Modulus of a number (absolute value of a number), definitions, examples, properties. The absolute value of a number
Module or absolute value a real number is called the number itself if X 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) xy| = |dg| |g/1;
- 2>- -H;
Uat
- 3) |x+r/|
- 4) |dt-g/|
Modulus of the difference of two numbers X - A| is the distance between points X And A on the number line (for any X And A).
It follows from this, in particular, that the solutions to the inequality X - A 0) are all points X interval (A- g, a + c), i.e. numbers satisfying the inequality a-d + G.
This interval (A- 8, A+ d) is called the 8-neighborhood of a point A.
Basic properties of functions
As we have already stated, all quantities in mathematics are divided into constants and variables. Constant value A quantity that retains the same value is called.
Variable value is a quantity that can take on different numerical values.
Definition 10.8. Variable value at called function from a variable value x, if, according to some rule, each value x e X assigned a specific value at e U; the independent variable x is usually called an argument, and the domain X its changes are called the domain of definition of the function.
The fact that at there is a function otx, most often expressed symbolically: at= /(x).
There are several ways to specify functions. The main ones are considered to be three: analytical, tabular and graphical.
Analytical way. This method consists of specifying the 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 the formula, for example, at= 2x + 1, at= tgx, etc.
Tabular The way to specify a function is that the function is specified by a table containing the values of the argument x and the corresponding values of the function /(.r). Examples include tables of the number of crimes for a certain period, tables of experimental measurements, and a table of logarithms.
Graphic way. Let a system of Cartesian rectangular coordinates be given on the plane xOy. The geometric interpretation of the function is based on the following.
Definition 10.9. Schedule function is called the geometric locus of points of the plane, coordinates (x, y) which satisfy the condition: U-Ah).
A function is said to be given graphically if its graph is drawn. The graphical method is widely used in experimental measurements using recording instruments.
Having a visual graph of a function before 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 a graph is the most important (usually the final) part of the study of a function.
Each method has both its advantages and disadvantages. Thus, the advantages of the graphic method include its clarity, and the disadvantages include its inaccuracy and limited presentation.
Let us now move on to consider the basic properties of functions.
Even and odd. Function y = f(x) called even, if for anyone X condition is met f(-x) = f(x). If for X from the domain of definition the condition /(-x) = -/(x) is satisfied, then the function is called odd. A function that is neither even nor odd is called a function general appearance.
- 1) y = x 2 is an even function, since f(-x) = (-x) 2 = x 2, i.e./(-x) =/(.r);
- 2) y = x 3 - an odd function, since (-x) 3 = -x 3, t.s. /(-x) = -/(x);
- 3) y = x 2 + x is a function of general form. 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 an odd function is symmetrical 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 /(x 2) > /(x,). Function at=/(x) is called decreasing, if 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 function that is monotonic in the strict sense. In general, monotonic functions include non-decreasing functions, i.e. such for which from x 2 > x, it follows/(x 2) >/(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 such a number exists M > 0, which |/(x)| M for any x e X.
For example, the function at =-
is bounded on the entire number line, so
Periodicity. Function at = f(x) called periodic, if such a number exists T^ Oh what f(x + T = f(x) for all X from the domain of the function.
In this case T is called the period of the function. Obviously, if T - period of the function 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 function / = cos.g has a period T= 2P, and the function y = tg Zx - period p/3.
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:
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;
§ 1 Modulus of a real number
In this lesson we will study the concept of “modulus” for any real number.
Let us write down the properties of the modulus of a real number:
§ 2 Solution of equations
Using the geometric meaning of the modulus of a real number, we solve several equations.
Therefore, the equation has 2 roots: -1 and 3.
Thus, the equation has 2 roots: -3 and 3.
In practice, various properties of modules are used.
Let's look at this in example 2:
Thus, in this lesson you studied the concept of “modulus of a real number”, its basic properties and geometric meaning. We also solved several typical problems using the properties and geometric representation of the modulus of a real number.
List of used literature:
- Mordkovich A.G. "Algebra" 8th grade. At 2 p.m. Part 1. Textbook for educational institutions / A.G. Mordkovich. – 9th ed., revised. – M.: Mnemosyne, 2007. – 215 p.: ill.
- Mordkovich A.G. "Algebra" 8th grade. At 2 p.m. Part 2. Problem book for educational institutions / A.G. Mordkovich, T.N. Mishustina, E.E. Tulchinskaya.. – 8th ed., – M.: Mnemosyne, 2006. – 239 p.
- Algebra. 8th grade. Tests for students of educational institutions of L.A. Alexandrov, ed. A.G. Mordkovich 2nd ed., erased. - M.: Mnemosyne, 2009. - 40 p.
- Algebra. 8th grade. Independent work for students of educational institutions: to the textbook by A.G. Mordkovich, L.A. Alexandrov, ed. A.G. Mordkovich, 9th ed., erased. - M.: Mnemosyne, 2013. - 112 p.
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Goals:
Equipment: projector, screen, personal computer, multimedia presentation
During the classes
1. Organizational moment.
2. Updating students' knowledge.
2.1. Answer students' questions about homework.
2.2. Solve the crossword puzzle (repetition of theoretical material) (Slide 2):
- A combination of mathematical symbols expressing something
– After solving the crossword puzzle, read the name of the topic of today’s lesson in the highlighted vertical column. (Slides 3, 4)
3. Explanation of a new topic.
3.1. – Guys, you have already met the concept of a module, you have used the notation | a| . Previously, we were talking only about rational numbers. Now we need to introduce the concept of modulus for any real number.
Each real number corresponds to a single point on the number line, and, conversely, each point on the number line corresponds to a single real number. All the basic properties of operations on rational numbers are preserved for real numbers.
The concept of the modulus of a real number is introduced. (Slide 5).
Definition. Modulus of a non-negative real number x call this number itself: | x| = x; modulus of a negative real number X call the opposite number: | x| = – x .
– Write down the topic of the lesson and the definition of the module in your notebooks:
In practice, various module properties, For example. (Slide 6) :
Complete orally No. 16.3 (a, b) – 16.5 (a, b) to apply the definition, properties of the module. (Slide 7) .
3.4. For any real number X can be calculated | x| , i.e. we can talk about function y = |x| .
Task 1. Construct a graph and list the properties of the function y = |x| (Slides 8, 9).
One student on the board is graphing a function
Fig 1.
Properties are listed by students. (Slide 10)
1) Domain of definition – (– ∞; + ∞) .
2) y = 0 at x = 0; y > 0 at x< 0 и x > 0.
3) The function is continuous.
4) y naim = 0 for x = 0, y naib does not exist.
5) The function is limited from below, not limited from above.
6) The function decreases on the ray (– ∞; 0) and increases on the ray )
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