Rational numbers and actions on them

The concept of numbers refers to the abstractions that characterize an object from a quantitative point of view. Even in the primitive society, people had a need for the counting of objects, so numerical notations appeared. Later they became the basis of mathematics as a science.

In order to operate with mathematical concepts, it is necessary first of all to imagine what kind of numbers there are. There are several basic types of numbers. It:

1. Natural - those that we get when numbering items (their natural account). Their set is denoted by the Latin letter N.

2. Integer (their set is denoted by the letter Z). This includes natural, opposite negative integers and zero.

3. Rational numbers (letter Q). These are those that can be represented in the form of fractions, the numerator of which is equal to an integer, and the denominator to a natural number. All integers and natural numbers are rational.

4. Valid (they are denoted by the letter R). They include rational and irrational numbers. Irrational are the numbers obtained from the rational by various operations (calculating the logarithm, extracting the root), which themselves are not rational.

Thus, any of the listed sets is a subset of the following. An illustration of this thesis is a diagram in the form of so-called. Circles of Euler. The figure represents several concentric ovals, each of which is located inside the other. The inner, smallest oval (area) denotes the set of natural numbers. It completely embraces and includes an area that symbolizes a set of integers, which in turn lies within the realm of rational numbers. The outer, largest oval, including all the others, denotes an array of real numbers.

In this article, we consider the set of rational numbers, their properties and singularities. As already mentioned, all existing numbers (positive, negative and zero) belong to them. Rational numbers constitute an infinite series having the following properties:

- this set is ordered, that is, taking any pair of numbers from this series, we can always find out which of them is larger;

- taking any pair of such numbers, we can always put between them at least one more, and, consequently, a whole series of such - thus, rational numbers represent an infinite series;

- All four arithmetic operations over such numbers are possible, the result of them is always a certain number (also rational); The exception is the division by 0 (zero) - it is impossible;

- Any rational numbers can be represented in the form of decimal fractions. These fractions can be either finite or infinite periodic.

To compare two numbers related to a set of rational, it is necessary to remember:

Any positive number is greater than zero;

- Any negative number is always less than zero;

- when two negative rational numbers are compared, there are more of them, whose absolute value (module) is less.

How are actions performed with rational numbers?

To add two such numbers with the same sign, you need to add their absolute values and put a common sign before the sum. To add numbers with different signs, it follows from the larger value to subtract the smaller and put the sign of the one of them whose absolute value is greater.

To subtract one rational number from another it is sufficient to add the opposite of the second to the first number. To multiply two numbers, multiply the values of their absolute values. The result obtained will be positive if the factors have the same sign, and negative, if different.

The division is carried out in a similar way, that is, there is a partial absolute value, and before the result the sign "+" is put in case of the divisible and divisor signs and the "-" sign in case of their mismatch.

Degrees of rational numbers look like products of several co-factors, equal to each other.