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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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    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.2011-09-25
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    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.2011-09-25
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    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.2011-09-25
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    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. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html2011-09-25
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    @Ted- Its interesting, can you add more about what are the other possible extensions of Q?2011-09-25
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    @Ted- A link or a pdf will be also appreciated.2011-09-25
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    A simple example of an extension of $\mathbb Q$ that differs from $\mathbb R$ is the algebraic numbers, defined as all complex numbers that are roots of polynomials with integer coefficients. This does not include all reals, but does include some numbers that are not real, such as $i$, which is root in $x^2+1$.2011-09-25

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