The concept of numbers refers to abstractions,characterizing an object from a quantitative point of view. Even in primitive society, people had a need for counting items, therefore, numerical designations appeared. Later they became the basis of mathematics as a science.
To operate with mathematical concepts, it is necessary, first of all, to imagine what kind of numbers there are. There are several main types of numbers. It:
1. Natural - those that we get with the numbering of objects (their natural count). Their set is denoted by the Latin letter N.
2. Integers (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 as a fraction, the numerator of which is an integer, and the denominator - 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 numbers that are obtained from rational by various operations (calculation of the logarithm, extraction of the root), which themselves are not rational.
Thus, any of the listed setsis a subset of the following. An illustration of this thesis is a diagram in the form of so-called. Euler circles. The figure represents several concentric ovals, each of which is located inside the other. The innermost smallest oval (region) denotes the set of natural numbers. It completely covers and includes the area symbolizing the set of integers, which, in turn, is enclosed within the area of rational numbers. The outer, largest oval, which includes all the others, denotes an array of real numbers.
In this article we will look at manyrational numbers, their properties and features. As already mentioned, all existing numbers belong to them (positive as well as negative and zero). 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 greater;
- taking any pair of such numbers, we can always put at least one more between them, and, consequently, a whole series of such numbers - thus, rational numbers are an infinite series;
- all four arithmetic operations on such numbers are possible, the result is always a certain number (also rational); the exception is the division by 0 (zero) - it is impossible;
- any rational numbers can be presented in the form of decimal fractions. These fractions can be either finite or infinite periodic.
To compare two numbers related to the set of rational, it is necessary to remember:
- any positive number is greater than zero;
- any negative number is always less than zero;
- when comparing two negative rational numbers, the one whose absolute value (modulus) is less is greater.
How are actions performed with rational numbers?
To add two such numbers having the samesign, you need to add their absolute values and put in front of the sum of the total sign. To add numbers with different signs, it is necessary to subtract the smaller value from the larger value and put the sign of the one whose absolute value is greater.
To subtract one rational number fromthe other is enough to add to the first number the opposite of the second. To multiply two numbers you need to multiply the values of their absolute values. The result will be positive if the factors have the same sign, and negative if they are different.
The division is made similarly, that is, there is a quotient of absolute values, and the result is preceded by the “+” sign in the case of the coincidence of the signs of the dividend and the divisor and the sign “-” in the event of their mismatch.
The degrees of rational numbers look like the products of several factors equal to each other.
