First we define the expression sign under the module sign, and then we expand the module:
- if the value of the expression is greater than zero, then we simply remove it from under the modulus sign,
- if the expression is less than zero, then we remove it from under the modulus sign, changing the sign, as we did earlier in the examples.
Well, shall we try? Let's evaluate:
(Forgot, Repeat.)
If so, what sign does it have? Well, of course, !
And, therefore, we expand 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 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 example:
What if we need to divide two numbers (expressions) under the modulus sign?
Yes, the same as with multiplication! Let's break it down 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 modulus is always positive. But under the modulus sign there can be any number: both positive and negative. Let's 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:
Everything seems clear with this property, let’s look at a couple more useful properties of the module.
What if we have this expression:
What can we do with this expression? The value of x is unknown to us, but we already know what, which means.
The number is greater than zero, which means you can simply write:
So we come to another property, which in general can be represented as follows:
What does this expression equal:
So, we need to define the sign under the modulus. 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. So what happens? Here's what:
Great, right? Quite convenient. And now a specific example to reinforce:
Well, why the doubts? Let's act boldly!
Have you figured it all out? Then go ahead and practice with examples!
1. Find the value of the expression if.
2. Which numbers have the same modulus?
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 and into the expression
Solution 2:
As we remember, opposite numbers are equal in modulus. This means that the modulus value is equal to two numbers: and.
Solution 3:
A)
b)
V)
G)
Did you catch everything? Then it's time to move on to something more complex!
Let's try to simplify the expression
Solution:
So, we remember that the modulus value cannot be less than zero. If the modulus sign has a positive number, then we can simply discard the sign: the modulus of the number will be equal to this number.
But if there is a negative number under the modulus sign, then the modulus value 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 or negative value.
It turns out that 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 that when 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 this expression in its entirety:
Module of 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 example:
Example:
Simplify the expression.
Solution:
Basic properties of the module
For all:
Example:
Prove property No. 5.
Proof:
Let us assume that there are such that
Let's 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 contradicts the definition of a module.
Consequently, such people do not exist, which means that the inequality holds for all
Examples for independent solutions:
1) Prove property No. 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 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: ;
- The modules of opposite numbers are equal: ;
- The modulus of the product of two (or more) numbers is equal to the product of their moduli: ;
- The modulus of the quotient of two numbers is equal to the quotient of their moduli: ;
- 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 multiplier can be taken out of the modulus sign: at;
At school, in math class every year, students study new topics. 6th grade usually studies the modulus of number - this is an important concept in mathematics, work with which is later encountered 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 complete other topics.
To begin with, you should understand that the absolute value is a parameter in statistics (measured quantitatively) that characterizes the phenomenon being studied in terms of its volume. In this case, the phenomenon must occur within a certain time frame and with a certain location. There are different meanings:
- total – suitable for a group of units or the entire population;
- individual - suitable only for working with a unit of a certain aggregate.
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 point X, which is measured in conventional 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 is highlighted graphically on paper 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. 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 geometry, module can be considered using 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, has a value of 6. Upon detailed study of the drawing, it will become clear that the distance from A to B is 5 units from zero, i.e. the 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 indicated as follows: | 5 | = 5. That is, the distance from a point to the origin is the modulus of a 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 will be itself, for example, the modulus of the number 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 geometric terms, 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| = |-a|;
- The modulus of zero is zero, provided that the real number is zero. This property is confirmed by the fact that zero is the origin. Graphically this 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 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 modulus depends on knowledge of these properties, which will help anyone 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 module is present, you must always remember that in order to obtain the final correct result, you should open the brackets, i.e. open module sign. Often, this is 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, if A is greater than or equal to zero and 5-A, if A is less than zero;
- Square brackets should most often be expanded regardless of the value of the variable, for example, if the brackets enclose a squared expression, since expansion will result in a positive number in any case.
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: the modulus 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
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 when solving problems;
know the properties of the module and be able to apply it when solving problems;
be able to imagine the distance between two points on a 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 the number itself, if, and its opposite number, if< 0.
The modulus of the number is denoted and written:
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 moduli based on the geometric meaning of the modulus. Using the concept of “the distance between two points of a coordinate line,” you 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 , it means that it is necessary to find the coordinates of such points, the distance from which to points with coordinate 1 is equal to 2.
In short, on a 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. Let us mark a point on the coordinate line whose coordinate is equal to 1 (Fig. 6). The points whose coordinates are equal to -1 and 3 are two units away from this point. This means that the required set of coordinates of points is a set consisting of the 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 , is 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 if or ;
12. then only when ;
13. then only if or ;
14. then only when ;
11. then only when .
Practical part
Exercise 1. Take a blank sheet of paper and write down the answers to all of the speaking exercises below.
Check your answers with the answers or brief instructions located 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:
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 use 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 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; e) |3 X– 7| = – 9.
7. What can we say 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 you say about the number? at, if 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. List all values of a for which the equality holds:
a) | A| = A; b) | A| = –A; V) A – |–A| =0; d) | A|A= –1; d) = 1.
12. Find all values 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 types of tasks in mathematics lessons. Decide for yourself which of the following tasks you need to complete. If you have any difficulties, please refer to the “Your Helper” section, 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 |. Using this, solve the equation.
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 provide graphic illustrations. At the same time, let's look at various examples of finding the modulus of a number by definition. After this, we will list and justify the main properties of the module. At the end of the article, we’ll talk about how the modulus of a complex number is determined and found.
Page navigation.
Number module - definition, notation and examples
First we introduce number modulus designation. We will write the modulus of the number a as , that is, to the left and right of the number we will put vertical dashes to form the modulus sign. Let's give a couple of examples. For example, module −7 can be written as ; module 4.125 is written as , and the module has a notation of the form .
The following definition of modulus refers to , and therefore to , and to integers, and to rational, and to irrational numbers, as constituent parts of the set of real numbers. We will talk about the modulus of a complex number in.
Definition.
Modulus of number a– this 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 entry means that if a>0 , if a=0 , and if a<0
.
The record can be presented 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 the entry 
. Here we should separately explain the case when a=0. In this case we have , but −0=0, since zero is considered a number that is opposite to itself.
Let's give examples of finding the modulus of a number using a stated definition. For example, let's find the modules of the numbers 15 and . Let's start by finding . Since the number 15 is positive, its modulus, by definition, is equal to this number itself, that is, . What is the modulus of a number? Since is a negative number, its modulus is equal to the number opposite to the number, that is, the number 
. Thus, .
To conclude this point, we present one conclusion that is very convenient to use 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 modulus sign without taking into account its sign, and from the examples discussed above this is very clearly visible. The stated statement explains why the module of a number is also called 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 give determining the modulus of a number through distance.
Definition.
Modulus of number a– this 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 clarify 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, therefore the distance from the origin to the point with coordinate 0 is equal to zero (you do not need to set aside a single unit segment and not a single segment that makes up any fraction of a unit segment in order to get from point O to a 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 this 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 equal to 9, since the distance from the origin to the point with coordinate 9 is equal to nine. Let's give another example. The point with coordinate −3.25 is located at a distance of 3.25 from point O, so 
.
The stated definition of the modulus of a number is a special case of the definition of the modulus of the difference of two numbers.
Definition.
Modulus of the difference 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 (origin) as point B, then we get the definition of the modulus of a number given at the beginning of this paragraph.
Determining the modulus of a number using the arithmetic square root
Occasionally occurs determining modulus via arithmetic square root.
For example, let's calculate the moduli of the numbers −30 and based on this definition. We have. Similarly, we calculate the module of two thirds: 
.
The definition of the modulus of a number through 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 a negative number. Then 
And 
, if a=0 , then 
.
Module properties
The module has a number of characteristic results - module properties. Now we will present the main and most frequently used of them. When justifying 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 property of the module - The 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 a distance, and distance cannot be expressed as a negative number.
Let's move on to the next module property. The modulus of a number is 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 different from the origin. And the distance from the origin to any point other than point O is not zero, since the distance between two points is zero if and only if these points coincide. The above reasoning proves that only the modulus of zero is equal to zero.
Go ahead. Opposite numbers have equal modules, that is, for any number a. Indeed, two points on the coordinate line, the coordinates of which are opposite numbers, are at the same distance from the origin, which means the modules of the opposite numbers are equal.
The following property of the module is: The modulus of the product of two numbers is equal to the product of the moduli of these numbers, that is, . By definition, the modulus of the product of numbers a and b is equal to either a·b if , or −(a·b) if . From the rules of multiplication of real numbers it follows that the product of the moduli of numbers a and b is equal to either a·b, , or −(a·b) if , which proves the property in question.
The modulus of the quotient of a divided by b is equal to the quotient of the modulus of a number divided by the modulus of b, that is, . 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 
. All that remains is to use the equality , which is valid by virtue of the definition of the modulus of a number.
The following property of a module 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 points A(a), B(b), C(c) on the coordinate line, and consider a 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, then the inequality is true 
, therefore, the inequality is also true.
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 follows directly from the inequality if we put −b instead of b and take c=0.
Modulus of a complex number
Let's give definition of the modulus of a complex number. May it be given to us 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 the imaginary unit.
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 p. 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 obtain 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. Graph 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. Graph the function: 2. Solve the equation: y = |2x+1| |2x+1|=5 y = |4x+1| |4x+1|=3
15 Advice from the great 1) |-3| 2)Number opposite to number (-6) 3) Expression opposite to expression) |- 4: 2| 5) Expression opposite to expression) |3 - 2| 7) |- 3 2| 8) | 7 - 5| Possible answers: __ _ AEGZHIKNTSHEYA
