module or absolute value a real number is called the number itself, if X is non-negative, and the opposite number, i.e. -x if X negative:
Obviously, but by definition, |x| > 0. The following properties of absolute values are known:
- 1) hu| = |dg| |r/1;
- 2>--H;
Atat
- 3) |x+r/|
- 4) |dt-g/|
Difference modulus of two numbers X - a| is the distance between the points X and a on the number line (for any X and a).
From this it follows, in particular, that the solutions of the inequality X - a 0) are all points X interval (a- g, a + c), i.e. numbers satisfying the inequality a-d + G.
Such an interval (a- 8, a+ d) is called the 8-neighbourhood of the point a.
Basic properties of functions
As we have already stated, all quantities in mathematics are divided into constants and variables. Constant value is called a quantity that retains the same value.
variable is a quantity that can take on various numerical values.
Definition 10.8. variable at called function of the variable x, if, according to some rule, each value of x e X assigned a specific value at e U; 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 in symbolic notation: at= /(x).
There are several ways to define functions. Three are considered to be the main ones: analytical, tabular and graphic.
Analytical way. This method consists in setting the relationship between the argument (independent variable) and the function in the form of a formula (or formulas). Usually /(x) is some analytic 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 to define a function is that the function is defined by a table containing the values of the argument x and the corresponding values of the function f(.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 ho. 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, the coordinates (x, y) which satisfy the condition: w-ah).
A function is said to be given graphically if its graph is drawn. The graphical method is widely used in experimental measurements using self-recording devices.
Having a visual graph of functions in front of your eyes, it is not difficult 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 the function.
Each method has both its advantages and disadvantages. So, the advantages of the graphical method include its visibility, the disadvantages - its inaccuracy and limited presentation.
Let us now turn to the consideration of the main properties of functions.
Even and odd. Function y = f(x) called even, if for any X the condition f(-x) = f(x). If for X from the domain of definition, the condition f(-x) = -/(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) =/(.r);
- 2) y= x 3 - odd function, since (-x) 3 \u003d -x 3, t.s. /(-x) = -/(x);
- 3) y= x 2 + x is a general function. Here / (x) \u003d x 2 + x, / (-x) \u003d (-x) 2 +
- (-x) \u003d x 2 - x, / (-x) * / (x); / (-x) - / "/ (-x).
The graph of an even function is symmetrical about the axis Oh, and the graph of an odd function is symmetrical with respect to 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 / (x 2) > / (x,). Function at=/(x) is called waning, if from x 2 > x, it follows / (x 2) (x,).
The function is called monotonous in between x, if it either increases over this entire interval or decreases over it.
For example, the function y= x 2 decreases by (-°°; 0) and increases by (0; +°°).
Note that we have given the definition of a monotonic function in the strict sense. In general, monotonic functions include nondecreasing functions, i.e. those for which from x 2 > x, it follows / (x 2) > / (x,), and non-increasing functions, i.e. those 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 entire number line, so
Periodicity. Function at = f(x) called periodical 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 2T, 3 T etc. Therefore, usually the period of a function is the smallest positive period (if it exists). For example, the functions / = cos.r has a period T= 2P, and the function y= tg Zx - period p/3.
Your goal:
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 in solving problems;
be able to understand the distance between two points of a coordinate line and be able to use it in solving problems.
input information
The concept of the modulus of a real number. The modulus of a real number is called this number itself, if , and the number opposite to it, if< 0.
The modulus of a number is denoted and written down:
Geometric interpretation of the module . 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 modules based on the geometric meaning of the module. Using the concept of “distance between two points of a coordinate line”, one can solve equations of the form or inequalities of the form , where any of the signs can be used instead of the sign.
Example. Let's solve the equation.
Decision. Let us reformulate the problem geometrically. Since is the distance on the coordinate line between points with coordinates and , it means that it is required to find the coordinates of such points, the distance from which to points with coordinate 1 is equal to 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 equal to 2.
Let's solve this problem. We mark a point on the coordinate line, the coordinate of which is equal to 1 (Fig. 6). Points whose coordinates are equal to -1 and 3 are removed two units from this point. Hence, the required 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 on a coordinate line. A 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 modulus of a real number:
3. 
;
7. 
;
8. 
;
9. 
;
When we have:
11. then only when or ;
12. then only when ;
13. then only when or ;
14. then only when ;
11. then only when .
Practical part
Exercise 1. Take a blank piece of paper and on it write down the answers to these oral exercises below.
Check your answers with the answers or brief instructions placed at the end of the learning element under the heading “Your Helper”.
1. Expand module sign:
a) |–5|; b) |5|; c) |0|; d) |p|.
2. Compare the numbers:
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; f) | a| and 0; h) 2| a| and |2 a|.
3. How, using the modulus sign, to write that at least one of the numbers a, b or with different from zero?
4. How to use the equal sign to write that each of the numbers a, b and with equal to zero?
5. Find the value 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 can be said 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 can be said about the number at if the equality holds:
a) i Xï = at; b) i Xï = – at ?
10. Solve the inequality:
a) | X| > X; c) | X| > –X; e) | X| £ X;
b) | X| ³ X; d) | X| ³ – X; f) | X| £ – X.
11. List all values of a for which equality holds:
a) | a| = a; b) | a| = –a; in) a – |–a| =0; d) | a|a= –1; e) = 1.
12. Find all values b, for which the following inequality holds:
a) | b| ³ 1; b) | b| < 1; в) |b| £0; d) | b| ³ 0; e) 1< |b| < 2.
You may have come across some of the following assignments in math classes. Decide which of the following tasks you need to complete. In case of difficulty, refer to the section “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 dots representing real numbers α and β on the coordinate line, is equal to | α – β |. Use this to solve the equation.
At school, in the math lesson every year, students analyze new topics. Grade 6 usually studies the modulus of a number - this is an important concept in mathematics, work with which is found later 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), which characterizes the phenomenon under study in terms of its volume. In this case, the phenomenon must be carried out within a certain time frame and with a certain location. Distinguish values:
- summary - suitable for a group of units or the entire population;
- individual - suitable only for working with a unit of a certain population.
The concepts are widely used in statistical measurements, the result of which are indicators characterizing 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 the 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, one should not take into account its sign (minus, plus), but it should be remembered that it can never be negative. This value on paper is highlighted graphically in the form of square brackets - |a|. In this case, the mathematical definition is:
|x| = x if x is greater than or equal to zero and -x if less than zero.
The English scientist R. Kotes was the person who first applied this concept in mathematical calculations. But K. Weierstrass, a mathematician from Germany, invented and put into use a graphic symbol.
In the module geometry, we can consider 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 located 6 units from the origin. We can conclude that module points, A = 5, and points B = 6. Graphically, this can be denoted as follows: | 5 | = 5. That is, the distance from the point to the origin is the modulus of the given point.
Useful video: what is the modulus of a real number?
Properties
Like any mathematical concept, module has its own mathematical properties:
- It is always positive, so the modulus of a positive value is itself, for example, the modulus of 6 and -6 is 6. Mathematically, this property can be written as |a| = a, for a> 0;
- The 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, this is written as follows: |a| = |-a|;
- 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, this is written as follows: |0| = 0;
- If you want 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|.
Successful solution of equations with modulus 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 because it defines the distance, which is always positive.
In the equations
In the case of working and solving mathematical inequalities in which the module is present, it is always necessary to remember that in order to obtain the final correct result, you should open the brackets, i.e. open sign module . Often, this is the meaning of the equation.
It is worth remembering that:
- if an expression is written in square brackets, it must be solved: |A + 5| \u003d A + 5, when A is greater than or equal to zero and 5-A, in the case of A less than zero;
- square brackets most often should be expanded regardless of the variable's values, for example, if the expression in the square is enclosed in brackets, since the expansion will be a positive number anyway.
It is very easy to solve equations with module by entering values into the coordinate system, because then it is easy to see visually the values and their indicators.
Useful video: real number modulus and its properties
Conclusion
The principle of understanding such a mathematical concept as a 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
In this article, we will analyze in detail the absolute value of a number. We will give various definitions of the modulus of a number, introduce notation and give graphic illustrations. In this case, we consider various examples of finding the modulus of a number by definition. After that, we list and justify the main properties of the module. At the end of the article, we will talk about how the modulus of a complex number is determined and found.
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Modulus of number - definition, notation and examples
First we introduce modulus designation. The module of the number a will be written as , that is, to the left and to the right of the number we will put vertical lines that form the sign of the module. Let's give a couple of examples. For example, modulo -7 can be written as ; module 4,125 is written as , and module is written as .
The following definition of the module refers to, and therefore, to, and to integers, and to rational and irrational numbers, as to the constituent parts of the set of real numbers. We will talk about the modulus of a complex number in.
Definition.
Modulus of a is either the number a itself, if a is a positive number, or the number −a, the opposite of the number a, if a is a negative number, or 0, if a=0.
The voiced definition of the modulus of a number is often written in the following form 
, this notation means that if a>0 , if a=0 , and if a<0
.
The record can be represented in a more compact form 
. This notation means that if (a is greater than or equal to 0 ), and if a<0
.
There is also a record 
. Here, the case when a=0 should be explained separately. In this case, we have , but −0=0 , since zero is considered a number that is opposite to itself.
Let's bring examples of finding the modulus of a number with a given definition. For example, let's find modules of numbers 15 and . Let's start with finding . Since the number 15 is positive, its modulus is, by definition, equal to this number itself, that is, . What is the modulus of a number? Since is a negative number, then its modulus is equal to the number opposite to the number, that is, the number 
. Thus, .
In conclusion of this paragraph, we give one conclusion, which is very convenient to apply in practice when finding the modulus of a number. From the definition of the modulus of a number it follows that the modulus of a number is equal to the number under the sign of the modulus, regardless of its sign, and from the examples discussed above, this is very clearly visible. The voiced statement explains why the modulus of a number is also called the absolute value of the number. So the modulus of a number and the absolute value of a number are one and the same.
Modulus of a number as a distance
Geometrically, the modulus of a number can be interpreted as distance. Let's bring determination of the modulus of a number in terms of distance.
Definition.
Modulus of a is the distance from the origin on the coordinate line to the point corresponding to the number a.
This definition is consistent with the definition of the modulus of a number given in the first paragraph. Let's explain this point. The distance from the origin to the point corresponding to a positive number is equal to this number. Zero corresponds to the origin, so the distance from the origin to the point with coordinate 0 is zero (no single segment and no segment that makes up any fraction of the unit segment needs to be postponed in order to get from point O to the point with coordinate 0). The distance from the origin to a point with a negative coordinate is equal to the number opposite to the coordinate of the given point, since it is equal to the distance from the origin to the point whose coordinate is the opposite number.
For example, the modulus of the number 9 is 9, since the distance from the origin to the point with coordinate 9 is nine. Let's take another example. The point with coordinate −3.25 is at a distance of 3.25 from point O, so 
.
The sounded definition of the modulus of a number is a special case of defining the modulus of the difference of two numbers.
Definition.
Difference modulus of two numbers a and b is equal to the distance between the points of the coordinate line with coordinates a and b .
That is, if points on the coordinate line A(a) and B(b) are given, then the distance from point A to point B is equal to the modulus of the difference between the numbers a and b. If we take point O (reference point) as point B, then we will get the definition of the modulus of the number given at the beginning of this paragraph.
Determining the modulus of a number through the arithmetic square root
Sometimes found determination of the modulus through the arithmetic square root.
For example, let's calculate the modules of the numbers −30 and based on this definition. We have . Similarly, we calculate the modulus of two-thirds: 
.
The definition of the modulus of a number in terms of the arithmetic square root is also consistent with the definition given in the first paragraph of this article. Let's show it. Let a be a positive number, and let −a be negative. Then 
and 
, if a=0 , then 
.
Module properties
The module has a number of characteristic results - module properties. Now we will give the main and most commonly used of them. When substantiating these properties, we will rely on the definition of the modulus of a number in terms of distance.
Let's start with the most obvious module property − modulus of a number cannot be a negative number. In literal form, this property has the form for any number a . This property is very easy to justify: the modulus of a number is the distance, and the distance cannot be expressed as a negative number.
Let's move on to the next property of the module. The modulus of a number is equal to zero if and only if this number is zero. The modulus of zero is zero by definition. Zero corresponds to the origin, no other point on the coordinate line corresponds to zero, since each real number is associated with a single point on the coordinate line. For the same reason, any number other than zero corresponds to a point other than the origin. And the distance from the origin to any point other than the point O is not equal to zero, since the distance between two points is equal to zero if and only if these points coincide. The above reasoning proves that only the modulus of zero is equal to zero.
Move on. Opposite numbers have equal modules, that is, for any number a . Indeed, two points on the coordinate line, whose coordinates are opposite numbers, are at the same distance from the origin, which means that the modules of opposite numbers are equal.
The next module property is: the modulus of the product of two numbers is equal to the product of the modules of these numbers, i.e, . By definition, the modulus of the product of numbers a and b is either a b if , or −(a b) if . It follows from the rules of multiplication of real numbers that the product of moduli of numbers a and b is equal to either a b , , or −(a b) , if , which proves the considered property.
The modulus of the quotient of dividing a by b is equal to the quotient of dividing the modulus of a by the modulus of b, i.e, . Let us justify this property of the module. Since the quotient is equal to the product, then . By virtue of the previous property, we have 
. It remains only to use the equality , which is valid due to the definition of the modulus of the number.
The following module property is written as an inequality: 
, a , b and c are arbitrary real numbers. The written inequality is nothing more than triangle inequality. To make this clear, let's take the points A(a) , B(b) , C(c) on the coordinate line, and consider the degenerate triangle ABC, whose vertices lie on the same line. By definition, the modulus of the difference is equal to the length of the segment AB, - the length of the segment AC, and - the length of the segment CB. Since the length of any side of a triangle does not exceed the sum of the lengths of the other two sides, the inequality 
, therefore, the inequality also holds.
The inequality just proved is much more common in the form 
. The written inequality is usually considered as a separate property of the module with the formulation: “ The modulus of the sum of two numbers does not exceed the sum of the moduli of these numbers". But the inequality directly follows from the inequality , if we put −b instead of b in it, and take c=0 .
Complex number modulus
Let's give determination of the modulus of a complex number. Let us be given complex number, written in algebraic form , where x and y are some real numbers, representing, respectively, the real and imaginary parts of a given complex number z, and is an imaginary unit.
First, we define the sign of the expression under the sign of the module, and then expand the module:
- if the value of the expression is greater than zero, then we simply take it out from under the module sign,
- if the expression is less than zero, then we take it out from under the sign of the module, while changing the sign, as we did earlier in the examples.
Well, shall we try? Let's estimate:
(Forgot, repeat.)
If so, what is the sign? Well, of course, !
And, therefore, we reveal the sign of the module by changing the sign of the expression:
Got it? Then try it yourself:
Answers:
What other properties does the module have?
If we need to multiply the numbers inside the modulo sign, we can safely multiply the modulus of these numbers!!!
In mathematical terms, the modulus of the product of numbers is equal to the product of the modules of these numbers.
For example:
But what if we need to divide two numbers (expressions) under the modulo sign?
Yes, the same as with multiplication! Let's break it into two separate numbers (expressions) under the module sign:
provided that (since you cannot divide by zero).
It is worth remembering one more property of the module:
The module of the sum of numbers is always less than or equal to the sum of the modules of these numbers:
Why is that? Everything is very simple!
As we remember, the modulus is always positive. But under the sign of the module can be any number: both positive and negative. Assume that the numbers and are both positive. Then the left expression will be equal to the right expression.
Let's look at 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 of the 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:
What is the meaning of this expression:
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 squared is always greater than zero! If you don't remember, see the topic. And what happens? And here's what:
It's great, right? Quite convenient. Now for a specific example:
Well, why doubt? Let's act boldly!
Did you understand everything? Then go ahead and practice with examples!
1. Find the value of the if expression.
2. What numbers have the module equal?
3. Find the meaning of expressions:
If not everything is clear yet and there are difficulties in making decisions, then let's figure it out:
Solution 1:
So, let's substitute the values in the expression
Solution 2:
As we remember, opposite numbers are modulo equal. This means that the value of the modulus is equal to two numbers: and.
Solution 3:
a)
b)
in)
G)
Did you catch everything? Then it's time to move on to something more complicated!
Let's try to simplify the expression
Decision:
So, we remember that the modulus value cannot be less than zero. If the number under 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 modulus sign is a negative number, then the value of the module 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 a negative one.
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, when expanding the sign of the modulus of the first expression, we must take this expression with the “-” sign. Like this:
In the second case, we simply drop the modulo sign:
Let's simplify this expression in its entirety:
Modulus of a number and its properties (strict definitions and proofs)
Definition:
The modulus (absolute value) of a number is the number itself if, and the number if:
For example:
Example:
Simplify the expression.
Decision:
Basic properties of the module
For all:
Example:
Prove property #5.
Proof:
Let us assume that there are
Let's square the left and right parts of the inequality (this can be done, since both parts of the inequality are always non-negative):
and this contradicts the definition of a module.
Consequently, there are no such ones, which means that for all the inequality
Examples for an independent solution:
1) Prove property #6.
2) Simplify the expression.
Answers:
1) Let's use property No. 3: , and since, then
To simplify, you need to expand the modules. And to expand the 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:
We add up 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 module of the quotient of two numbers is equal to the quotient of their modules: ;
- The module of the sum of numbers is always less than or equal to the sum of the modules of these numbers: ;
- A constant positive factor can be taken out of the modulus sign: at;
