The concept of numbers refers to abstractions that characterize an object from a quantitative point of view. Even in primitive society, people had a need for counting objects, so 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 numbers are. The main types of numbers are few. It:
1. Natural - those that we get when numbering objects (their natural count). Their many are denoted by the Latin letter N.
2. Integers (their set is denoted by the letter Z). This includes positive integers, negative integers opposite them, and zero.
3. Rational numbers (letter Q). These are those that can be represented in the form of a fraction, the numerator of which is equal to an integer, and the denominator is natural. All integers and natural numbers are rational.
4. Valid (they are denoted by the letter R). They include rational and irrational numbers. Irrational numbers are those obtained from rational ones 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. Euler circles. The figure represents several concentric ovals, each of which is located inside the other. The inner, smallest oval (region) denotes a set of natural numbers. It completely covers and includes the area symbolizing the set of integers, which, in turn, is enclosed within the field of rational numbers. The outer, largest oval, including all the others, denotes an array of real numbers.
In this article we will consider the set of rational numbers, their properties and features. As already mentioned, all existing numbers belong to them (positive, as well as negative and zero). Rational numbers make up 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 one is more;
- taking any pair of such numbers, we can always place between them at least one more, and, consequently, a whole series of those - thus, rational numbers are an infinite series;
- all four arithmetic operations on such numbers are possible, their result is always a certain number (also rational); the exception is the division by 0 (zero) - it is impossible;
- any rational numbers can be represented as decimal fractions. These fractions can be either finite or infinite periodic.
To compare two numbers that belong to the set of rational, you need to remember:
- any positive number is greater than zero;
- any negative number is always less than zero;
- when comparing two negative rational numbers, the larger is the one whose absolute value (modulus) is less.
How are rational numbers performed?
To add two such numbers having the same sign, you need to add their absolute values and put the total sign in front of 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 whose absolute value is greater.
To subtract one rational number from another, it is enough to add the opposite to the second to the first number. 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 different.
The division is carried out similarly, that is, the quotient of the absolute values is found, and the result is preceded by a “+” sign in case of coincidence of the signs of the dividend and divider and a “-” sign in case of their mismatch.
The degrees of rational numbers look like the products of several factors equal to each other.