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.