Views On Number System

Many mathematical concepts need to be understood from an altitude, like one will do with philosophy. In fact, in many areas in professional Maths such as analysis, many proofs are fostered in this way, too. Unfortunately, these understandings are rarely given by textbooks and teachers, but very useful for those who want to learn Maths further in the university or ... just to get more interested in Maths. So here I will talk about one example of this kind of understanding on number system.

When it comes to number system, there are several basic concepts: C (complex numbers), R (real numbers), Q (rational numbers) and Z (integers). These groups of numbers, regardless of their different power to include "numbers" created by human, were the consequence of extensions during the history. For examples, Q (rational numbers) can be viewed as the extension of Z (integers). Interestingly, the extensions between these groups (C,R,Q,Z) are due to some specific arithmetic operations.

Rational numbers are actually the fraction of integers, i.e. m/n & Q, where m, n & Z. Similarly, real numbers are created to carry out the solution to the equations like x²=3, after it is proved that the solution must not be a rational number. Complex numbers are the result of doing square roots of negative numbers.

Every time the number system was expanded, it brought differences to the number axis. With only Z existing, you may imagine the number axis as dots, with the same distances ... pretty big distances between each other. After rational numbers were introduced, the spaces seemed to be filled out, which we call it "dense". But the density is not the "perfect" end of expanding the number system. By expanding the number system to R, we found that numbers on the axis finally "tightly" connected to each other. Complex numbers then offered another thoughts to the numbers, by adding an imaginary axis beyond the real axis.

That`s a brief introduction about my views on number system. Some of you might find it a bit hard but definitely you can find some new angles of thinking on this. Some of the concept and lexis I used was beyond A-level, some even beyond my level of study but that is totally fine. What I want to convey is that, Maths is not something "unique". It actually shares a lot of thinking methods with other subjects. Sometimes it is rather boring to deal with bunches of Maths problems and do all the calculations but I want to justify that it`s never the whole picture. The reason of having difficulties in Maths is highly possible to be understanding Maths in very limited ways. I think that`s where you might need my help.

If you are going to have tutorials with me I`d love to share some of these thoughts with you. Maths is beautiful and its beauty deserves discovering. Check the first chapter of Rudin`s Real Analysis if you are interested in what I mentioned above about the construction of number system.