Nothing could be simpler, clearer and more exciting mathematics. We just need to thoroughly understand its fundamentals.
That will help this article, which details and easily reveals the essence of rational and irrational numbers.
It's much easier than you think!
From abstract mathematical concepts sometimesso it blows cold and detachment that one might think, "Why all this?". However, despite the first impression, all theorem, arithmetic operations, functions, etc. - No more than a desire to satisfy the basic needs. Especially clearly it can be seen on the example of the emergence of different sets.
It all started with the appearance of natural numbers. And, although hardly anyone is now able to answer exactly how it was, but most likely, the legs of the Queen of Sciences growing somewhere out of the cave. Here, analyzing the number of skins, stones and fellow man opened many "numbers for the account." And that was enough for him. Up to a certain point, of course.
Then it took the skins and the stones and shareconsuming. So there was a need for arithmetic operations, and together with them and rational numbers, which can be defined as a fraction of type m / n, where, for example, m - number of skins, n - number of tribesmen.
It would seem, it has an open mathematicalthe device is enough to enjoy life. But it soon turned out that there are cases where the result is something that is not an integer, but did not roll! And, indeed, the square root of two is not no other way to express with the help of the numerator and denominator. Or, for example, the well-known number Pi, open the ancient Greek scientist Archimedes, just not rational. And these discoveries eventually became so much that all the intractable "rationalization" of united and called irrational.
sets discussed previously belong to the setfundamental concepts of mathematics. This means that they can not be defined through a simple mathematical objects. But it can be done with the help of categories (from the Greek. "Statements") or postulates. In this case, the best thing would designate properties of the data sets.
o ;; Irrational numbers determine Dedekind cuts in the set of rational numbers, in which the lower class is not the greatest, and at the top there is the smallest number.
o; Each transcendental number is irrational.
o; every irrational number is either algebraic or transcendental.
o; The set of irrational numbers is dense on the number line: there is an irrational number between any two numbers.
o; set of irrational numbers is uncountable, it is a set of second Baire category.
o;.. This set is ordered, ie for every two different rational numbers a and b, you can specify which of them are smaller than the other.
o; between any two different rational numbers there is at least one rational number, and consequently, an infinite set of rational numbers.
o; Arithmetic operations (addition, subtraction,multiplication and division) on any two rational numbers are always possible and result in a certain rational same number. An exception is a division by zero, which is impossible.
o; every rational number can be expressed as a decimal fraction (finite or infinite periodic).