# Useless math that became useful

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

• What about math that was once useful but now useless? Like all of the tricks engineers had to use to multiply using slide rules... Dec 17, 2012 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. Dec 17, 2012 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. Dec 17, 2012 at 21:39 • 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. Jan 15, 2013 at 13:18 • Of course we're more likely to hear about a bit of math that used to be useless if it is now useful, so there's a selection effect here: a cherrypicked list of candidates doesn't tell you much about the actual conditional probability that something will end up being useful given that it currently looks useless. Jan 4, 2014 at 21:01 ## 15 Answers Number theory, in particular investigations related to prime numbers, was famously considered useless (e.g., by Hardy) for practical matters. Now, since "everybody" needs some cryptography it is quite useful to know how to generate primes (e.g., for an RSA key) and alike, sometimes involving prior 'useless' number theory results. • This answer being given, let me add that I am not convinced your idea regarding the article is a good one. – user9072 Dec 17, 2012 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? Dec 17, 2012 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? Dec 17, 2012 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... Dec 17, 2012 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, 2012 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). • Fantastic answer, I used Radon transforms in CV quite recently Feb 4, 2019 at 3:35 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 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, jstor, reprint - doi: 10.1007/978-1-4612-5559-8_21 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. EDIT 2. However most histories of Greek mathematics say that conic sections were invented/discovered by Menaechmus, as a tool for doubling the cube, which is of course a useless problem from our modern point of view. EDIT 3. Parabolic mirrors is not a real application. Of course, this is a nice property of parabola, but conic sections have many other nice properties. The legend of Archimedes burning ships with them is a legend, nothing more. This is impossible, even with modern technology. And MAKING a parabolic mirror is another great technological challenge, absolutely out of reach for the ancients. Most reflecting telescopes were made with spherical mirrors, for exactly this reason: nobody knew how to make a parabolic one. To be sue, Diocles wrote a book On burning mirrors in 3d or 2nd century BC (the book did not survive), but this was pure mathematics. There was no real applications of parabolic mirrors in antiquity because they did not now how to make them. • 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. Dec 18, 2012 at 6:57 • en.wikipedia.org/wiki/Menaechmus (cites Boyer's and Cooke's history of math texts) www-history.mcs.st-andrews.ac.uk/Biographies/Menaechmus.html www-history.mcs.st-and.ac.uk/HistTopics/Sundials.html W.W. Dolan: Early Sundials and the Discovery of the Conic Sections, Mathematics Magazine, 1972. 45(1): p. 8-12. Dec 18, 2012 at 16:02 • Sorry about the formatting there---I tried to make it nice . . . Dec 18, 2012 at 16:08 • Conic sections were apparently used by the Greeks (possibly Archimedes) in real life: en.wikipedia.org/wiki/Parabolic_reflector Jan 16, 2013 at 0:53 • Doubling the cube is an absolutely useless problem? I guess you don't care whether the citizens of Delos are able to defeat the plague sent by Apollo! en.wikipedia.org/wiki/Doubling_the_cube#History Aug 24, 2017 at 22:40 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. http://en.wikipedia.org/wiki/Non-Euclidean_geometry) and Gauss also raised his hand, but still not useful for practical problems, until Einstein developed a non-euclidean explanation of the universe. 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. http://www.math.ethz.ch/education/bachelor/seminars/fs2008/nas/woerner.pdf • FFT was inented by Gauss as a computational tool in an applied problem (in astronomy). So it was certainly useful (though perhaps too shallow for Gauss to pubish). Aug 25, 2017 at 0:08 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. • 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. Feb 19, 2014 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. Apr 20, 2016 at 12:15 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. • Useless concept for Euler? Is my sarcasm detector broken? Dec 17, 2012 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). Dec 17, 2012 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. Dec 17, 2012 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. Dec 17, 2012 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. Dec 17, 2012 at 21:17 The answer to this question depends on what one means by "useful". We have to distinguish between "useful" for other branches of mathematics or useful for Theoretical Physics or even Physics in general. 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 extensively been used in String Theory and Supergravity (for example in the classification of certain flux compactification backgrounds), 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. On the other hand, essentially every piece of "reasonable" Mathematics finds its place and application in Physics. Dealing with Mathematics' applications to Physics, there are many examples of mathematical theories that in the beginning seemed useless to Physics and only years later were found to have remarkable applications in physics. 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 describe it. 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 Riemannian manifolds. To the best of my knowledge, when this theory was developed in the 50's it was completely 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 of certain String compactifications is again of special holonomy (tipically Kahler, Hyper-Kahler and Quaternionic-Kahler). This manifold encodes, in a way which is not completely understood, the local moduli space of the corresponding compactification.

• Kodaira's classification of singularities in elliptic fibrations. This is really a shocking example, again 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 an elliptically fibered Calabi-Yau manifold as a compactification space space. The singularities of the fibration crucially inform on the matter content of the theory.

• 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 (who else): many Supergravity solutions, like the self-dual string, are in fact particular instances of gerbes.

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

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.

• People are also looking at applications of knot theory to protein folding. Knot theory belongs to algebraic and geometric (low-dimensional) topology. Aug 3, 2016 at 19:37
• I have to disagree with this. Algebraic topology has long had a history intertwined with physics. Simplicial homology tells you plenty about de Rham cohomology and to quote Tao "The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds." Aug 24, 2017 at 15:38
• Graph theory? Useless? It was developed to solve the bridges of Koenigsberg problem! :-) Aug 24, 2017 at 16:05
• I disagree with this evaluation of early topology: early topologists like Poincare and Bohl had very specific applications to differential equations and mechanics in mind. Aug 25, 2017 at 0:10

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.

• 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? Jan 4, 2014 at 18:39
• -1, which shows that negative numbers can be useful.
– Joël
Sep 12, 2017 at 16:46
• I can explain the -1. 1830-1940 is not "almost two centuries". Moreover, finite fields have had a central place in geometry, algebra, and number theory even before Galois invented them -- with Gauss for instance.
– Joël
Sep 12, 2017 at 16:48

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

• Um, Leibniz was trying to invent computers. See, for example, gwleibniz.com/calculator/calculator.html -- I think he didn't use binary under the hood because base change is "computationally expensive". Jan 3, 2014 at 11:29

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.

"The Recursive Least Squares (RLE) adaptive filter algorithm that recursively finds the coefficients that minimize a weighted linear least squares cost function relating to the input signals.
It was discovered by Gauss in 1821, but lay dormant until 1950 when Plackett rediscovered the original work of Gauss" (cited from the wikipedia article).

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.

• 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, 2013 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, 2013 at 16:35
• "save as a tool for solving differential equations" - some might say this is was a pretty important and heavy use! Jan 15, 2013 at 19:01

The Logarithmic Barrier Approach to solving Linear Programs

I just read a paper explaining interior point methods in which on finds the following statement:

"The logarithmic barrier approach to solving a linear program dates back to the work of Fiacco and McCormick in 1967 in their book Sequential Unconstrained Minimization Techniques, also known simply as SUMT. The method was not believed then to be either practically or theoretically interesting, when in fact today it is both! The method was re-born as a consequence of Karmarkar’s interior-point method, and has been the subject of an enormous amount of research and computation, even to this day." (emphases by me)