There is nothing more simply, more clear and more fascinating than mathematics. It is necessary to understand its bases only properly. The present article in which in detail and easily the essence of rational and irrational numbers reveals will also help with it.
It is simpler and simpler, than it seems!
From abstractness of mathematical concepts sometimes so blows as cold and dispassionateness that involuntarily there is a thought: "Why all this?". But, despite the first impression, all theorems, arithmetic operations, functions, etc. – no more, than desire to satisfy urgent needs. Especially accurately it can be noticed on the example of emergence of various sets.
Everything began with emergence of natural numbers. And, though, hardly now someone will be able to answer how precisely it was, but most likely, legs at the queen of sciences grow from where from a cave. Here, analyzing the number of skins, stones and tribespeople, the person opened a set of "numbers for the account". And it to it there was enough. Till some moment, of course.
It was required to divide and take away skins and stones further. So there was a need for arithmetic operations, and together with them and rational numbers which can be defined as fraction like m/n where, for example, m is the number of skins, n there is a number of tribesmen. It would seem, already open mathematical apparatus is quite enough to be happy life. But soon it turned out that there are cases when result not that not an integer, but even not fraction! And, really, a square root from two differently not to express by means of numerator and a denominator in any way. Or, for example, the known number of Pi opened by the Ancient Greek scientist Archimedes is also not all rational. And such opening became so much over time that all numbers resistant to "rationalization" united and called irrational.
Properties
The sets considered earlier belong to a set of fundamental concepts of mathematics. It means that they will not manage to be defined through simpler mathematical objects. But it can be done by means of categories (from Greek "statement") or postulates. In this case it was best of all to designate properties of these sets.o Irrational numbers Dedekind sections in a set of rational numbers which in the lower class do not have the greatest define, and in top there is no smallest number.o Each transcendental number is irrational. o Each irrational number is either algebraic, or transcendental. o the Set of irrational numbers is dense on a numerical straight line everywhere: between any two numbers there is an irrational number.o the Set of irrational numbers incalculably, This set ordered is a set of the second category of Rem.o, i.e. for each two various rational numbers a ib it is possible to specify what of them is less than another. o Between each two various rational numbers exists still at least one rational number and consequently, and an infinite set of rational numbers. o Arithmetic actions (addition, subtraction, multiplication and division) over any two rational numbers are always possible and give as a result a certain rational number. An exception is division into zero which is impossible. o Each rational number can be presented in the form of decimal fraction (final or infinite periodic).