What are rational numbers?Older students and students of mathematical specialties are likely to easily answer this question. But those who are far from this by profession will be more difficult. What is it really?
By rational numbers is meant suchwhich can be represented as an ordinary fraction. Positive, negative, and also zero are included in this set. The numerator of the fraction in this case must be an integer, and the denominator must be a natural number.
This set in mathematics is denoted as Q andcalled the "field of rational numbers." This includes all integers and integers, denoted Z and N, respectively. The set Q itself is included in the set R. It is this letter that denotes the so-called real or real numbers.
As already mentioned, rational numbers areset, which includes all integer and fractional values. They can be presented in different forms. First, in the form of an ordinary fraction: 5/7, 1/5, 11/15, etc. Of course, whole numbers can also be written in a similar form: 6/2, 15/5, 0/1, - 10/2, etc. Secondly, another type of representation is a decimal fraction with a final fractional part: 0.01, -15.001006, etc. This is perhaps one of the most common forms.
Но есть еще и третья - периодическая дробь.This type is not very common, but still used. For example, the fraction 10/3 can be written as 3.33333 ... or 3, (3). In this case, different representations will be considered similar numbers. Equal fractions will also be called, for example, 3/5 and 6/10. It seems that it became clear that such rational numbers. But why is this term used to designate them?
The word "rational" in modern Russiangenerally has a slightly different meaning. It is rather "reasonable", "considered". But mathematical terms are close to the direct meaning of this borrowed word. In Latin, "ratio" is a "relation," "fraction," or "division." Thus, the name reflects the essence of what is rational numbers. However, the second value
When solving mathematical problems, we constantlywe encounter rational numbers without knowing it ourselves. And they possess a number of interesting properties. All of them follow either from the definition of a set or from actions.
First, rational numbers have the propertyrelationship order. This means that between two numbers there can be only one relationship - they are either equal to each other, or one is greater or less than the other. Ie:
or a = b; or a> b, or a
In addition, the transitivity of the relation also follows from this property. That is, if a more at, at more fromthen a more from. In the language of mathematics, it looks like this:
(a> b) ^ (b> c) => (a> c).
Secondly, there are arithmetic operations withrational numbers, that is, addition, subtraction, division, and, of course, multiplication. In the process of transformation, you can also select a number of properties.
Когда же речь идет об обыкновенных, а не decimal, fractions or integers, actions with them can cause certain difficulties. Thus, addition and subtraction are possible only with equal denominators. If they are initially different, you should find the common one, using the multiplication of the whole fraction by certain numbers. Comparison is also most often possible only if this condition is met.
Деление и перемножение обыкновенных дробей produced in accordance with fairly simple rules. Reduction to a common denominator is not necessary. The numerators and denominators are multiplied separately, while in the process of performing the action, if possible, the fraction should be reduced and simplified as much as possible.
As for division, this action is similar to the first with a small difference. For the second fraction it is necessary to find the return, that is
Finally, another property inherent in rationalnumbers, called the axiom of Archimedes. Often in the literature also found the name "principle". It is valid for the whole set of real numbers, but not everywhere. So, this principle does not work for some sets of rational functions. In fact, this axiom means that with the existence of two quantities a and b one can always take a sufficient amount of a to exceed b.
So, those who have learned or remembered whatrational numbers, it becomes clear that they are used everywhere: in accounting, economics, statistics, physics, chemistry and other sciences. Naturally, they also have a place in mathematics. Not always knowing that we are dealing with them, we constantly use rational numbers. Even young children, learning to count objects, cutting an apple into pieces or performing other simple actions, encounter them. They literally surround us. And yet for solving some problems they are not enough, in particular, using the example of the Pythagorean theorem, one can understand the necessity of introducing the concept of irrational numbers.
