At the beginning of the lesson, we repeat the basic properties of square roots, and then consider several complex examples To simplify expressions containing square roots.

Subject:Function. Properties square root

Lesson:Transformation and simplification more complex expressions with roots

1. Repeat the properties of square roots

Briefly repeat the theory and remind the basic properties of square roots.

Properties of square roots:

1., therefore;

3. ;

4. .

2. Examples for simplifying expressions with roots

Let us turn to the examples of using these properties.

Example 1. Simplify expression .

Decision. To simplify the number 120, it is necessary to decompose on simple factors:

Square amount will be revealed according to the corresponding formula:

Example 2. Simplify expression .

Decision. We take into account that this expression makes sense not with all possible values \u200b\u200bof the variable, since there is square roots and fractions, which leads to the "narrowing" of the area of \u200b\u200bpermissible values. OTZ: ().

We give the expression in brackets to the general denominator and with a spinner of the last fraction as a difference of squares:

Answer. at.

Example 3. Simplify expression .

Decision. It can be seen that the second numerator bracket has an uncomfortable look and needs to be simplified, try to decompose it for multipliers using the grouping method.

For the possibility of making a common factor, we simplified the roots by decomposition of multipliers. We substitute the resulting expression in the original fraction:

After cutting the fraction, apply the formula of the difference in squares.

3. Example on getting rid of irrationality

Example 4. Frequently from irrationality (roots) in the denominator: a); b).

Decision. a) in order to get rid of irrationality in the denominator, applied standard method Domination and numerator and denominator of the fraction on the factor conjugate to the denominator (the same expression, but with a reverse sign). This is done to supplement the denominator of the fraction to the difference in squares, which allows the root from the root in the denominator. Perform this technique in our case:

b) Perform similar actions:

4. Example for evidence and on the release of a full square in a complex radical

Example 5. Prove equality .

Evidence. We use the definition of the square root, from which it follows that the square of the right expression should be equal to the guided expression:

. We will reveal brackets by the Square Formula:

Required true equality.

Proved.

Example 6. Simplify the expression.

Decision. The specified expression is customary called a complex radical (root under the root). IN this example It is necessary to guess to allocate a full square from the feeding expression. To do this, we note that of the two components is a challenge for the role of a doubled work in the formula of the square of the difference (difference, since there is a minus). We bring it in the form of such a work:, then the role of one of the complications of the full square is claimed, and on the role of the second - 1.

We will substitute this expression on the root.

Expressions, transformation of expressions

Powerful expressions (expressions with degrees) and their conversion

In this article we will talk about transforming expressions with degrees. First we will focus on transformations that are performed with expressions of any species, including powerful expressions, such as disclosing brackets, bringing similar terms. And then we will analyze the transformation inherent in expressions with degrees: work with the basis and indicator of the degree, the use of the properties of degrees, etc.

Navigating page.

What are power expressions?

The term "powerful expressions" practically does not occur to school textbooks of mathematics, but it often appears in collections of tasks, especially designed to prepare for EGE and OGE, for example,. After analyzing the tasks in which any actions are required with power expressions, it becomes clear that under power expressions understand the expressions containing in their degree records. Therefore, it is possible to accept such a definition for yourself:

Definition.

Power expressions - These are expressions containing degrees.

Here examples of power expressions. Moreover, we will submit them according to how the development of views on degree with a natural indicator to the degree with the actual indicator occurs.

As you know, first the acquaintance with the degree of number with a natural figure, at this stage the first simplest power expressions of type 3 2, 7 5 +1, (2 + 1) 5, (-0,1) 4, 3 · A 2 appear -A + a 2, x 3-1, (a 2) 3, etc.

A little later, the degree of number with an integer is studied, which leads to the emergence of power expressions with whole negative degrees, like the following: 3 -2, , A -2 + 2 · b -3 + C 2.

In high school, returned to degrees again. There is a degree with a rational indicator, which entails the appearance of appropriate power expressions: , , etc. Finally, discusses degrees with irrational indicators and comprising their expressions: ,.

The case listed by power expressions is not limited to: the variable penetrates further in terms of the extent, and there are such expressions 2 x 2 +1 or . And after acquaintance, expressions with degrees and logarithms begin to meet, for example, x 2 · LGX -5 · X LGX.

So, we dealt with the question, which represents powerful expressions. We will continue to learn to convert them.

The main types of transformations of power expressions

With power expressions you can perform any of the main identity transformations of expressions. For example, you can reveal brackets, replace numerical expressions by their values, bring similar terms, etc. Naturally, it should be necessary to comply with the procedure for performing actions. We give examples.

Example.

Calculate the value of the power expression 2 3 · (4 2 -12).

Decision.

According to the procedure for performing actions, first perform actions in brackets. There, firstly, we replace the degree 4 2 of its value 16 (see if necessary), and secondly, we calculate the difference 16-12 \u003d 4. Have 2 3 · (4 2 -12) \u003d 2 3 · (16-12) \u003d 2 3 · 4.

In the resulting expression, we replace degree 2 3 of its value 8, after which we calculate the product 8 · 4 \u003d 32. This is the desired value.

So, 2 3 · (4 2 -12) \u003d 2 3 · (16-12) \u003d 2 3 · 4 \u003d 8 · 4 \u003d 32.

Answer:

2 3 · (4 2 -12) \u003d 32.

Example.

Simplify expressions with degrees 3 · a 4 · b -7 -1 + 2 · a 4 · b -7.

Decision.

It is obvious that this expression contains similar terms 3 · a 4 · b -7 and 2 · a 4 · b -7, and we can lead them :.

Answer:

3 · a 4 · b -7 -1 + 2 · a 4 · b -7 \u003d 5 · a 4 · b -7 -1.

Example.

Present an expression with degrees in the form of a work.

Decision.

Credit with the task allows the representation of the number 9 in the form of degree 3 2 and the subsequent use of the formula of the abbreviated multiplication. Square differences:

Answer:

There is also a number of identical transformations inherent in power expressions. Then we will discern them.

Work with the basis and indicator of the degree

There are extent, at the base and / or indicator of which are not just numbers or variables, but some expressions. As an example, give the record (2 + 0.3 · 7) 5-3.7 and (a · (a + 1) -a 2) 2 · (x + 1).

When working with similar expressions, it is possible as an expression at the base of the degree and the expression in the indicator to replace identically equal expression On the odd of its variables. In other words, we can separately convert the rootation of degree to us separately, and separately the indicator. It is clear that as a result of this transformation, an expression will be identically equal to the initial one.

Such transformations make it possible to simplify expressions with degrees or reach other purposes we need. For example, in the above-mentioned power expression (2 + 0.3 · 7) 5-3.7, it is possible to perform actions with numbers at the base and indicator, which will allow you to move to the degree of 4.1 1.3. And after the disclosures of the brackets and bringing similar terms at the base of the degree (A · (A + 1) -a 2) 2 · (x + 1) we get a power expression more simple view a 2 · (x + 1).

Use the properties of degrees

One of the main tools for transforming expressions with degrees is equality reflecting. Recall the main of them. For any positive numbers a and b and arbitrary valid numbers R and S are fair the following properties degrees:

• a r · a s \u003d a r + s;
• a R: A S \u003d A R-S;
• (a · b) r \u003d a r · b r;
• (A: B) R \u003d A R: B R;
• (a r) s \u003d a r · s.

Note that with natural, integers, as well as the positive indicators of the degree of restriction on the number A and B may not be as strict. For example, for natural numbers M and N Equality A m · a n \u003d a m + n is true not only for positive A, but also for negative, and for a \u003d 0.

At school, the focus on the transformation of power expressions is focused on the ability to select a suitable property and apply it correctly. At the same time, the bases of degrees are usually positive, which allows the use of the properties of degrees without restrictions. The same applies to the transformation of expressions containing variables in the bases of degrees - the area of \u200b\u200bpermissible values \u200b\u200bof variables is usually that the bases are taken only positive meaningsthat allows you to freely use the properties of degrees. In general, it is necessary to constantly wonder if it is possible to use any property of degrees in this case, because the inaccient use of properties can lead to a narrowing of OTZ and other troubles. In detail and on examples, these moments are disassembled in the article transformation of expressions using the properties of degrees. Here we will restrict ourselves to the consideration of several simple examples.

Example.

Prepare an expression A 2.5 · (A 2) -3: A -5.5 as a degree with a base a.

Decision.

First, the second factor (A 2) -3 is converting the exercise in the degree in the degree in the degree: (a 2) -3 \u003d a 2 · (-3) \u003d a -6. The initial power expression takes the form A 2.5 · A -6: A -5.5. Obviously, it remains to take advantage of the properties of multiplication and division of degrees with the same basis, we have
a 2.5 · A -6: A -5.5 \u003d
a 2.5-6: A -5.5 \u003d A -3,5: A -5.5 \u003d
a -3.5 - (- 5.5) \u003d a 2.

Answer:

a 2.5 · (A 2) -3: A -5.5 \u003d A 2.

The properties of degrees when converting power expressions are used both from left to right and right to left.

Example.

Find the value of a power expression.

Decision.

Equality (a · b) r \u003d a r · b R, applied to the right left, allows from the initial expression to move to the product and further. And when multiplying degrees with identical grounds Indicators fold: .

It was possible to perform the transformation of the initial expression and otherwise:

Answer:

.

Example.

The power expression A 1.5 -a 0.5 -6, enter a new variable T \u003d A 0.5.

Decision.

The degree A 1.5 can be represented as a 0.5 · 3 and on the database of the degree property to the degree (A R) S \u003d A R · S, applied to the right to left, convert it to the form (A 0.5) 3. In this way, a 1,5 -A 0.5 -6 \u003d (A 0.5) 3 -A 0.5 -6. Now it is easy to enter a new variable T \u003d A 0.5, we obtain T 3 -T-6.

Answer:

t 3 -T-6.

Transformation of fractions containing degrees

Powerful expressions may contain fractions with degrees or represent such fractions. Such fractions are fully applicable any of the main transformations of fractions that are inherent in fractions of any kind. That is, fractions that contain degrees can be reduced, lead to a new denominator, work separately with their numerator and separately with the denominator, etc. To illustrate the words, consider solutions of several examples.

Example.

Simplify power expression .

Decision.

This power expression is a fraction. We will work with its numerator and denominator. In the numerator, we will reveal the brackets and simplifies the expression obtained after this, using the properties of degrees, and in the denominator we will give similar terms:

And still change the sign of the denominator, placing minus before the fraction: .

Answer:

.

Bringing the degrees of fractions to a new denominator is carried out similarly to bringing rational fractions to a new denominator. At the same time, an additional factor is also located and multiplying the numerator and denominator of the fraction. Performing this action, it is worth remembering that bringing to a new denominator can lead to a narrowing of OTZ. To this not happen, it is necessary that the additional factor does not apply to zero at no matter what values \u200b\u200bof the variables from the odd variables for the initial expression.

Example.

Give fractions to a new denominator: a) to the denominator A, b) to the denominator.

Decision.

a) In this case, it is quite simple to imagine what an additional factor helps to achieve the desired result. This is a multiplier a 0.3, as a 0.7 · a 0.3 \u003d a 0.7 + 0.3 \u003d a. Note that on the area of \u200b\u200bpermissible values \u200b\u200bof the variable A (these are a plurality of all positive valid numbers) degree A 0.3 does not appeal to zero, therefore, we have the right to multiply the numerator and denominator of the specified fraction on this additional factor:

b) looking more closely to the denominator, it can be found that

And the multiplication of this expression on will give the amount of cubes and, that is,. And this is the new denominator to which we need to bring the original fraction.

So we found an additional factor. On the area of \u200b\u200bpermissible values \u200b\u200bof the variables x and y, the expression does not apply to zero, therefore, we can multiply the numerator and denominator of the fraction:

Answer:

but) b) .

There is nothing new in the reduction of fractions containing degrees, there is nothing new: the numerator and the denominator are represented as a number of multipliers, and the same multipliers of the numerator and the denominator are reduced.

Example.

Reduce the fraction: a) , b).

Decision.

a) firstly, the numerator and denominator can be reduced to numbers 30 and 45, which is equal to 15. Also, obviously, you can make a reduction on x 0.5 +1 and . That's what we have:

b) In this case, the same multipliers in the numerator and the denominator cannot be immediately visible. To get them, you will have to perform preliminary transformations. In this case, they are concluded in the expansion of the denominator for multipliers using the formula of the square difference:

Answer:

but)

b) .

Bringing fractions to a new denominator and the reduction of fractions is mainly used to perform action with fractions. Actions are performed according to the well-known rules. When adding (subtracting) fractions, they are given to a shared denominator, after which they are completed (subtracted) numerals, and the denominator remains the same. As a result, it turns out a fraction, the numerator of which is the product of numerals, and the denominator is a product of denominators. The division of the fraction is multiplication by fraction, inverse it.

Example.

Follow the steps .

Decision.

First, we perform the subtraction of fractions located in brackets. To do this, bring them to a common denominator who has , after which we subtract the numbers:

Now we multiply the fractions:

Obviously, it is possible to reduce the degree of x 1/2, after which we have .

You can still simplify the power expression in the denominator, using the formula of the square difference: .

Answer:

Example.

Simplify power expression .

Decision.

Obviously, this fraction can be reduced by (x 2.7 +1) 2, it gives a fraction . It is clear that you need to do something else with the degrees of ICA. To do this, we transform the resulting fraction into the work. This gives us the opportunity to take advantage of the property of degrees with the same grounds: . And in conclusion proceed from the last work To fraction.

Answer:

.

And I also add that it is possible and in many cases, it is desirable to transfer multiple degree rates from the numerator to a denominator or from the denominator to a numerator, changing the indicator sign. Such transformations often simplify further actions. For example, a power expression can be replaced by.

Transformation of expressions with roots and degrees

Often in expressions that require some transformations, along with degrees with fractional indicators there are roots. To convert a similar expression to listeningIn most cases, it is enough to go only to roots or only to degrees. But since it is more convenient to work with degrees, usually go from roots to degrees. However, it is advisable to exercise such a transition when the OTZ variables for the initial expression makes it possible to replace the roots by degrees without having to turn to the module or split OTZ to several gaps (we disassembled in detail the transition from the roots to the degrees and back after exploring the degree with a rational indicator The degree with the irrational indicator is introduced, which allows you to talk about the degree with an arbitrary real indicator. At this stage, the school begins to study exponential function which is analyzically defined by the degree in which the number is located, and in the indicator - the variable. So we are confronted with the powerful expressions containing the number in the foundation of the degree, and in the indicator - expressions with variables, and naturally there is a need to perform transformations of such expressions.

It should be said that the transformation of the expressions of the specified species usually has to be performed when solving indicatory equations and indicative inequalities And these transformations are quite simple. In the overwhelming number of cases, they are based on the degree properties and are aimed for the most part to enter a new variable in the future. Demonstrate them will allow the equation 5 2 · x + 1 -3 · 5 x · 7 x -14 · 7 2 · x-1 \u003d 0.

Firstly, the degrees in the indicators of which there is a sum of some variable (or expressions with variables) and the numbers are replaced by the works. This applies to the first and last term expressions from the left side:
5 2 · x · 5 1 -3 · 5 x · 7 x -14 · 7 2 · x · 7 -1 \u003d 0,
5 · 5 2 · x -3 · 5 x · 7 x -2 · 7 2 · x \u003d 0.

Further, the division of both parts of equality is performed on the expression 7 2 · x, which only positive values \u200b\u200btake on the source equation to the source equation (this is the standard reception of solving equations of this type, it is not about him now, so focus on subsequent transformations of expressions with degrees ):

Now the fractions are reduced with degrees, which gives .

Finally, the ratio of degrees with the same indicators is replaced by degrees of relations, which leads to the equation That is equivalent . Transformations made allow you to enter a new variable, which reduces the solution of the original indicative equation to solve the square equation

I. V. Boykov, L. D. Romanova Collection of tasks for preparation for the exam. Part 1. Penza 2003.

I. Expressions in which, along with letters, the numbers, marks of arithmetic action and brackets can be used, are called algebraic expressions.

Examples of algebraic expressions:

2m -n; 3. · (2a + b); 0.24X; 0,3A -B. · (4a + 2b); a 2 - 2ab;

Since the letter in algebraic expression can be replaced by some various numbers, the letter is called a variable, and the algebraic expression itself is an expression with a variable.

II. If in algebraic expression letters (variables), replace them with values \u200b\u200band perform these actions, then the resulting number is called an algebraic expression value.

Examples. Find the value of the expression:

1) a + 2b -c at a \u003d -2; b \u003d 10; C \u003d -3.5.

2) | x | + | Y | - | z | at x \u003d -8; y \u003d -5; z \u003d 6.

Decision.

1) a + 2b -c at a \u003d -2; b \u003d 10; C \u003d -3.5. Instead of variables, we substitute their values. We get:

— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.

2) | x | + | Y | - | z | at x \u003d -8; y \u003d -5; z \u003d 6. Substitute specified values. Remember that the module negative number It is equal to the opposite number, and the module of a positive number is equal to the very number. We get:

|-8| + |-5| -|6| = 8 + 5 -6 = 7.

III. The values \u200b\u200bof the letter (variable), under which the algebraic expression makes sense, is called the allowable values \u200b\u200bof the letter (variable).

Examples. Under what values \u200b\u200bof the variable expression does not make sense?

Decision. We know that it is impossible to divide to zero, therefore, each of these expressions will not make sense in the value of the letter (variable), which draws the denomoter of the fraction in zero!

In Example 1) this value is a \u003d 0. Indeed, if instead and substitute 0, then you need to share the number 6 to 0, and this can not be done. Answer: Expression 1) does not make sense at a \u003d 0.

In Example 2) the denominator x - 4 \u003d 0 at x \u003d 4, therefore, this value x \u003d 4 and cannot be taken. Answer: Expression 2) does not make sense at x \u003d 4.

In Example 3) denominator x + 2 \u003d 0 at x \u003d -2. Answer: Expression 3) does not make sense at x \u003d -2.

In example 4) denominator 5 - | x | \u003d 0 with | x | \u003d 5. And since | 5 | \u003d 5 and | -5 | \u003d 5, then it is impossible to take x \u003d 5 and x \u003d -5. Answer: Expression 4) does not make sense at x \u003d -5 and at x \u003d 5.
IV. Two expressions are identically equal, if with any valid values \u200b\u200bof the variables, the corresponding values \u200b\u200bof these expressions are equal.

Example: 5 (a - b) and 5a - 5b are shadely equal, since equality 5 (a - b) \u003d 5a - 5b will be faithful at any values \u200b\u200bof A and b. Equality 5 (A - B) \u003d 5A - 5B There is a identity.

Identity - This is equality, just with all the permissible values \u200b\u200bof the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, the distribution property.

The replacement of one expression to another, identically equal to it by the expression, is called identical conversion or simply by the transformation of the expression. Identical transformations Expansions with variables are performed based on the properties of actions above the numbers.

Examples.

a) Convert the expression to identically equal, using the distribution property of multiplication:

1) 10 · (1.2x + 2.3,); 2) 1.5 · (A -2B + 4C); 3) A · (6m -2n + k).

Decision. Recall the distribution property (law) of multiplication:

(A + B) · C \u003d A · C + B · C (The distribution law of multiplication relative to addition: to multiply the amount of two numbers to the third number, you can multiply each component to this number and folded the results).
(A-B) · C \u003d A · C-B · C (Distribution law of multiplication relative to subtraction: To multiply the difference between two numbers to multiply by the third number, you can multiply by this number reduced and subtractable separately and from the first result of subtracting the second one).

1) 10 · (1.2x + 2,31) \u003d 10 · 1.2x + 10 · 2.3U \u003d 12x + 23W.

2) 1.5 · (A -2B + 4C) \u003d 1,5A -3B + 6C.

3) A · (6m -2n + k) \u003d 6am -2an + AK.

b) Convert the expression to identically equal, using the Moveless and Fashion Properties (laws) of addition:

4) x + 4.5 + 2x + 6.5; 5) (3a + 2,1) + 7.8; 6) 5.4C -3 -2.5 -2.3C.

Decision. Apply the laws (properties) of addition:

a + b \u003d b + a (Movement: the amount does not change from the rearrangement of the terms).
(A + B) + C \u003d A + (B + C) (Combining: To add a third number to the sum of the two components, you can add the second and third amount to the first number).

4) x + 4.5 + 2x + 6.5 \u003d (x + 2x) + (4.5 + 6.5) \u003d 3x + 11.

5) (3a + 2,1) + 7.8 \u003d 3A + (2.1 + 7.8) \u003d 3a + 9.9.

6) 6) 5.4C -3 -2.5 -2.3c \u003d (5.4C -2.3C) + (-3 -2.5) \u003d 3.1С -5.5.

in) Convert the expression to identically equal, using the Multiplication Multiplication: Multiplication:

7) 4 · H. · (-2,5); 8) -3,5 · 2ow · (-one); 9) 3A. · (-3) · 2c.

Decision. Apply the laws (properties) of multiplication:

a · b \u003d b · a (Movement: From the permutation of multipliers, the work does not change).
(A · b) · C \u003d A · (B · C) (Combining: To multiply the work of two numbers to the third number, you can multiply the first number to the work of the second and third).

7) 4 · H. · (-2,5) = -4 · 2,5 · x \u003d -10x.

8) -3,5 · 2ow · (-1) \u003d 7U.

9) 3A. · (-3) · 2C \u003d -18As.

If the algebraic expression is given in the form of a reduced fraction, then using the crushing rule, it can be simplified, i.e. Replace identically equal to a simpler expression.

Examples. Simplify using the reduction of fractions.

Decision. Reduce the fraction - this means dividing its numerator and denominator to the same number (expression), different from zero. Fraction 10) will reduce on 3b.; fraction 11) will reduce on but and fraction 12) will reduce on 7N.. We get:

Algebraic expressions are used to compile formulas.

The formula is an algebraic expression recorded in the form of equality and expressing the relationship between two or several variables. Example: Formula Formula you know s \u003d V · T (S is the path traveled, V is speed, t - time). Remember what other formulas you know.

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In the fifth century BC, the ancient Greek philosopher Zenon Elayky formulated his famous apiorials, the most famous of which is Achilles and Turtle Aritia. This is how it sounds:

Suppose Achilles runs ten times faster than the turtle, and is behind it at a distance of a thousand steps. For the time, for which Achilles is running through this distance, a hundred steps will crash in the same side. When Achilles runs a hundred steps, the turtle will crawl about ten steps, and so on. The process will continue to infinity, Achilles will never catch up to the turtle.

This reasoning has become a logical shock for all subsequent generations. Aristotle, Diogen, Kant, Hegel, Hilbert ... All of them somehow considered the Apriology of Zenon. Shock turned out to be so strong that " ... Discussions continue and at present, to come to the general opinion on the essence of paradoxes to the scientific community has not yet been possible ... A mathematical analysis, the theory of sets, new physical and philosophical approaches was involved in the study of the issue; None of them became a generally accepted issue of the issue ..."[Wikipedia," Yenon Apriya "]. Everyone understands that they are blocked, but no one understands what deception is.

From the point of view of mathematics, Zeno in his Aproria clearly demonstrated the transition from the value to. This transition implies application instead of constant. As far as I understand, the mathematical apparatus of the use of variables of units of measurement is either yet not yet developed, or it was not applied to the Aporition of Zenon. The use of our ordinary logic leads us to a trap. We, by inertia of thinking, use permanent time measurement units to the inverter. From a physical point of view, it looks like a slowdown in time to its complete stop at the moment when Achilles is stuffed with a turtle. If time stops, Achilles can no longer overtake the turtle.

If you turn the logic usually, everything becomes in place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on its overcoming, ten times less than the previous one. If you apply the concept of "infinity" in this situation, it will correctly say "Achilles infinitely will quickly catch up the turtle."

How to avoid this logical trap? Stay in permanent time measurement units and do not move to reverse values. In the language of Zenon, it looks like this:

For that time, for which Achilles runs a thousand steps, a hundred steps will crack the turtle to the same side. For the next time interval, equal to the first, Achilles will run another thousand steps, and the turtle will crack a hundred steps. Now Achilles is an eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. On the Zenonian Agrac of Achilles and Turtle is very similar to the statement of Einstein on the irresistibility of the speed of light. We still have to study this problem, rethink and solve. And the decision should be sought not in infinitely large numbers, but in units of measurement.

Another interesting Yenon Aproria tells about the flying arrows:

The flying arrow is still, since at every moment she rests, and since it rests at every moment of time, it always rests.

In this manor, the logical paradox is very simple - it is enough to clarify that at each moment the flying arrow is resting at different points of space, which, in fact, is the movement. Here you need to note another moment. According to one photo of the car on the road, it is impossible to determine the fact of its movement, nor the distance to it. To determine the fact of the car's motion, you need two photos made from one point at different points in time, but it is impossible to determine the distance. To determine the distance to the car you need two photos made from different points Spaces at one point in time, but it is impossible to determine the fact of movement (naturally, additional data is still needed for calculations, trigonometry to help you). What I want to pay special attention is that two points in time and two points in space are different things that do not be confused, because they provide different opportunities for research.

wednesday, July 4, 2018

Very good differences between many and multiset are described in Wikipedia. We look.

As you can see, "there cannot be two identical elements in a set", but if identical elements are in the set there are, such a set is called "Mix". A similar logic of absurd reasonable beings never understand. This is the level of speaking parrots and trained monkeys, which are missing from the word "at all." Mathematics act as ordinary trainers, preaching our absurd ideas.

Once the engineers who built the bridge during the tests of the bridge were in the boat under the bridge. If the bridge collapsed, the talentless engineer died under the wreckage of his creation. If the bridge has withstood the load, a talented engineer built other bridges.

As math did not hide behind the phrase "Chur, I am in a house", more precisely, "mathematics studies abstract concepts," there is one umbilical cord, which inextricably binds them with reality. This umbilical cord is money. Apply mathematical theory of sets to mathematics themselves.

We taught mathematics very well and now we sit at the checkout, we issue a salary. That comes to us the mathematician for your money. We count on it the entire amount and lay out on your table on different stacks, in which we add bills of one dignity. Then we take from each stack on one bill and hand the mathematics of his "mathematical set of salary". Explain the mathematics that the rest of the bills will receive only when it proves that the set without the same elements is not equal to the set with the same elements. Here the most interesting will begin.

First of all, the logic of deputies will work: "It is possible to apply it to others, to me - low!". There will be further assurances of us that there are different numbers on bills of equal dignity, which means that they cannot be considered the same elements. Well, count the salary with coins - there are no numbers on the coins. Here the mathematician will begin to dislike physics: on different coins there is a different amount of dirt, the crystal structure and the location of atoms each coin is unique ...

And now I have the most interesting question: where is the line, behind which the elements of the multisament turn into elements of the set and vice versa? Such a face does not exist - everyone solves the shamans, the science here and not lying close.

Here are looking. We take football stadiums with the same field area. The field area is the same - it means we have a multipart. But if we consider the names of the same stadiums - we have many, because the names are different. As you can see, the same set of elements is both set and multiset. How correct? And here the mathematician-shaman-shuller pulls out the trump ace from the sleeve and begins to tell us either about the set or about the multiset. In any case, he will convince us of her right.

To understand how modern shamans operate the theory of sets, tie it to reality, it is enough to answer one question: how are the elements of one set differ from the elements of another set? I will show you, without any "imaginable as not a single whole" or "not thoughtful as a whole."

sunday, March 18, 2018

The amount of numbers is a dance of shamans with a tambourine, which does not have any relation to mathematics. Yes, in the lessons of mathematics, we are taught to find the amount of numbers of numbers and use it, but they are shamans to train your descendants to their skills and wisdoms, otherwise the shamans will simply be cleaned.

Do you need evidence? Open Wikipedia and try to find the number of numbers page. It does not exist. There is no formula in mathematics at which you can find the amount of numbers of any number. After all, the numbers are graphic symbols, with which we write numbers and in mathematics language, the task sounds like this: "Find the sum of graphic characters depicting any number". Mathematics can not solve this task, but shamans are elementary.

Let's deal with what and how we do in order to find the amount of the numbers of the specified number. And so, let us have a number of 12345. What should be done in order to find the amount of numbers of this number? Consider all the steps in order.

1. Record the number on the piece of paper. What did we do? We transformed the number in the graphic symbol of the number. This is not a mathematical action.

2. We cut one image obtained into several pictures containing individual numbers. Cutting pictures is not a mathematical action.

3. We convert individual graphic characters in numbers. This is not a mathematical action.

4. We fold the numbers. This is already mathematics.

The amount of numbers of 12345 is 15. These are the "cutters and sewing courses" from the shamans apply mathematicians. But that's not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems Number of numbers of numbers of the same number will be different. In mathematics, the number system is indicated in the form of the lower index to the right of the number. FROM large number 12345 I do not want to fool my head, consider the number 26 of the article about. We write this number in binary, octal, decimal and hexadecimal number systems. We will not consider every step under the microscope, we have already done. Let's look at the result.

As you can see, in different number systems, the sum of the numbers of the same number is obtained different. This result for mathematics has nothing to do. It is like determining the area of \u200b\u200bthe rectangle in meters and centimeters you would get completely different results.

Zero in all surge systems looks the same and the amount of numbers does not have. This is another argument in favor of what. Question to mathematicians: how in mathematics is indicated that is not a number? What, for mathematicians, nothing but numbers does not exist? For shamans, I can be allowed, but for scientists - no. Reality consists not only of numbers.

The result obtained should be considered as proof that the number systems are units of numbers. After all, we can not compare the numbers with different units Measurements. If the same action with different units of measurement of the same value lead to different results after their comparison, it means that it has nothing to do with mathematics.

What is real mathematics? This is when the result of mathematical action does not depend on the value of the number used by the unit of measurement and on who performs this action.

Plate on doors Opens the door and says:

Oh! Isn't that a female toilet?
- Girl! This is a laboratory for the study of the Indefile Holiness of the Souls in Ascension to Heaven! Nimbi from above and arrow up. What else toilet?

Female ... Nimbi from above and arrogant down - it's a male.

If you in front of your eyes several times a day flashes this is the work of designer art,

Then it is not surprising that in your car you suddenly find a strange icon:

Personally, I am doing an effort on myself to be in a cuffing person (one picture), to see the minus four degrees (a composition of several pictures: a minus sign, a number four, designation of degrees). And I do not think this girl is a fool who does not know physics. It is simply an arc stereotype of the perception of graphic images. And mathematics we are constantly taught. Here is an example.

1A is not "minus four degrees" or "One A". This is a "cuffing person" or the number of "twenty-six" in a hexadecimal number system. Those people who constantly work in this number system automatically perceive the figure and letter as one graphic symbol.

§ 1 The concept of simplifying the letter expression

In this lesson, we will get acquainted with the concept of "similar terms" and on the examples will learn to carry out the alignment of such terms, simplifying, so literal expressions.

Let's find out the meaning of the concept of "simplification". The word "simplification" is formed from the word "simplify". Simplify - it means to make it simple, easier. Therefore, simplify the letter expression is to make it shorter, with minimum quantity actions.

Consider the expression 9x + 4x. This is an alphabet expression that is the amount. The components here are presented in the form of works of the number and letters. The numeric factor of such terms is called the coefficient. In this expression, the coefficients will be numbers 9 and 4. Pay attention, the multiplier submitted by the letter is the same in both terms of this amount.

Recall the distribution law of multiplication:

To multiply the amount by the number, you can multiply on this number each component and the obtained works are folded.

IN general It is written as follows: (a + b) ∙ C \u003d AC + BC.

This law is performed in both sides of AC + BC \u003d (A + B) ∙ with

Apply it to our letter expression: the amount of the works of 9x and 4x is equal to the work, the first factor of which is equal to the amount of 9 and 4, the second factor - x.

9 + 4 \u003d 13, it turns out to be 13x.

9x + 4 x \u003d (9 + 4) x \u003d 13x.

Instead of three actions, one action remains in expression - multiplication. So we made our letter expression easier, i.e. Simplified it.

§ 2 Bringing similar terms

The components of 9x and 4x differ only on their coefficients - such components are called similar. The alphabet part of the same components is the same. A similar term also includes numbers and equal terms.

For example, in expression 9a + 12 - 15, these terms will be 12 and -15, and in the amount of the work 12 and 6a, the number 14 and the works 12 and 6a (12 ∙ 6a + 14 + 12 ∙ 6a) are similar to those who are equal to the components presented work 12 and 6a.

It is important to note that the terms that are equal to the coefficients, and the letter multipliers are different, although they are sometimes useful to apply the distribution law of multiplication, for example, the amount of works 5x and 5th is equal to the product of the number 5 and the sum of x and y

5x + 5y \u003d 5 (x + y).

We simplify expression -9a + 15a - 4 + 10.

Similar terms in this case are the components -9A and 15a, as they differ only on their coefficients. The letter factor they have the same, are also similar to the components -4 and 10, since they are numbers. We fold similar terms:

9a + 15a - 4 + 10

9a + 15a \u003d 6a;

We get: 6A + 6.

Simplifying the expression, we found the sums of such terms, in mathematics it is called the lifting of similar terms.

If the creation of such terms is difficult, you can come up with words to them and put items.

For example, consider the expression:

For each letter, we take your item: B-apple, s-pear, then it turns out: 2 apples minus 5 pears plus 8 pears.

Can you make patties of pears from apples? Of course not. But to minus 5 pears add 8 pears we can.

We give similar terms -5 pears + 8 pears. Such alkaline part is the same, therefore, when you bring such terms, it is enough to complete the addition of coefficients and add the letter part to the result:

(-5 + 8) Pear - it will turn out 3 pears.

Returning to our letter expression, we have -5 C + 8C \u003d 3C. Thus, after bringing such terms, we obtain the expression 2b + 3c.

So, in this occupation you met with the concept of "similar terms" and learned to simplify alphabetic expressions by bringing similar terms.

List of references:

1. Mathematics. Grade 6: Pounding plans for the textbook I.I. Zubareva, A.G. Mordkovich // Author-compiler L.A. Topil. Mnemozina 2009.
2. Mathematics. Grade 6: Textbook for students of general educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemozina, 2013.
3. Mathematics. Grade 6: Textbook for general education institutions / GV. Dorofeev, I.F. Sharygin, S.B. Suvorov et al. / Edited GV Dorofeeva, I.F. Sharygin; ROS.Akad. Nauk, Ros.Akad.D.Fortation. M.: "Enlightenment", 2010.
4. Mathematics. Grade 6: studies. We are general formation. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzbord. - M.: Mnemozina, 2013.
5. Mathematics. 6 cl.: Tutorial / G.K. Muravin, O.V. Moravin. - M.: Drop, 2014.

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