At school, in a math lesson, students analyze new topics every year. Grade 6 usually studies the module of number - this is an important concept in mathematics, work with which is found further in algebra and higher mathematics. It is very important to initially correctly understand the explanation of the term and understand this topic in order to successfully pass other topics.
To begin with, it should be understood that the absolute value is a parameter in statistics (measured quantitatively) that characterizes the phenomenon under study in terms of its volume. In this case, the phenomenon should be carried out within a certain time frame and with a certain location. Distinguish between values:
- total - suitable for a group of units or the entire population;
- individual - suitable only for working with a unit of a certain set.
The concepts are widely used in statistical measurements, the result of which are indicators that characterize the absolute dimensions of each unit of a certain phenomenon. They are measured in two indicators: natural, i.e. physical units (pieces, people) and conditionally natural. A module in mathematics is a display of these indicators.
What is the modulus of a number?
Important! This definition of "module" is translated from Latin as "measure" and means the absolute value of any natural number.
But this concept also has a geometric explanation, since a module in geometry is equal to the distance from the origin of the coordinate system to the point X, which is measured in the usual units of measurement.
In order to determine this indicator for a number, you should not take into account its sign (minus, plus), but you should remember that it can never be negative. This value on paper is highlighted graphically in the form of square brackets - | a |. Moreover, the mathematical definition is as follows:
| x | = x if x is greater or is zero and -x if less than zero.
The English scientist R. Cotes was the person who first applied this concept in mathematical calculations. But K. Weierstrass, a mathematician from Germany, invented and introduced a graphic symbol.
In the geometry module can be considered by the example of a coordinate line, on which 2 arbitrary points are plotted. Suppose one - A has a value of 5, and the second - B - 6. Upon a detailed study of the drawing, it will become clear that the distance from A to B is 5 units from zero, i.e. origin, and point B is 6 units from the origin. It can be concluded that module points, A = 5, and points B = 6. Graphically, this can be denoted as follows: | 5 | = 5. That is, the distance from a point to the origin is the modulus of this point.
Helpful video: what is a real number module?
Properties
Like any mathematical concept, module has its own mathematical properties:
- It is always positive, so the modulus of a positive value will be itself, for example, the modulus of 6 and -6 is 6. Mathematically, this property can be written as | a | = a, for a> 0;
- Indicators of opposite numbers are equal to each other. This property is clearer in a geometric presentation, since on a straight line these numbers are located in different places, but at the same time they are separated from the origin by an equal number of units. Mathematically, it is written like this: | a | = | -а |;
- The modulus of zero is zero, provided that the real number is zero. This property is supported by the fact that zero is the origin. Graphically it is written like this: | 0 | = 0;
- If you need to find the modulus of two multiplying digits, you should understand that it will be equal to the resulting product. In other words, the product of the quantities A and B = AB, provided that they are positive or negative, and then the product is equal to -AB. Graphically, this can be written as | A * B | = | A | * | B |.
The successful solution of equations with a module depends on knowledge of these properties, which will help anyone to correctly calculate and work with this indicator.
Module properties
Important! The exponent cannot be negative, since it defines the distance, which is always positive.
In equations
In the case of working and solving mathematical inequalities in which module is present, you must always remember that to get the final correct result, you should open the brackets, i.e. open the module sign. This is often the point of the equation.
It is worth remembering that:
- if an expression is written in square brackets, it must be solved: | A + 5 | = A + 5, when A is greater than or equal to zero and 5-A, in case A is less than zero;
- square brackets are most often required to expand regardless of the value of the variable, for example, if the expression in a square is enclosed in parentheses, since the expansion will anyway be a positive number.
It is very easy to solve equations with module by entering values into a coordinate system, since then it is easy to visually see the values and their indicators.
Useful video: module of a real number and its properties
Conclusion
The principle of understanding such a mathematical concept as module is extremely important, since it is used in higher mathematics and other sciences, so you need to be able to work with it.
In contact with
3 NUMBERS positive non-positive negative non-negative Modulus of a real number
4 X if X 0, -X if X 
5 1) | a | = 5 a = 5 or a = - 5 2) | x - 2 | = 5 x - 2 = 5 or x - 2 = - 5 x = 7 3) | 2 x + 3 | = 4 2 x + 3 = or 2 x + 3 = 2 x = x = 4) | x - 4 | = - 2 x =, 5- 3.5 Modulus of a real number
6 X if X 0, -X if X 
7 Working with the textbook on page Formulate the properties of the module 2. What is the geometric meaning of the module? 3. Describe the properties of the function y = | x | according to plan 1) D (y) 2) Zeros of the function 3) Boundedness 4) y n / b, y n / m 5) Monotonicity 6) E (y) 4. How to get from the graph of the function y = | x | graph of the function y = | x + 2 | y = | x-3 | ?
8 X if X 0, -X if X 
13 Independent work"2 - 3" 1. Construct a graph of the function y = | x + 1 | 2. Solve the equation: a) | x | = 2 b) | x | = 0 "3 - 4" 1. Build a graph of the function: 2. Solve the equation: Option 1 Option 2 y = | x-2 | | x-2 | = 3 y = | x + 3 | | x + 3 | = 2 "4 - 5" 1. Build the graph of the function: 2. Solve the equation: y = | 2x + 1 | | 2x + 1 | = 5 y = | 4x + 1 | | 4x + 1 | = 3
15 Great tips 1) | -3 | 2) Number opposite to number (-6) 3) Expression opposite to expression) | - 4: 2 | 5) The opposite expression) | 3 - 2 | 7) | - 3 2 | 8) | 7 - 5 | Answer options: __ _ AEGZHIKNTSHEYA
First, we determine the sign of the expression under the sign of the module, and then we expand the module:
- if the value of the expression is greater than zero, then we simply move it out from under the modulus sign,
- if the expression is less than zero, then we take it out from under the sign of the modulus, while changing the sign, as we did earlier in the examples.
Well, shall we try? Let's estimate:
(Forgot, repeat.)
If, then what sign does it have? Well, of course, !
And, therefore, we expand the sign of the module, changing the sign of the expression:
Understood? Then try it yourself:
Answers:
What other properties does the module have?
If we need to multiply the numbers inside the modulus sign, we can easily multiply the moduli of these numbers !!!
In mathematical terms, the modulus of the product of numbers is equal to the product of the moduli of these numbers.
For instance:
What if we need to separate two numbers (expressions) under the modulus sign?
Yes, the same as with multiplication! Let's split into two separate numbers (expressions) under the modulus sign:
provided that (since you cannot divide by zero).
It is worth remembering one more property of the module:
The modulus of the sum of numbers is always less than or equal to the sum of the moduli of these numbers:
Why is that? Everything is very simple!
As we remember, the module is always positive. But the modulus sign can contain any number: both positive and negative. Suppose the numbers and are both positive. Then the left expression will equal the right expression.
Let's take an example:
If, under the modulus sign, one number is negative and the other is positive, the left expression will always be less than the right one:
It seems that everything is clear with this property, let's consider a couple more useful properties module.
What if we have this expression:
What can we do with this expression? We do not know the value of x, but we already know what, which means.
The number is greater than zero, which means you can simply write:
So we came to another property, which in general can be represented as follows:
And what is this expression equal to:
So, we need to define the sign under the module. Is it necessary to define a sign here?
Of course not, if you remember that any number in a square is always greater than zero! If you don't remember, see the topic. And what happens? Here's what:
Great, huh? Quite convenient. And now a concrete example to fix:
Well, why doubts? We act boldly!
Did you figure it out? Then go ahead and train with examples!
1. Find the value of the expression if.
2. Which numbers have the module equal to?
3. Find the meaning of the expressions:
If not everything is clear yet and there are difficulties in solutions, then let's figure it out:
Solution 1:
So, let's substitute the values into the expression
Solution 2:
As we remember, opposite numbers are equal in absolute value. This means that the value of the modulus is equal to two numbers: and.
Solution 3:
a)
b)
v)
G)
Did you catch everything? Then it's time to move on to the more difficult!
Let's try to simplify the expression
Solution:
So, we remember that the modulus value cannot be less than zero. If the modulus sign is positive, then we can simply discard the sign: the modulus of the number will be equal to this number.
But if under the module sign negative number , then the value of the modulus is equal to the opposite number (that is, the number taken with the "-" sign).
In order to find the modulus of any expression, you first need to find out whether it takes a positive value or negative.
It turns out the value of the first expression under the module.
Therefore, the expression under the modulus sign is negative. The second expression under the modulus sign is always positive, since we are adding two positive numbers.
So, the value of the first expression under the modulus sign is negative, the second is positive:
This means, expanding the modulus sign of the first expression, we must take this expression with the "-" sign. Like this:
In the second case, we simply discard the modulus sign:
Let's simplify the whole expression:
Modulus of a number and its properties (rigorous definitions and proofs)
Definition:
The modulus (absolute value) of a number is the number itself, if, and the number, if:
For instance:
Example:
Simplify the expression.
Solution:
Basic properties of the module
For all:
Example:
Prove property # 5.
Proof:
Suppose there are such that
Let us square the left and right sides of the inequality (this can be done, since both sides of the inequality are always non-negative):
and this is contrary to the definition of a module.
Consequently, such do not exist, and therefore, for all, the inequality
Examples for an independent solution:
1) Prove property 6.
2) Simplify the expression.
Answers:
1) Let's use property # 3:, and since, then
To keep things simple, you need to expand the modules. And in order to expand modules, you need to find out whether the expressions under the module are positive or negative?
a. Let's compare the numbers and and:
b. Now let's compare and:
Add the values of the modules:
The absolute value of a number. Briefly about the main thing.
The modulus (absolute value) of a number is the number itself, if, and the number, if:
Module properties:
- The modulus of a number is a non-negative number:;
- Modules of opposite numbers are equal:;
- The module of the product of two (or more) numbers is equal to the product of their modules:;
- The modulus of the quotient of two numbers is equal to the quotient of their modules:;
- The modulus of the sum of numbers is always less than or equal to the sum of the moduli of these numbers:;
- A constant positive factor can be taken outside the sign of the modulus: at;
Your aim:
clearly know the definition of the modulus of a real number;
understand the geometric interpretation of the modulus of a real number and be able to apply it in solving problems;
know the properties of the module and be able to apply when solving problems;
be able to understand the distance between two points of the coordinate line and be able to use it when solving problems.
Input information
The concept of the modulus of a real number. The modulus of a real number is called the number itself, if, and the opposite number is the number if< 0.
The modulus of the number is denoted and recorded:
Geometric module interpretation . Geometrically the modulus of a real number is the distance from the point representing the given number on the coordinate line to the origin.
Solving equations and inequalities with moduli based geometric meaning module. Using the concept of "distance between two points of the coordinate line", one can solve equations of the form or inequalities of the form, where any of the signs can be used instead of a sign.
Example. Let's solve the equation.
Solution. Let us reformulate the problem geometrically. Since is the distance on the coordinate line between points with coordinates and, therefore, it is required to find the coordinates of such points, the distance from which to points with coordinate 1 is 2.
In short, on the coordinate line, find the set of coordinates of points, the distance from which to the point with coordinate 1 is 2.
Let's solve this problem. Let's mark a point on the coordinate line, the coordinate of which is 1 (Fig. 6) Two units from this point are points whose coordinates are equal to -1 and 3. This means that the desired set of coordinates of points is a set consisting of numbers -1 and 3.
Answer: -1; 3.
How to find the distance between two points of a coordinate line. Number expressing the distance between points and , called the distance between numbers and .
For any two points and a coordinate line, the distance
.
Basic properties of the real number module:
3. 
;
7. 
;
8. 
;
9. 
;
When we have:
11. if only when or;
12. if only when;
13. then only when or;
14. if only when;
11. if only when.
Practical part
Exercise 1. Take clear sheet paper and write your answers to these oral exercises below on it.
Check your answers against the answers or short instructions at the end of the learning element under the heading “Your helper”.
1. Expand the module sign:
a) | –5 |; b) | 5 |; c) | 0 |; d) | p |.
2. Compare the numbers with each other:
a) || and -; c) | 0 | and 0; e) - | –3 | and –3; g) –4 | a| and 0;
b) | –p | and p; d) | –7.3 | and –7.3; e) | a| and 0; h) 2 | a| and | 2 a|.
3. How to write with the modulus sign that at least one of the numbers a, b or With nonzero?
4. How to write with the equal sign that each of the numbers a, b and With is equal to zero?
5. Find the meaning of the expression:
a) | a| – a; b) a + |a|.
6. Solve the equation:
a) | X| = 3; c) | X| = –2; e) | 2 X– 5| = 0;
b) | X| = 0; d) | X- 3 | = 4; f) | 3 X– 7| = – 9.
7. What about numbers X and at, if:
a) | X| = X; b) | X| = –X; c) | X| = |at|?
8. Solve the equation:
a) | X– 2| = X- 2; c) | X– 3| =|7 – X|;
b) | X– 2| = 2 – X; d) | X– 5| =|X– 6|.
9. What about the number at if the equality holds:
a) ï Xï = at; b) ï Xï = – at ?
10. Solve the inequality:
a) | X| > X; c) | X| > –X; e) | X| £ X;
b) | X| ³ X; d) | X| ³ – X; e) | X| £ – X.
11. Specify all values of a for which equality holds:
a) | a| = a; b) | a| = –a; v) a – |–a| = 0; d) | a|a= –1; e) = 1.
12. Find all meanings b, for which the inequality holds:
a) | b| ³ 1; b) | b| < 1; в) |b| £ 0; d) | b| ³ 0; e) 1< |b| < 2.
You may have encountered some of the following activities in your math class. Decide yourself which of the following tasks you need to complete. In case of difficulty, refer to the heading "Your assistant", for advice from a teacher or for help from a friend.
Task 2. Based on the definition of the modulus of a real number, solve the equation:
Task 4. Distance between points representing real numbers α and β on the coordinate line is equal to | α – β |. Use this to solve the equation.
Module or absolute value a real number is called this number itself if X is non-negative, and the opposite number, i.e. -x if X negatively:
Obviously, but the definition, | x | > 0. The following properties of absolute values are known:
- 1) hu| = | dg | | g / 1;
- 2> - -H;
Haveat
- 3) | x + r / |
- 4) | dt-g / |
Difference modulus of two numbers X - a| there is a distance between points X and a on the number line (for any X and a).
This implies, in particular, that the solutions of the inequality X - a 0) are all points X interval (a- d, a + s), i.e. numbers satisfying the inequality a-d + G.
Such an interval (a- 8, a+ d) is called an 8-neighborhood of the point a.
Basic properties of functions
As we have already stated, all quantities in mathematics are divided into constants and variables. Constant is called a quantity that retains the same value.
Variable is called a quantity that can take on various numerical values.
Definition 10.8. Variable at called function on a variable x if, according to some rule, each value x e X assigned a specific value at f y; the independent variable x is usually called the argument, and the scope X its change is called the scope of the function.
The fact that at there is a function otx, most often expressed by a symbolic notation: at= / (x).
There are several ways to define functions. Three are considered to be the main ones: analytical, tabular and graphical.
Analytical way. This method consists in defining a relationship between an argument (independent variable) and a function in the form of a formula (or formulas). Usually, f (x) is some analytical expression containing x. In this case, the function is said to be defined by a formula, for example, at= 2x + 1, at= tgx etc.
Tabular The way of defining the function is that the function is defined by a table containing the values of the argument x and the corresponding values of the function /(.r). Examples are tables of the number of crimes for a certain period, tables of experimental measurements, a table of logarithms.
Graphic way. Let a system of Cartesian rectangular coordinates be given on the plane hoy. The geometric interpretation of the function is based on the following.
Definition 10.9. Schedule function is called the locus of points of the plane, coordinates (x, y) which satisfy the condition: u-ah).
A function is called graphically defined if its graph is drawn. The graphical method is widely used in experimental measurements using recorders.
Having before your eyes a visual graph of functions, it is easy to imagine many of its properties, which makes the graph an indispensable tool for studying a function. Therefore, plotting is the most important (usually final) part of the study of a function.
Each method has both advantages and disadvantages. So, the advantages of the graphic method include its clarity, the disadvantages - its inaccuracy and limited presentation.
Let us now proceed to consider the main properties of functions.
Even and odd parity. Function y = f (x) called even, if for any X the condition is satisfied f (-x) = f (x). If for X from the domain of definition, the condition f (- x) = - f (x) is satisfied, then the function is called odd. A function that is not even or odd is called a function general view.
- 1) y = x 2 is an even function, since f (-x) = (-x) 2 = x 2, i.e. / (- x) = / (. d);
- 2) y = x 3 - odd function, since (-x) 3 = -x 3, i.e. / (- x) = - / (x);
- 3) y = x 2 + x is a general function. Here / (x) = x 2 + x, / (- x) = (-x) 2 +
- (-x) = x 2 - x, / (- x) * / (x); / (- x) - / "/ (- x).
The graph of an even function is symmetrical about the axis Oh, and the graph of the odd function is symmetric about the origin.
Monotone. Function at= / (x) is called increasing in between X, if for any x, x 2 e X from the inequality x 2> x, it follows that f (x 2)> f (x,). Function at= / (x) is called decreasing if from x 2> x, it follows / (x 2) (x,).
The function is called monotonous in between X, if it either increases throughout this interval, or decreases in it.
For example, the function y = x 2 decreases on (- °°; 0) and increases on (0; + °°).
Note that we have given the definition of a function that is monotone in the strict sense. In general, monotone functions include non-decreasing functions, i.e. such that from x 2> x, it follows f (x 2)> f (x,), and non-increasing functions, i.e. such for which from x 2> x, it follows / (x 2)
Limitation. Function at= / (x) is called limited in between X, if there is such a number M> 0 such that | / (x) | M for any x e X.
For example, the function at =-
bounded on the whole number line, so
Periodicity. Function at = f (x) called periodic if there is such a number T^ Oh what f (x + T = f (x) for all X from the scope of the function.
In this case T is called the period of the function. Obviously if T - function period y = f (x), then the periods of this function are also 2Г, 3 T etc. Therefore, the period of a function is usually called the smallest positive period (if it exists). For example, the functions f = cos.r has a period T = 2P, and the function y = tg Zx - period n / 3.