of the effective action is again of special holonomy (tipically Kahler, Hyper-Kahler and Quaternionic-Kahler), which encodes the corresponding moduli space.

  • Kodaira's classification of singularities in elliptic fibrations. This is really a shocking example from the 50's. In principle completely unrelated to physics, it found its realization again through String Theory, and in particular through F-theory, which requires a singular elliptically fibered Calabi-Yau manifold as a compactification space space.

  • Gerbes. A gerbe is a particularly abstract construction introduced by Jean Giraud in the 70's. Surprisingly, it has recently found its realization in Physics through again String Theory: many Supergravity solutions, like the self-dual string, are in fact particular instances of gerbes.

There are more examples. By the way, there also various examples of the opposite: mathematical theories first found by physicists and then formalized by mathematicians.

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Fourier transforms were not useful in the 19th century, save as a tool for solving differential equations or for obtaining other theoretic results. Today of course they are ubiquitous.

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I find this answer quite surprising. In particular a quotation of Jacobi comes to mind 'M. Fourier avait l'opinion que le but principal des mathématiques était l'utilité publique et l'explication des phénomènes naturels ; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c’est l'honneur de l'esprit humain, et que sous ce titre, une question de nombres vaut autant qu'une question du système du monde.' – user9072 Jan 15 '13 at 16:31
The first part of which roughly translates to: "Mr. Fourier held the opinion that the principal goal of mathematics was the public good and the explication of natural phenomena;" Thus, and in general, I am unconvinced that it ever was mainstream opinion that this type of mathematics was 'useless'. – user9072 Jan 15 '13 at 16:35
"save as a tool for solving differential equations" - some might say this is was a pretty important and heavy use! – Yemon Choi Jan 15 '13 at 19:01

Real numbers: Kroneckers "God created the integers, everything else is man made" is a prototypical continuation of the way the Pythagoreans wished the world to be, namely that everything can be measured in integers or their ratios.

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Binary numbers: discovered by Leibniz and of no use until the advent of computers.

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Um, Leibniz was trying to invent computers. See, for example, -- I think he didn't use binary under the hood because base change is "computationally expensive". – Sam Nead Jan 3 '14 at 11:29

The theory of finite fields, introduced by Galois (but as often, using former ideas) was considered useless and as a mathematical curiosity for almost 2 centuries, until they found a use in error-correcting codes for telecommunications.

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I think I must have missed something here. Obviously finite fields were useful to mathematicians far earlier than the 1940's (class field theory, curves and varieties over finite fields, the Frobenius, etc.). Since when did this question become a forum on practical applications in the usual sense? – Todd Trimble Jan 4 '14 at 18:39

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