Do you know what “rational” means and what numbers are called rational?

In the distant past, when the calculus system was not yet invented, people counted everything on their fingers. With the advent of arithmetic and the foundations of mathematics, it has become much simpler and more practical to keep records of goods, products, as well as household items. However, what does the modern calculus system look like: what types are the existing numbers divided into and what does the “rational type of numbers” mean? Let's figure it out.

How many kinds of numbers are there in mathematics?

The very concept of "number" means a certain unit of any subject, which characterizes its quantitative, comparative or ordinal indicators. In order to correctly calculate the number of certain things or to carry out certain mathematical operations with numbers (add, multiply, etc.), you should first familiarize yourself with the varieties of these same numbers.

So, existing numbers can be divided into the following categories:

1. Natural numbers are those numbers by which we count the number of objects (the smallest positive integer is 1, it is logical that the series of positive integers is infinite, that is, there is no largest integer). The set of natural numbers is usually denoted by the letter N.
2. Whole numbers. All natural numbers belong to this set, and negative values ​​are added to it, including the number "zero". The designation of the set of integers is written in the form of the Latin letter Z.
3. Rational numbers are those that we can mentally convert to a fraction, the numerator of which will belong to the set of integers, and the denominator to the natural ones. Below we will analyze in more detail what “rational number” means, and give a few examples.
4. Real numbers are a set that includes all rational and irrational numbers. This set is denoted by the letter R.
5. Complex numbers contain part of the real and part of the variable number. Complex numbers are used in solving various cubic equations, which, in turn, can have a negative expression in the formulas under the root sign (i ² = -1).

What does "rational" mean: we analyze

If those numbers that we can represent as an ordinary fraction are considered rational, then it turns out that all positive and negative integers are also included in the set of rational ones. After all, any integer, for example 3 or 15, can be represented as a fraction, where the unit is in the denominator.

Fractions: -9/3; 7/5, 6/55 are examples of rational numbers.

What does rational expression mean?

Move on. We have already figured out what the rational form of numbers means. Let us now imagine a mathematical expression that consists of the sum, difference, product or quotient of various numbers and variables. Here is an example: a fraction, in the numerator of which is the sum of two or more integers, and the denominator contains both an integer and a certain variable. This expression is called rational. Based on the rule “it is impossible to divide by zero”, one can guess that the value of this variable cannot be such that the value of the denominator becomes zero. Therefore, when solving a rational expression, you must first determine the range of the variable. For example, if the denominator has the following expression: x + 5-2, then it turns out that "x" cannot be equal to -3. Indeed, in this case the whole expression turns to zero, therefore, when solving, it is necessary to exclude the integer -3 for this variable.

How to solve rational equations?

Rational expressions can contain quite a large number of numbers and even 2 variables, so sometimes their solution becomes difficult. To facilitate the solution of this expression, it is recommended to perform certain operations in a rational way. So, what does it mean in a “rational way” and what rules should be applied when deciding?

1. The first view, when it is enough to simplify the expression. To do this, you can resort to the operation of reducing the numerator and denominator to an irreducible value. For example, if the numerator has the expression 18x, and the denominator has 9x, then reducing both indicators by 9x, we get just an integer equal to 2.
2. The second method is practical when we have a monomial in the numerator and a polynomial in the denominator. Let's take an example: in the numerator we have 5x, and in the denominator - 5x + 20x ² . In this case, it is best to put the variable in the denominator out of brackets, we get the following kind of denominator: 5x (1 + 4x). And now you can use the first rule and simplify the expression by reducing 5x in the numerator and denominator. As a result, we get a fraction of the form 1/1 + 4x.

What actions can be performed with rational numbers?

The set of rational numbers has a number of its features. Many of them are very similar to the characteristic present in integers and natural numbers, since the latter are always included in many rational numbers. Here are a few properties of rational numbers, knowing which you can easily solve any rational expression.

1. The commutativity property allows you to sum two or more numbers, regardless of their order. Simply put, the amount does not change due to a change in the places of the terms.
2. The distributive property allows us to solve problems using the distribution law.
3. And finally, the operations of addition and subtraction.

Even schoolchildren know what the "rational type of numbers" means and how to solve problems on the basis of such expressions, so an adult educated person just needs to remember at least the basics of many rational numbers.