Solving problems and some expressions does not always lead to pure numerical answers. Even in the case of trivial calculations, you can come up with a certain construction, called a variable expression.
For example, consider two practical tasks. In the first case, we have a certain factory that produces 5 tons of milk every day. It is necessary to find how much milk is produced by the plant in r days.
In the second case, there is a rectangle that is 5 cm wide and p cm long. Find the area of the figure.
Of course, if a plant produces five tons a day, then in r days, according to the simplest mathematical logic, it will produce 5r tons of milk. On the other hand, the area of a rectangle is equal to the product of its sides - that is, in this case, it is 5p. In other words, in two trivial problems with different conditions, the answer is one whole expression - 5p. Such monomials are called an expression with a variable, since in addition to the numeric part, they contain a certain letter called an unknown, or variable. Such an element is denoted lowercase letters Latin alphabet, most often, x or y, although this is not essential.
A feature of a variable is that it can take any value in practice. Substituting different numbers, we will receive the final solution for our tasks, for example, for the first:
p = 2 days, the plant produces 5p = 10 tons of milk;
p = 4 days, the plant produces 5p = 20 tons of milk;
Or for the second one:
p = 10 cm, the area of the figure is 5p = 50 cm2
p = 20 cm, the area of the figure is 5p = 100 cm2
It is important to understand that p is not a set of some individual values, but the whole set that will mathematically correspond to the condition of the problem. The main role of the variable is to replace the missing element in the condition. Any math problem should include some constructs and show the relationship between these constructs in the condition. If the value of an object is not enough, then a variable is entered instead. Moreover, it is an abstract replacement of the very element of the condition (the amount of something represented by a number or expression), and not functional connections.
If we consider an expression of the form 5p, as a neutral and independent object, then the value of p in it can take any values, in fact, p here is equal to the set of all real numbers.
But in our problems, certain mathematical restrictions are imposed on the answer in the form of 5p, which follow from the conditions. For example, days and days cannot be negative, so p in both problems is always is zero or more. In addition, days cannot be fractional - only those p values that are positive integers are valid for the first problem.
In the first problem: p is equal to the finite set of all positive integers;
In the second problem: p is equal to the finite set of all positive numbers.
Expressions can include two variables at once, for example:
In this case, a bin is represented by two monomials, each of which has a variable in its composition, and these variables are different, that is, independent of each other. The value of this expression can be fully calculated only if the value of both variables is present. For example, if x = 2 and y = 4, then:
2x + 3y = 4 + 12 = 16 (for x = 2, y = 4)
It is worth noting that in this expression there are no mathematical or logical restrictions on the values of the variable - both x and y belong to the entire set of real numbers.
In general, the set of all numbers, when substituted instead of a variable, the expression retains meaning and validity, is called the domain (or value) of the variable.
In abstract examples that are not related to real problems, the scope of a variable is most often either equal to the entire set of real numbers or limited by some constructions, for example, a fraction. As you know, when the divisor is zero, the entire fraction loses its meaning. Therefore, a variable in an expression of the form:
cannot be equal to five, because then:
7x / (x - 5) = 7x / 0 (at x = 5)
And the fraction will lose its meaning. Therefore, for this expression, the variable x has a domain of definition - the set of all numbers with the exception of 5.
In our video tutorial, we also noted a special case of using variables when they denote a number of the same order. For example, the numbers 54, 30, 78 can be specified through the variable a, or through the construction ab (with a horizontal bar above, to distinguish from the product), where b specifies units (4, 0, 8, respectively), and tens (respectively, 5, 3, 7).
Consider a small problem that is often found in various magazines and tricks.
The magician invites you to guess a certain number. Then he asks to multiply it by three, and add six to the result. Then he asks to divide the resulting amount by three and subtract the resulting number from the result. Then he tells you the correct answer.
How does this happen, is it really magic?
No, in fact, everything is simpler. Let us think of the number 5. Now we will perform all the actions that the magician suggested to us.
- 1. 5*3=15.
- 2. 15+6=21.
- 3. 21:3=7.
- 4. 7-5=2.
Received a two in response. We could write this solution in the form of a numerical expression (5 * 3 + 6): 3 - 5. And its value would be the number 2.
Now, let's say we conceived the number 3. The result would be a numerical expression (3 * 3 + 6): 3 - 3. And its value would be the number 2.
Deuce again. The thought arises that there is no focus here, and in any case the number 2 will be obtained. Let's try to check it. Let's denote the number that we conceived by the letter x, and write down all the actions that the magician asked to do in the necessary order.
- We get (x * 3 + 6): 3 -x.
- (x * 3 + 6): 3 -x = x + 2-x = 2.
It turns out that the number we have conceived does not play any role at all, it will be reduced in any case.
In the analysis of the problem, we received the expression (x * 3 + 6): 3 –x, which is written using a letter denoting any number, numbers 3 and 6, brackets and action signs. Such an expression is called an algebraic expression or a variable expression.
Defining a variable expression
- An algebraic expression or an expression with a variable is called any meaningful notation consisting of letters denoting any number, numbers and action signs.
For example, the following entries would be algebraic expressions:
- 2 * (x + y),
- 34 * a-13 * a * x,
- (123-65 * a): 3 +4.
If instead of each letter that is included in the algebraic expression, you substitute some numerical value, and then perform all the actions, then the result will be a certain number. This number is called value algebraic expression.
For example, the value of the algebraic expression 5 * a + 2 * x-7 with a = 2 and x = 3 will be the number 9, since 5 * 2 + 2 * 3 -7 = 9.
In the problem that we considered at the beginning, the value of the algebraic expression (x * 3 + 6): 3 - x will be the number 2, for any value of the variable x.
Expressions made up of numbers, action signs, and parentheses are called numerical expressions... The number that is the result of performing all actions in numerical expression is called the value of a numeric expression... O numerical expressions who don't matter say they don't make sense.
To compare numbers, signs are used, 
,
,
,
,
... In this case, double inequalities of the form 
etc. Inequalities in which signs are used 
and 
are called strict which use the signs 
and 
,
–not strict.
Expressions made up of numbers, letters, action marks, and parentheses are called literal expressions, or variable expressions or with variables... The set of variable values for which the expression with the variable has a numeric value (makes sense) is called range of valid values variable of the given expression.
Variable expressions are used to write numbers a certain kind... For example, the entry 
means any three-digit number for which 
hundreds, 
dozens and 
units, i.e. 
... With the help of literal expressions, it is convenient to write down mathematical rules, laws, definitions. For example, module definition(absolute value) numbers
can be written like this: 
.
Elements of statistics
A series of numbers obtained as a result of statistical research is called statistical sampling or simply sampling, and each number of this series is variant sampling... The number of numbers in a row is called volume sampling. The record of a sample, when the next option is not less than the previous one, is called ordered series of data(or variation series).
The arithmetic mean of the sample is called the quotient of the sum of all sample variants and the number of variants (i.e., the quotient of the sum of all variants and volume sampling). The number of occurrences of the same variant in the sample is called frequency this option. The sample with the highest frequency is called fashion sampling... The difference between the largest and smallest sample variants is called sweep sampling... If there is an odd number of variants in an ordered series of data, then the average variant is called median... If there is an even number of variants in the ordered row, then the arithmetic mean of the two averages for the account is called median.
Preparatory option
Writing the conditions of problems using the notation accepted in mathematics leads to the appearance of the so-called mathematical expressions, which are simply called expressions. In this article, we will talk in detail about numeric, literal and variable expressions: we will give definitions and give examples of expressions of each type.
Page navigation.
Numeric expressions - what are they?
Acquaintance with numerical expressions begins almost from the very first lessons of mathematics. But their name - numerical expressions - they officially acquire a little later. For example, if you follow the course of M.I. Moro, then this happens on the pages of a mathematics textbook for 2 grades. There, the idea of numerical expressions is given as follows: 3 + 5, 12 + 1−6, 18− (4 + 6), 1 + 1 + 1 + 1 + 1, etc. - it's all numeric expressions, and if the specified actions are performed in the expression, then we will find expression value.
It can be concluded that at this stage of the study of mathematics, numerical expressions are called records that have a mathematical meaning, composed of numbers, brackets, and addition and subtraction signs.
A little later, after getting acquainted with multiplication and division, the records of numerical expressions begin to contain the signs "·" and ":". Here are some examples: 6 4, (2 + 5) 2, 6: 2, (9 3): 3, etc.
And in high school, the variety of notation for numerical expressions grows like a snowball rolling down a mountain. Ordinary and decimals, mixed numbers and negative numbers, degrees, roots, logarithms, sines, cosines and so on.
Let's summarize all the information in the definition of a numeric expression:
Definition.
Numeric expression is a combination of numbers, arithmetic signs, fractional bars, root signs (radicals), logarithms, symbols for trigonometric, inverse trigonometric and other functions, as well as brackets and other special mathematical symbols, compiled in accordance with the rules accepted in mathematics.
Let's explain all the constituent parts of the sounded definition.
Absolutely any numbers can participate in numerical expressions: from natural to real, and even complex. That is, in numerical expressions you can find
With the signs of arithmetic operations, everything is clear - these are the signs of addition, subtraction, multiplication and division, which have the form "+", "-", "·" and ":", respectively. Numeric expressions may contain one of these characters, some of them or all at once, and more than once. Here are examples of numerical expressions with them: 3 + 6, 2.2 + 3.3 + 4.4 + 5.5, 41−2 4: 2−5 + 12 3 2: 2: 3: 12−1 / 12.
As for parentheses, there are both numeric expressions that contain parentheses and expressions without them. If there are parentheses in a numeric expression, then they are basically
And sometimes parentheses in numeric expressions have some specific, separately indicated special purpose. For example, you can find square brackets representing the integer part of the number, so the numeric expression +2 means that the number 2 is added to the integer part of 1.75.
It can also be seen from the definition of a numeric expression that the expression may contain,, log, ln, lg, designations, or the like. Here are examples of numerical expressions with them: tgπ, arcsin1 + arccos1 − π / 2 and 
.
Numeric divisions can be indicated with. In this case, there are numerical expressions with fractions. Let us give examples of such expressions: 1 / (1 + 2), 5+ (2 3 + 1) / (7−2,2) +3 and 
.
As special mathematical symbols and designations that can be found in numerical expressions, we give. For example, let's show a numeric expression with the module 
.
What are literal expressions?
The concept of literal expressions is introduced almost immediately after acquaintance with numeric expressions. It is introduced like this. In some numerical expression, one of the numbers is not written, but a circle (or a square, or something similar) is put in its place, and it is said that a number can be substituted for the circle. Let's take a record as an example. If instead of a square you put, for example, the number 2, you get the numerical expression 3 + 2. So instead of circles, squares, etc. agreed to write down letters, and such expressions with letters were called letter expressions... Let's go back to our example, if in this entry instead of a square we put the letter a, then we get an alphabetic expression of the form 3 + a.
So, if we assume in the numerical expression the presence of letters that denote some numbers, then we get the so-called literal expression. Let us give an appropriate definition.
Definition.
An expression containing letters that denote some numbers is called literal expression.
From this definition it is clear that a literal expression is fundamentally different from a numerical expression in that it can contain letters. Usually, in literal expressions, small letters of the Latin alphabet (a, b, c,…) are used, and when denoting angles, small letters of the Greek alphabet (α, β, γ,…).
So, literal expressions can be composed of numbers, letters and contain all mathematical symbols that can occur in numerical expressions, such as brackets, root signs, logarithms, trigonometric and other functions, etc. We emphasize separately that the literal expression contains at least one letter. But it can also contain several identical or different letters.
Now we will give some examples of literal expressions. For example, a + b is a literal expression with the letters a and b. Here is another example of a literal expression 5 · x 3 −3 · x 2 + x − 2.5. And let's give an example of a literal expression complex kind: 
.
Variable expressions
If in a literal expression a letter denotes a quantity that does not take on any one specific value, but can take different meanings, then this letter is called variable and the expression is called variable expression.
Definition.
Variable expression Is a literal expression in which letters (all or some) denote quantities that take on different meanings.
For example, suppose that in the expression x 2 -1 the letter x can take any natural values from the interval from 0 to 10, then x is a variable, and the expression x 2 -1 is an expression with the variable x.
It is worth noting that there can be several variables in an expression. For example, if we consider x and y as variables, then the expression 
is an expression with two variables x and y.
In general, the transition from the concept of an alphabetic expression to an expression with variables occurs in the 7th grade, when they begin to study algebra. Up to this point, literal expressions have modeled some specific tasks. In algebra, however, they begin to look at the expression more generally, without being tied to a specific problem, with the understanding that given expression suitable for a huge number of tasks.
In conclusion of this point, let us pay attention to one more point: by appearance literal expression it is impossible to know whether the letters included in it are variables or not. Therefore, nothing prevents us from considering these letters as variables. In this case, the difference between the terms "literal expression" and "expression with variables" disappears.
Bibliography.
- Maths... 2 cl. Textbook. for general education. institutions with adj. to the electron. carrier. At 2 pm Part 1 / [M. I. Moro, MA Bantova, GV Beltyukova and others] - 3rd ed. - M .: Prosveshenie, 2012 .-- 96 p .: ill. - (School of Russia). - ISBN 978-5-09-028297-0.
- Maths: textbook. for 5 cl. general education. institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., Erased. - M .: Mnemosina, 2007 .-- 280 p .: ill. ISBN 5-346-00699-0.
- Algebra: study. for 7 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M.: Education, 2008 .-- 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.