What is the rational numbers?Senior pupils and students of mathematical specialties, probably easy to answer this question.But those who by profession is far from this, it will be harder.What it actually is?
essence and designation
Under rational numbers mean those which can be represented as a common fraction.Positive, negative, and zero are also included in this set.The numerator of the fraction thus must be an integer, and the denominator - is a natural number.
This set of mathematics is referred to as Q and is called the "field of rational numbers."They include all whole and natural, are respectively as Z and N. The very same set Q is included in the set R. It is this letter designates the so-called real or real numbers.
Presentation
As already mentioned, the rational numbers - this set, which includes all the integer and fractional values.They can be presented in different forms.Firstly, a common fraction: 5/7, 1/5, and 11/15 m. E. Of course, the integers may also be recorded in a similar way: 6/2, 15/5, 0/1, -10/2, and so on. d. Second, another kind of representation - with a finite decimal fractional part: 0.01, -15.001006 and so. on. This is perhaps one of the most common forms.
But there is a third - periodic fraction.This species is not very common, but still used.For example, the fraction 10/3 can be written as 3.33333 ... or 3, (3).The different views will be considered the same numbers.The same will be called to each other and equal fractions, such as 3/5 and 6/10.It seems that it became clear that a rational number.But why refer to them using this term?
origin of the name The word "rational" in the modern Russian language in general carries a slightly different meaning.It's more of a "reasonable", "deliberate".But mathematical terms close to the literal sense of the word borrowed.In Latin "ratio" - is the "attitude", "roll" or "division."Thus, the name reflects the essence of what is rational.However, the second meaning is gone far from the truth.
Actions them
In solving mathematical problems, we are constantly confronted with rational numbers, without knowing it.And they have a number of interesting properties.They all follow a plurality of definitions, either of action.
First, the rational numbers have the property relations of the order.This means that the two numbers may be only one ratio - they are either equal, or more or less than one another.Ie .:
or a = b; or a & gt;b, or a & lt;b.
In addition, this property also follows transitive relation.That is if a longer b , b longer c , the a longer c .In the language of mathematics is as follows:
(a & gt; b) ^ (b & gt; c) = & gt;(a & gt; c).
Secondly, there are arithmetic operations with rational numbers, that is, addition, subtraction, division, and, of course, multiplication.In the process of transformation can also highlight a number of properties.
- a + b = b + a (change of places terms commutative);
- 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 (wherein a is not 0);
- (a + b) c = ac + ab;
- (a & gt; b) ^ (c & gt; 0) = & gt;(ac & gt; bc).
When it comes to ordinary rather than decimal, fractions and integers, actions with them may cause some difficulties.For addition and subtraction only possible with equal denominators.If they are different initially, should be to find a common, all fractions using multiplication to certain numbers.Compare also often possible only under this condition.
multiplication and division of fractions are produced in accordance with fairly simple rules.Bringing to a common denominator is needed.Separately, multiply the numerators and denominators, while in the course of the action as possible fraction needed to minimize and simplify.
As for the division, then it is similar to the first with a slight difference.For the second shot must find the inverse, that is, to "turn" it.Thus, the numerator of the first fraction need to be multiplied with the denominator of the second and vice versa.
Finally, another property inherent in rational numbers, called the axiom of Archimedes.Often in the literature also found the name of "principle."It is valid for the entire set of real numbers, but not everywhere.So, this principle does not apply to certain sets of rational functions.In essence, this axiom is that the existence of two variables a and b, you can always take a sufficient amount, to surpass b.
Scope
So, those who knew or thought that a rational number, it becomes clear that they are used everywhere: in accounting, economics, statistics, physics, chemistry and other sciences.Of course, 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 apart an apple or performing other simple steps to face them.They literally surround us.Yet for certain tasks they are insufficient, in particular, the example of the Pythagorean theorem can understand the need to introduce the concept of irrational numbers.
