In the distant past, when it was not yet inventedthe calculus system, people counted everything on their fingers. With the advent of arithmetic and the foundations of mathematics, it became much easier and more practical to keep records of goods, products, and household items. However, what does the modern system of calculus look like: what kinds of numbers do exist and what does the "rational kind of numbers" mean? Let's figure it out.
How many varieties of numbers exist in mathematics?
The very concept of "number" means a certain unitAny object that characterizes its quantitative, comparative or ordinal values. In order to correctly calculate the number of certain things or conduct some mathematical operations with numbers (add, multiply, etc.), first of all you should familiarize yourself with the varieties of these very numbers.
So, the existing numbers can be divided into the following categories:
- Natural - these are the numbers that wewe count the number of objects (the smallest natural number is 1, it is logical that the series of natural numbers is infinite, that is, there is no greatest natural number). The set of natural numbers is usually denoted by N.
- Whole numbers. This set includes all natural numbers, 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.
- Rational numbers are those that we mentallycan be converted into a fraction whose numerator will belong to the set of integers, and the denominator - to natural numbers. A little later we will discuss in more detail what "rational number" means, and give some examples.
- Real numbers are a set in which all rational and irrational numbers enter. The given set is denoted by the letter R.
- Complex numbers contain a partreal and part of a variable number. Complex numbers are used in solving various cubic equations, which, in turn, can have in the formulas under the root sign a negative expression (i2= -1).
What does "rational" mean: we analyze by examples
If the numbers we are rational are considered to becan be represented in the form of an ordinary fraction, it turns out that all positive and negative integers also enter into the set of rational fractions. After all, any integer, for example 3 or 15, can be represented in the form of a fraction, where there is a unit in the denominator.
Fractions: -9/3; 7/5, 6/55 - these are examples of rational numbers.
What does "rational expression" mean?
Go ahead. We have already analyzed what the rational form of numbers means. Let us now imagine a mathematical expression, which consists of the sum, difference, product or particular different numbers and variables. Here is an example: a fraction in the numerator of which the sum of two or several integers, and the denominator contains both an integer and a certain variable. It is this expression that is called rational. Based on the rule "you can not divide by zero," you can guess that the value of this variable can not be such that the denominator value is zero. Therefore, when solving a rational expression, you must first determine the range of the variable. For example, if the following expression is in the denominator: x + 5-2, then it turns out that "x" can not be -3. After all, in this case, the whole expression turns to zero, so when solving it is necessary to exclude an integer -3 for a given variable.
How to solve rational equations correctly?
Rational expressions can containquite a lot of numbers and even 2 variables, so sometimes their solution becomes difficult. To facilitate the solution of such an expression, it is recommended to perform certain operations in a rational way. So, what does "rational way" mean and what rules should be applied in the solution?
- The first kind, when it is enough just to simplifyexpression. To do this, one can resort to an operation of reducing the numerator and denominator to an uncontractable value. For example, if the numerator has the expression 18x, and in the denominator 9x, then, by reducing both indicators by 9x, we get just an integer equal to 2.
- The second method is practical when in the numerator we have a monomial, and in the denominator we have a polynomial. Let's analyze by an example: in the numerator we have 5x, and in the denominator we have 5x + 20x2. In this case, it is best to take a variable indenominator in brackets, we obtain the following denominator: 5x (1 + 4x). And now you can use the first rule and simplify the expression, reducing 5x in the numerator and in the denominator. As a result, we obtain 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 itsfeatures. Many of them are very similar to the characteristic present in integers and natural numbers, because the latter always enter into a set of rational numbers. Here are some properties of rational numbers, knowing which, you can easily solve any rational expression.
- The commutativity property allows you to sum two or more numbers, regardless of their order. Simply put, the amount does not change from changing the places of the terms.
- The distributivity property allows solving problems using distributive law.
- And, finally, the operations of addition and subtraction.
Even schoolchildren know what "rationalkind of numbers "and how to solve problems based on such expressions, so an adult educated person simply needs to recall at least the basics of a set of rational numbers.</ p>