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The term (module) literally translated from Latin means “measure”. This concept was introduced into mathematics by the English scientist R. Cotes. And the German mathematician K. Weierstrass introduced the modulus sign - a symbol that denotes this concept when writing.
First this concept studied in mathematics according to the 6th grade program high school. According to one definition, the modulus is an absolute value real number. In other words, to find out the modulus of a real number, you need to discard its sign.
Graphically absolute value A denoted as |a|.
Main distinguishing feature This concept is that it is always a non-negative quantity.
Numbers that differ from each other only in sign are called opposite numbers. If a value is positive, then its opposite is negative, and zero is its opposite.
Geometric meaning
If we consider the concept of module from the standpoint of geometry, then it will denote the distance that is measured in unit segments from the origin of coordinates to given point. This definition fully reveals the geometric meaning of the term being studied.
Graphically this can be expressed as follows: |a| = OA.
Properties of absolute value
All will be discussed below mathematical properties this concept and ways of writing it in the form literal expressions:
Features of solving equations with modulus
If we talk about the decision mathematical equations and inequalities that contain module, then you must remember that to solve them you will need to open this sign.
For example, if the sign of the absolute value contains some mathematical expression, then before opening the module, it is necessary to take into account the current mathematical definitions.
|A + 5| = A + 5, if, A is greater than or equal to zero.
5-A, if, A value less than zero.
In some cases, the sign can be revealed unambiguously for any value of the variable.
Let's look at another example. Let's construct a coordinate line on which we mark everything numeric values absolute value of which there will be 5.
First you need to draw a coordinate line, mark the origin of coordinates on it and set the size of a unit segment. In addition, the straight line must have a direction. Now on this straight line it is necessary to apply markings that will be equal to the size of a unit segment.
Thus, we can see that on this coordinate line there will be two points of interest to us with values 5 and -5.
Lesson Objectives
Introduce students to this mathematical concept, as a modulus of a number;
To teach schoolchildren the skills of finding modules of numbers;
Reinforce the learned material by completing various tasks;
Tasks
Strengthen children's knowledge about the modulus of numbers;
Using the solution test tasks check how students have mastered the material studied;
Continue to instill interest in mathematics lessons;
Educate schoolchildren logical thinking, curiosity and perseverance.
Lesson Plan
1. General concepts and definition of the modulus of a number.
2. Geometric meaning of the module.
3. The modulus of a number and its properties.
4. Solving equations and inequalities that contain the modulus of a number.
5. Historical reference about the term “modulus of a number”.
6. Assignment to consolidate knowledge of the covered topic.
7. Homework.
General concepts about the modulus of a number
The modulus of a number is usually called the number itself if it does not have negative value, or the same number is negative, but with the opposite sign.
That is, the modulus of a non-negative real number a is the number itself:
And, the modulus of a negative real number x will be the opposite number:
In recording it will look like this:
For a more accessible understanding, let's give an example. So, for example, the modulus of the number 3 is 3, and also the modulus of the number -3 is 3.
It follows from this that the modulus of a number means an absolute value, that is, its absolute value, but without taking into account its sign. To put it even more simply, it is necessary to remove the sign from the number.
The module of a number can be designated and look like this: |3|, |x|, |a| etc.
So, for example, the modulus of the number 3 is denoted |3|.
Also, it should be remembered that the modulus of a number is never negative: |a|≥ 0.
|5| = 5, |-6| = 6, |-12.45| = 12.45, etc.
Geometric meaning of the module
The modulus of a number is the distance that is measured in unit segments from the origin to the point. This definition reveals a module with geometric point vision.
Let's take a coordinate line and designate two points on it. Let these points correspond to numbers such as −4 and 2.
Now let's pay attention to this figure. We see that point A indicated on the coordinate line corresponds to the number -4 and if you look carefully, you will see that this point is located at a distance of 4 from the reference point 0 single segments. It follows that the length of segment OA is equal to four units. In this case, the length of the segment OA, that is, the number 4, will be the modulus of the number -4.
Identified and recorded in in this case modulus of the number in this way: |−4| = 4.
Now let’s take and designate point B on the coordinate line.
This point B will correspond to the number +2, and, as we see, it is located at a distance of two unit segments from the origin. It follows from this that the length of segment OB is equal to two units. In this case, the number 2 will be the modulus of the number +2.
In the recording it will look like this: |+2| = 2 or |2| = 2.
Now let's summarize. If you and I take some unknown number a and denote it on the coordinate line by point A, then in this case the distance from point A to the origin, that is, the length of the segment OA, is precisely the modulus of the number “a”.
In writing it will look like this: |a| = OA.
The modulus of a number and its properties
Now let's try to highlight the properties of the module, consider all possible cases and write them using literal expressions:
Firstly, the modulus of a number is a non-negative number, which means that the modulus of a positive number is equal to the number itself: |a| = a, if a > 0;
Secondly, the modules that consist of opposite numbers are equal: |a| = |–a|. That is, this property tells us that opposite numbers always have equal modules, as on the coordinate line, although they have opposite numbers, they are at the same distance from the reference point. It follows from this that the modules of these opposite numbers are equal.
Thirdly, the modulus of zero is equal to zero if this number is zero: |0| = 0 if a = 0. Here we can say with confidence that the modulus of zero is zero by definition, since it corresponds to the origin of the coordinate line.
The fourth property of a modulus is that the modulus of the product of two numbers equal to the product moduli of these numbers. Now let's take a closer look at what this means. If we follow the definition, then you and I know that the modulus of the product of numbers a and b will be equal to a b, or −(a b), if a b ≥ 0, or – (a b), if a b is greater than 0. B recording it will look like this: |a b| = |a| |b|.
The fifth property is that the modulus of the quotient of numbers equal to the ratio moduli of these numbers: |a: b| = |a| : |b|.
AND following properties module number:
Solving equations and inequalities that involve the modulus of a number
When starting to solve problems that have a number modulus, you should remember that in order to solve such a task, it is necessary to reveal the sign of the modulus using knowledge of the properties to which this problem corresponds.
Exercise 1
So, for example, if under the module sign there is an expression that depends on a variable, then the module should be expanded in accordance with the definition:
Of course, when solving problems, there are cases when the module is revealed uniquely. If, for example, we take
, here we see that such an expression under the modulus sign is non-negative for any values of x and y.
Or, for example, let’s take
, we see that this modulus expression is not positive for any values of z.
Task 2
A coordinate line is shown in front of you. On this line it is necessary to mark the numbers whose modulus will be equal to 2.
Solution
First of all, we must draw a coordinate line. You already know that to do this, first on the straight line you need to select the origin, direction and unit segment. Next, we need to place points from the origin that are equal to the distance of two unit segments.
As you can see, there are two such points on the coordinate line, one of which corresponds to the number -2, and the other to the number 2.
Historical information about the modulus of numbers
The term "module" comes from Latin name modulus, which translated means the word “measure”. This term was coined by the English mathematician Roger Cotes. But the modulus sign was introduced thanks to the German mathematician Karl Weierstrass. When written, a module is denoted using the following symbol: | |.
Questions to consolidate knowledge of the material
In today's lesson, we got acquainted with such a concept as the modulus of a number, and now let's check how you have mastered this topic by answering the questions posed:
1. What is the name of the number that is the opposite of a positive number?
2. What is the name of the number that is the opposite of a negative number?
3. Name the number that is the opposite of zero. Does such a number exist?
4. Name a number that cannot be a modulus of a number.
5. Define the modulus of a number.
Homework
1. In front of you are numbers that you need to arrange in descending order of modules. If you complete the task correctly, you will find out the name of the person who first introduced the term “module” into mathematics.
2. Draw a coordinate line and find the distance from M (-5) and K (8) to the origin.
Equations with modules, solution methods. Part 1.
Before you begin directly studying techniques for solving such equations, it is important to understand the essence of the module, its geometric meaning. It is in understanding the definition of the module and its geometric meaning that the main methods for solving such equations are laid. The so-called method of intervals when opening modular brackets is so effective that using it it is possible to solve absolutely any equation or inequality with moduli. In this part we will study in detail two standard methods: interval method and method of replacing an equation with a set.
However, as we will see, these methods are always effective, but not always convenient and can lead to long and even not very convenient calculations, which naturally require more time to solve. Therefore, it is important to know those methods that significantly simplify the solution of certain equation structures. Squaring both sides of an equation, a method for introducing a new variable, graphic method, solving equations containing a modulus under the modulus sign. We will look at these methods in the next part.
Determination of the modulus of a number. Geometric meaning of the module.
First of all, let's get acquainted with geometric sense module:
Modulus of numbers a (|a|) call the distance on the number line from the origin (point 0) to the point A(a).
Based on this definition, let's look at some examples:
|7| - this is the distance from 0 to point 7, of course it is equal to 7. → | 7 |=7
|-5|- this distance from 0 to point -5 and it is equal to: 5. → |-5| = 5
We all understand that distance cannot be negative! Therefore |x| ≥ 0 always!
Let's solve the equation: |x |=4
This equation can be read like this: the distance from point 0 to point x is 4. Yeah, it turns out that from 0 we can move both to the left and to the right, which means moving to the left at a distance equal to 4 we will end up at the point: -4, and moving to the right we will end up at point: 4. Indeed, |-4 |=4 and |4 |=4.
Hence the answer is x=±4.
If you carefully study the previous equation, you will notice that: the distance to the right along the number line from 0 to the point is equal to the point itself, and the distance to the left from 0 to the number is opposite number! Realizing that to the right of 0 positive numbers, and to the left of 0 are negative, let us formulate definition of the modulus of a number: the modulus (absolute value) of a number X(|x|) is the number itself X, if x ≥0, and number – X, if x<0.
Here we need to find a set of points on the number line, the distance from 0 to which will be less than 3, let's imagine a number line, point 0 on it, go to the left and count one (-1), two (-2) and three (-3), stop. Next will be points that lie further than 3 or the distance to which from 0 is greater than 3, now we go to the right: one, two, three, stop again. Now we select all our points and get the interval x: (-3;3).
It is important that you see this clearly, if you still can’t, draw it on paper and look so that this illustration is completely clear to you, don’t be lazy and try to see the solutions to the following tasks in your mind:
|x |=11, x=? |x|=-5, x=?
|x |<8, х-? |х| <-6, х-?
|x |>2, x-? |x|> -3, x-?
|π-3|=? |-x²-10|=?
|√5-2|=? |2х-х²-3|=?
|x²+2|=? |x²+4|=0
|x²+3x+4|=? |-x²+9| ≤0
Did you notice the strange tasks in the second column? Indeed, the distance cannot be negative therefore: |x|=-5- has no solutions, of course it cannot be less than 0, therefore: |x|<-6 тоже не имеет решений, ну и естественно, что любое расстояние будет больше отрицательного числа, значит решением |x|>-3 are all numbers.
After you learn to quickly see pictures with solutions, read on.
Module Definition can be given as follows: The absolute value of the number a(modulus) is the distance from the point representing a given number a on the coordinate line, to the origin. From the definition it follows that:
Thus, in order to expand a module, it is necessary to determine the sign of the submodular expression. If it is positive, then you can simply remove the modulus sign. If the submodular expression is negative, then it must be multiplied by “minus”, and, again, the modulus sign should no longer be written.
Main properties of the module:
Some methods for solving equations with moduli
There are several types of modulus equations for which there is a preferred solution. However, this method is not the only one. For example, for an equation of the form:
The preferred solution would be to go to the aggregate:
And for equations of the form:
You can also move on to an almost similar set, but since the module takes only positive values, the right side of the equation must also be positive. This condition must be added as a general constraint for the entire example. Then we get the system:
Both of these types of equations can be solved in another way: by opening the module accordingly on intervals where the submodular expression has a certain sign. In this case, we will obtain a combination of two systems. Let us present the general form of the solutions obtained for both types of equations given above:
To solve equations that contain more than one module, use interval method, which is as follows:
- First, we find the points on the number axis at which each of the expressions under the module vanishes.
- Next, we divide the entire numerical axis into intervals between the resulting points and examine the sign of each of the submodular expressions on each interval. Note that to determine the sign of an expression, you must substitute any value into it x from the interval, except for the boundary points. Choose those values x, which are easy to substitute.
- Next, on each resulting interval, we open all the modules in the original equation in accordance with their signs on this interval and solve the resulting ordinary equation. In the final answer we write down only those roots of this equation that fall within the interval under study. Once again: we carry out this procedure for each of the resulting intervals.
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How to successfully prepare for the CT in physics and mathematics?
In order to successfully prepare for the CT in physics and mathematics, among other things, it is necessary to fulfill three most important conditions:
- Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on different topics and of varying complexity. The latter can only be learned by solving thousands of problems.
- Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty solving most of the CT at the right time. After this, you will only have to think about the most difficult tasks.
- Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.
Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.
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