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## 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

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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 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 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 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 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 at 13:18 show 5 more comments ## 7 Answers 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. - This answer being given, let me add that I am not convinced your idea regarding the article is a good one. – quid Dec 17 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 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 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 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.' :) – quid Dec 17 at 23:54 show 6 more comments 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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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

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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.' – quid Jan 15 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'. – quid Jan 15 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 at 19:01

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. This 18 centuries passed between math research and the first application!

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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 at 6:57
Thanks for this information. Could you cite some source of this information? Doubling the cube is not a serious application, and conics certainly do not help here, but the shade path on the sundial seems to be an application. Still hard to imagine that this application justified the treatise of Apollonius... – Alexandre Eremenko Dec 18 at 14:51
en.wikipedia.org/wiki/Menaechmus (cites Boyer's and Cooke's history of math texts) www-history.mcs.st-andrews.ac.uk/Biographies/… www-history.mcs.st-and.ac.uk/HistTopics/… W.W. Dolan: Early Sundials and the Discovery of the Conic Sections, Mathematics Magazine, 1972. 45(1): p. 8-12. – Miles Dec 18 at 16:02
Sorry about the formatting there---I tried to make it nice . . . – Miles Dec 18 at 16:08
Conic sections were apparently used by the Greeks (possibly Archimedes) in real life: en.wikipedia.org/wiki/Parabolic_reflector – YangMills Jan 16 at 0:53

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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P. S. How about CWing the question and tagging it big list? – Harun Šiljak Dec 17 at 18:11
Useless concept for Euler? Is my sarcasm detector broken? – Franz Lemmermeyer Dec 17 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 at 18:22
@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 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 at 21:17