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I'm looking for examples of ideas/discoveries/concepts in Maths or Science that had no practical application at first and were maybe considered as nothing but a theoritical concept but they turned out to be useful years after?

I think that imaginary numbers may be such a thing - it's written on Wikipedia that "The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless." - and now they are very common in Physics and Maths, right? What about some other examples?

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    $\begingroup$ Hi, welcome to hsm. Could you make your question a bit more specific by indicating the time period and/or the subject area that you are most interested in? As it is there are too many examples to cover in an answer of reasonable length. $\endgroup$ – Conifold Nov 19 '15 at 0:37
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    $\begingroup$ Number theory from hundreds of years ago had no practical use until it was applied in cryptography. Riemann's work in differential geometry was pure math for 60 years until Einstein found it to be the right framework for general relativity. Group representations were developed by Frobenius and others starting from the end of the 19th century and founds applications to physics (by Wigner) in the 1930s. In science, Maxwell predicted in the 1860s that there are electromagnetic waves traveling at the speed of light and Hertz found them 20 years later but had no idea what consequences (contd.) $\endgroup$ – KCd Nov 19 '15 at 1:51
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    $\begingroup$ his discovery could have ("None" is essentially what he said when asked). Of course the later rise of radio and electronic communication proved Hertz spectacularly wrong. $\endgroup$ – KCd Nov 19 '15 at 1:53
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I would probably say number theory, which had quite a lot of work on theorems regarding prime numbers that (as far as I know) did not get much use until the development of cryptography, such as the RSA algorithm from the 70's using Fermat's little theorem from the 1640's.

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  • $\begingroup$ This is probably the most epic example. Number theory was basically the definition of pure mathematics for literally thousands of years, and now it makes up a major part of the backbone of the internet. $\endgroup$ – PyRulez Jul 1 '16 at 16:36
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Radon's transform is a good recent example. It was discovered in 1917 by Johann Radon. Computer tomography is based on it.

But actually there are plenty of examples. The most famous is the theory of conic sections which had no practical purpose until Kepler found that they describe orbits of celestial bodies.

Linear programming can be traced back to Cauchy and Fourier, but real practical applications begin in the middle 20s century.

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Most of the stuff Einstein worked on can come under this category. For example, people did have any use of the relativity theory in daily life before Global Positioning System came around. Similarly, we do not yet know how to exploit the wormholes (which are allowed by modern physics) but one day might.

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I am not an expert on the history of the transistor, but I know enough for the purposes of this question (If anyone knows more, I would love to hear about).

The transistor (the piece of electronics found in effectively everything you use) came from an obscure branch of quantum mechanics which was called surface physics. The theoretical work began in the early 1900's during the beginning of quantum and had no immediate (or even foreseeable) application/use. By the late 1940's to early 1950's, the first prototypes of the transistor were developed (I believe the first was built at Bell Labs).

By the year 2006, the transistor alone would be responsible for over 30 trillion dollars (that is \$30,000,000,000,000 USD) in revenue (this comes from a 2007 MIT Masters thesis on a market evaluation of semiconductor technologies). A more recent report by McKinsey&Company found that search engines alone generate some $780 billion USD GDP annually across global markets. Thus, a seemingly intangible theory can result in very profitable applications.

There was a recent economic study [van Bochove, 2012] that showed the financial benefits of pure research. In that study, they state:

Directing basic research towards economic opportunities is detrimental to growth and may reduce the growth rate by as much as one half. The steady state is shown to be globally stable; in the steady state, the growth rate is independent of the research intensity, but the level of income depends on it. Given current OECD levels of R&D spending and saving, a one dollar increase of applied R&D spending will increase national income with 6-25 dollars and one dollar extra basic research by 20-100 dollars. These rates of return are ten and thirty times higher, respectively, than those on physical capital investment.

References
van Bochove, C.A. "Basic Research and Prosperity: Sampling and Selection of Technological Possibilities and of Scientific Hypotheses as an Alternative Engine of Endogenous Growth," CWTS Working Paper Series, Leiden University, 2012.

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Perhaps somewhat the other way around: Arthur C. Clarke (yes, the science fiction writer) invented communication satellites and published about them in 1945. He didn't bother to patent the idea, he thought it would take longer than the patent's duration. First communication satellite was launched in 1958.

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One mathematical idea that went unused in physics until decades later was the idea on non-Euclidean space. Gauss and Riemann among others had formulated the idea of space that did not follow Euclidean geometry and defined rules for how the two other geometries (hyperbolic and elliptic) would work. It was not until Einstein that it was utilized by physics to model the universe because it was Einstein that theorized/made popular that the universe is a non-Euclidean space, and that Euclidean geometry is simply a good enough model at the scales we had been looking at.

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