What are rational numbers? High school and math students are likely to easily answer this question. But for those who are far from this by profession, it will be more difficult. What is it really like?
The essence and designation
By rational numbers we mean those that can be represented as ordinary fractions. Positive, negative, and also zero are also included in this set. The numerator of the fraction should be integer, and the denominator should be a natural number.
This set in mathematics is denoted as Q and is called the "field of rational numbers." It includes all integers and integers, denoted respectively by Z and N. The set Q itself enters the set R. It is with this letter that the so-called real or real numbers are designated .
Representation
As already mentioned, rational numbers are a set that includes all integer and fractional values. They can be presented in different forms. Firstly, in the form of an ordinary fraction: 5/7, 1/5, 11/15, etc. Of course, integers 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 finite fractional part: 0.01, -15.001006, etc. This is perhaps one of the most common forms.
But there is also a third - periodic fraction. 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, various representations will be considered similar numbers. Equal fractions, for example 3/5 and 6/10, will also be called. It seems that it has become clear what rational numbers are. But why is this term used for their designation?
origin of name
The word "rational" in modern Russian generally has a slightly different meaning. It is rather "reasonable", "deliberate." But mathematical terms are close to the direct meaning of this borrowed word. In Latin, “ratio” is “ratio”, “fraction” or “division”. Thus, the name reflects the essence of what rational numbers are. However, the second value
not far from the truth.
Actions with them
When solving mathematical problems, we constantly encounter rational numbers, without knowing it ourselves. And they have a number of interesting properties. All of them follow either from the definition of a set or from actions.
First, rational numbers have the property of an order relation. This means that between two numbers there can only be one relationship - they are either equal to each other, or one is more or less than the other. T. e .:
either a = b; either a> b or a <b.
In addition, the transitivity of the relation also follows from this property. That is, if a is greater than b , b is greater than c , then a is greater than c . In the language of mathematics, it looks like this:
(a> b) ^ (b> c) => (a> c).
Secondly, there are arithmetic operations with rational numbers, that is, addition, subtraction, division, and, of course, multiplication. Moreover, in the process of transformations, a number of properties can also be distinguished.
- a + b = b + a (change of places of terms, commutativity);
- 0 + a = a + 0;
- (a + b) + c = a + (b + c) (associativity);
- a + (-a) = 0;
- ab = ba;
- (ab) c = a (bc) (distributivity);
- ax 1 = 1 xa = a;
- ax (1 / a) = 1 (while a is not equal to 0);
- (a + b) c = ac + ab;
- (a> b) ^ (c > 0) => (ac> bc).
When it comes to ordinary, but not decimal, fractions or integers, actions with them can cause certain difficulties. So, addition and subtraction are possible only if the denominators are equal. If they are initially different, you should find a common one, using the multiplication of the whole fraction by one or another number. Comparison is also most often possible only if this condition is met.
Division and multiplication of ordinary fractions are carried out in accordance with fairly simple rules. Reduction to a common denominator is not necessary. The numerators and denominators are separately multiplied, while in the process of performing the action, if possible, the fraction should be minimized and simplified as much as possible.
As for division, this action is similar to the first with a slight difference. For the second fraction, find the inverse, i.e.
"flip" her. Thus, the numerator of the first fraction will need to be multiplied with the denominator of the second and vice versa.
Finally, another property inherent in rational numbers is called the axiom of Archimedes. Often in the literature the name "principle" is also found. It is valid for the whole set of real numbers, but not everywhere. So, this principle does not apply to some sets of rational functions. In fact, this axiom means that with the existence of two quantities a and b, you can always take a sufficient amount of a to exceed b.
Application area
So, for those who have learned or remembered what rational numbers are, 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 small children, learning to count objects, cutting an apple into pieces or performing other simple actions, encounter them. They literally surround us. Nevertheless, to solve some problems, they are not enough, in particular, using the example of the Pythagorean theorem, one can understand the need to introduce the concept of irrational numbers.