I came across a term "Construction" in a mathematical analysis book. First of all numbers came into being because of counting. Hence man counted using natural numbers. Now given the system of counting numbers only, it is impossible to answer questions like what should you add to 3 so as to get 1. This is how perhaps the concept of negative integers came into existance. Now how come the rationals? To this I can say it might have been introduced so as to answer questions on property division. For example a man has 7 crores and he wants to divide it among his two sons equally. So with integers this was impossible and the purpose of numbers is to measure or count, which in essence means comparasion. so to facilitate equality in this kind of division problems what came up was the rational numbers. Then we tried to investigate that whether all rationals are product of two equal rationals and so searched whether there exists any such rational number which when multiplied with itself gives 2. Then someone came up with an idea (with certitude certified by a proof) that there cannot exist any such rational number. Now I guess the whole process of explaining the arrangement of rational and irrational numbers , proving or investigating their formation of a continuous system and unifying them into a system called " Real numbers" is called as construction of $\mathbb R$.

Now after sometime there came a question that whether there exists any square root of negative numbers in $\mathbb R$, propelled the birth of complex numbers. Now is there any question that can point out an inadequacy in $\mathbb C$, and hence give rise to a new system of numbers?

P.S.: for all those who are interested in commenting or answering to this question, I would like to request them to first comment on whether my comprehension of the term "construction" is correct.

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    $\begingroup$ There is unlimited scope for generalizations and extensions. Many of them branch away from the 'tower' of extensions that you refer to at an earlier point. For examples the rational numbers can be "completed" in infinitely many non-equivalent ways. It is also possible to go beyond the complex numbers. The problems requiring those extensions don't appear in the high school / early college level of applications of mathematics into physics, economics, et cetera, so you need to be a kind of specialist to learn about those. $\endgroup$ – Jyrki Lahtonen Sep 25 '11 at 7:31
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    $\begingroup$ What you describe is basically correct as a sequence of mathematical constructions (though the exact historical sequence may be more muddled with regard to when negative numbers were accepted). As far as extending $\mathbb{C}$ further, it's important to realize that when you extend number systems, sometimes you also lose existing features. For example, in $\mathbb{R}$ there is an ordering given by < , but in $\mathbb{C}$ there is not. There is a number system called $\mathbb{H}$, the quaternions, that is an "extension" of $\mathbb{C}$, but you lose commutativity of multiplication. $\endgroup$ – Ted Sep 25 '11 at 7:38
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    $\begingroup$ So you can always extend numbers further (even $\mathbb{R}$ isn't the only possible extension of $\mathbb{Q}$), the question is what properties you want to gain or to keep, compared to what you lose. $\endgroup$ – Ted Sep 25 '11 at 7:40
  • $\begingroup$ The historical sequence was indeed quite muddled. For example, in the 16th century Cardano made use of complex numbers, but didn't consider 0 to be a number. See e.g. www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html $\endgroup$ – Robert Israel Sep 25 '11 at 8:21
  • $\begingroup$ @Ted- Its interesting, can you add more about what are the other possible extensions of Q? $\endgroup$ – user16186 Sep 25 '11 at 14:44

First, I comment on the idea of 'construction.' When analysis books today refer to the construction of the real numbers (or rationals, etc), they usually refer to some sort of explicit extension of something number of axioms. Many people have no trouble 'believing' in the natural numbers, and so start there. Groups are lovely structures (a set with an associative operation, an identity, and inverses) that are well-studied, and so one might add zero and the negative numbers to get a group. But fields (groups with two operations, sort of, that commute and distribute) are great structures too, and there are ways to turn the familiar system of rationals numbers into a much bigger set - the real numbers. The general ideas of this construction often falls under the idea of making the rationals 'complete' - making it so that any Cauchy sequence of rationals (a sequence of rationals such that the differences between terms becomes arbitrarily small, sort of) converges. And then one can sort of complete this process with the complex numbers.

I say 'complete this process' because all of our standard mathematical operators are closed in the complex numbers. All polynomials completely factor, we can take roots and multiples, multiply by inverses, etc. without fear of getting something meaningless. And the constructions in analysis books usually go in specific ways that amount to assuming the least amount of things to get the greatest amount of information. But I want to point out that this isn't really how the systems themselves got started (except perhaps the complex numbers - they have their own history). These different systems had a much less clear progression. It is not as if one day, Pythagoras woke up and said, "Hmm. I wish that my triangles lengths would exist. So I will extend my normal ratios into something beautiful." Instead, Greeks panicked when they thought there were things so imperfect as an irrational number.

But one aspect is preserved from the original development of these number systems: the idea that some systems aren't closed under some sort of operation, and so we try to expand it. First came the addition of 0 (absent for much of history). Ratios are very old, and strangely enough are often older than the concept of a zero in various areas of the world. They're just so... useful. But then the Greeks found irrational numbers (Pythagoras, for instance, suggested that there was no common measure between the hypotenuse and leg of a 45-45-90 right triangle - i.e. it was irrational). This was a very big idea, and this took a very long time to sink in. Then things like Cardano's solution to the cubic used imaginary numbers in meaningful ways - whoa. Taking square roots of negative numbers is a new trick. And the funny thing is that he didn't even use his formula to produce imaginary numbers - they just happened to appear and then later cancel in the middle steps. Another big idea. But by then, our current formal style was starting to get around. Not too long afterwards, the idea of fields and field extensions came around, and so we began to consider what things were and weren't closed.

The basic idea of a field extension is that we take some field (like the rationals) and we take something not in that field (perhaps the root of a polynomial - the standard idea). And then we see how much larger we need to make that field to accommodate that extra bit. Sometimes, it's small (extending the rationals to hold the root of $x^2 - 2 = 0$, for instance, is a very small extension). Sometimes, it's big (to hold something like $\pi$, for instance). Now, the big idea is that no root of a polynomial with complex coefficients lies outside of the complex numbers. So one can't extend the complex numbers in that fashion. And in that sense, the complex numbers are as 'big' as it gets.

But one might ask whether or not it is possible to meaningfully go beyond the complex numbers. And the answer is... sort of. There is something called the Cayley-Dickson Construction that makes arbitrarily 'bigger' algebras than the complex numbers. But the problem is that all the nice algebraic properties quickly break down. The quaternions, the one 'after' the complex numbers, don't even commute!

And there are perhaps less-extreme extensions, but of a different calibre. For instance, one might do things called 'compactifications' that usually sort of amount to pretending that infinity was actually a number of sorts in our space. This is something from the realm of topology, so it's a bit more abstract. But one might give rise to the analysis of projective spaces or the Riemann Sphere in this way. But the 'operations' that these facilitate are not really numeric in nature, and so this is a fine line.

So in short - the complex numbers are adequate in almost every way we could hope for. But there are things that are bigger in some sense or another.


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