Comparison of standard numbers with negative powers. Negative power of a number: construction rules and examples

First level

The degree and its properties. Comprehensive guide (2019)

Why are degrees needed? Where will they be useful to you? Why do you need to take the time to study them?

To learn everything about degrees, what they are for, how to use your knowledge in Everyday life read this article.

And, of course, knowledge of degrees will bring you closer to a successful passing the exam or the Unified State Exam and admission to the university of your dreams.

Let "s go ... (Let's go!)

Important note! If instead of formulas you see gibberish, clear the cache. To do this, press CTRL + F5 (on Windows) or Cmd + R (on Mac).

FIRST LEVEL

Exponentiation is the same mathematical operation as addition, subtraction, multiplication, or division.

Now I will explain everything in human language in a very simple examples... Pay attention. The examples are elementary, but they explain important things.

Let's start with addition.

There is nothing to explain. You already know everything: there are eight of us. Each has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

The same cola example can be written differently:. Mathematicians are cunning and lazy people. They first notice some patterns, and then come up with a way to quickly "count" them. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table... You can, of course, do everything slower, harder and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful:

What else tricky tricks lazy mathematicians invented the accounts? Right - raising a number to a power.

Raising a number to a power

If you need to multiply a number by itself five times, then mathematicians say that you need to raise this number to the fifth power. For example, . Mathematicians remember that two to the fifth degree is. And they solve such problems in their heads - faster, easier and without mistakes.

All you need to do is remember what is highlighted in the table of powers of numbers... Believe me, this will make your life much easier.

By the way, why is the second degree called square numbers, and the third - cube? What does it mean? Highly good question... Now you will have both squares and cubes.

Life example # 1

Let's start with a square or the second power of a number.

Imagine a square meter-by-meter pool. The pool is in your country house. It's hot and I really want to swim. But ... a pool without a bottom! It is necessary to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the area of the bottom of the pool.

You can simply count, poking your finger, that the bottom of the pool consists of meter by meter cubes. If you have a tile meter by meter, you will need pieces. It's easy ... But where have you seen such tiles? The tile is more likely to be cm by cm. And then you will be tortured by the "finger count". Then you have to multiply. So, on one side of the bottom of the pool, we will fit tiles (pieces) and on the other, too, tiles. Multiplying by, you get tiles ().

Have you noticed that we multiplied the same number by ourselves to determine the area of the pool bottom? What does it mean? Once the same number is multiplied, we can use the "exponentiation" technique. (Of course, when you have only two numbers, you still multiply them or raise them to a power. But if you have a lot of them, then raising to a power is much easier and there are also fewer errors in calculations. For the exam, this is very important).
So, thirty in the second degree will be (). Or you can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. Conversely, if you see a square, it is ALWAYS the second power of a number. A square is a representation of the second power of a number.

Real life example # 2

Here's a task for you, count how many squares are on the chessboard using the square of the number ... On one side of the cells and on the other, too. To count their number, you need to multiply eight by eight or ... if you notice that Chess board is a square with a side, then you can square eight. You will get cells. () So?

Life example no. 3

Now the cube or the third power of the number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters... Unexpectedly, right?) Draw a pool: the bottom is a meter in size and a meter deep and try to calculate how many cubes in the meter by meter will go into your pool.

Point your finger and count! One, two, three, four ... twenty two, twenty three ... How much did it turn out? Not lost? Is it difficult to count with your finger? So that! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes ... Easier, right?

Now imagine how lazy and cunning mathematicians are if they simplified this too. They reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself ... What does that mean? This means that you can take advantage of the degree. So, what you once counted with your finger, they do in one action: three in a cube is equal. It is written like this:.

It only remains remember the table of degrees... Unless, of course, you are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can continue to count with your finger.

Well, to finally convince you that the degrees were invented by idlers and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Life example no. 4

You have a million rubles. At the beginning of each year, you make another million from every million. That is, your every million at the beginning of each year doubles. How much money will you have in years? If you are now sitting and “counting with your finger,” then you are a very hardworking person and .. stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two times two ... in the second year - what happened was two more, in the third year ... Stop! You noticed that the number is multiplied by itself once. So two to the fifth power is a million! Now imagine that you have a competition and those millions will be received by the one who calculates faster ... Is it worth remembering the degrees of numbers, what do you think?

Real life example no. 5

You have a million. At the beginning of each year, you earn two more on every million. Great, isn't it? Every million triples. How much money will you have in years? Let's count. The first year - multiply by, then the result by another ... It's already boring, because you already understood everything: three times is multiplied by itself. So the fourth power is equal to a million. You just need to remember that three to the fourth power is or.

Now you know that by raising a number to a power, you will greatly facilitate your life. Let's take a look at what you can do with degrees and what you need to know about them.

Terms and concepts ... so as not to get confused

So, first, let's define the concepts. What do you think, what is exponent? It is very simple - this is the number that is "at the top" of the power of the number. Not scientific, but understandable and easy to remember ...

Well, at the same time that such degree basis? Even simpler - this is the number that is below, at the base.

Here's a drawing to be sure.

Well, in general view, in order to summarize and better remember ... A degree with a base "" and an exponent "" is read as "in degree" and is written as follows:

Degree of number with natural exponent

You probably already guessed: because the exponent is natural number... Yes, but what is natural number? Elementary! Natural numbers are those that are used in counting when listing objects: one, two, three ... When we count objects, we do not say: "minus five", "minus six", "minus seven". We also do not say: "one third", or "zero point, five tenths." These are not natural numbers. What numbers do you think they are?

Numbers like "minus five", "minus six", "minus seven" refer to whole numbers. In general, whole numbers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and a number. Zero is easy to understand - this is when there is nothing. What do negative ("minus") numbers mean? But they were invented primarily to indicate debts: if you have rubles on your phone, it means that you owe the operator rubles.

Any fractions are rational numbers. How do you think they came about? Very simple. Several thousand years ago, our ancestors discovered that they lacked natural numbers to measure length, weight, area, etc. And they came up with rational numbers... Interesting, isn't it?

There are also irrational numbers. What are these numbers? In short, endless decimal... For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Summary:

Let us define the concept of a degree, the exponent of which is a natural number (that is, an integer and positive).

Any number in the first power is equal to itself:
To square a number is to multiply it by itself:
To cube a number is to multiply it by itself three times:

Definition. Raising a number to a natural power means multiplying the number by itself times:
.

Power properties

Where did these properties come from? I will show you now.

Let's see: what is and ?

A-priory:

How many factors are there in total?

It's very simple: we added multipliers to the multipliers, and the total is multipliers.

But by definition, it is the degree of a number with an exponent, that is, as required to prove.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule necessarily must have the same bases!
Therefore, we combine the degrees with the base, but remains a separate factor:

just for the product of degrees!

In no case can you write that.

2.that is -th power of a number

Just as with the previous property, let us turn to the definition of the degree:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In essence, this can be called "bracketing the indicator". But you should never do this in total:

Let's remember the abbreviated multiplication formulas: how many times did we want to write?

But this is not true, after all.

Degree with negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the foundation?

In degrees with natural rate the basis can be any number... Indeed, we can multiply any numbers by each other, be they positive, negative, or even.

Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, will the number be positive or negative? A? ? With the first, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But negative is a little more interesting. After all, we remember a simple rule from the 6th grade: "minus by minus gives a plus." That is, or. But if we multiply by, it works.

Decide on your own which sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Did you manage?

Here are the answers: In the first four examples, hopefully everything is clear? We just look at the base and exponent and apply the appropriate rule.

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive.

Well, unless the base is zero. The foundation is not equal, is it? Obviously not, since (because).

Example 6) is no longer so easy!

6 examples to train

Parsing the solution 6 examples

If we ignore the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

Let's take a close look at the denominator. It looks a lot like one of the multipliers in the numerator, but what's wrong? Wrong order of terms. If they were to be reversed, the rule could be applied.

But how to do that? It turns out to be very easy: here the even degree of the denominator helps us.

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

Whole we call the natural numbers opposite to them (that is, taken with the sign "") and the number.

positive integer, but it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at some new cases. Let's start with an indicator equal to.

Any number in the zero degree is equal to one:

As always, let us ask ourselves the question: why is this so?

Consider a degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and got the same as it was -. And what number should you multiply so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number in the zero degree is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it should be equal to any degree - no matter how much you multiply by yourself, you will still get zero, this is clear. But on the other hand, like any number in the zero degree, it must be equal. So which of this is true? Mathematicians decided not to get involved and refused to raise zero to zero. That is, now we cannot not only divide by zero, but also raise it to a zero power.

Let's go further. In addition to natural numbers and numbers, negative numbers also belong to integers. To understand what a negative power is, let's do the same as last time: multiply some normal number by the same negative power:

From here it is already easy to express what you are looking for:

Now we will extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number in the negative power is inverse to the same number in the positive power. But at the same time the base cannot be null:(because you cannot divide by).

Let's summarize:

I. Expression not specified in case. If, then.

II. Any number to the zero degree is equal to one:.

III. A number that is not equal to zero is in negative power inverse to the same number in a positive power:.

Tasks for independent solution:

Well, and, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are terrible, but on the exam you have to be ready for anything! Solve these examples or analyze their solution if you could not solve and you will learn how to easily cope with them on the exam!

Let's continue to expand the circle of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is Fractional degree, consider the fraction:

Let's raise both sides of the equation to the power:

Now let's remember the rule about "Degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the th root.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse operation of the exponentiation:.

It turns out that. Obviously this special case can be expanded:.

Now we add the numerator: what is it? The answer is easily obtained using the degree-to-degree rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, you cannot extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But this is where the problem arises.

The number can be represented as other, cancellable fractions, for example, or.

And it turns out that it does exist, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write. But if we write down the indicator in a different way, and again we get a nuisance: (that is, we got a completely different result!).

To avoid such paradoxes, we consider only positive radix with fractional exponent.

So if:

- natural number;
- an integer;

Examples:

Rational exponents are very useful for converting rooted expressions, for example:

5 examples to train

Analysis of 5 examples for training

And now the hardest part. Now we will analyze irrational degree.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception of

Indeed, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are whole numbers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms.

For example, a natural exponent is a number multiplied by itself several times;

...zero-degree number- it is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely the number;

...integer negative exponent- it was as if some kind of "reverse process" took place, that is, the number was not multiplied by itself, but divided.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number.

But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a power to a power:

Now look at the indicator. Does he remind you of anything? We recall the formula for abbreviated multiplication, the difference of squares:

In this case,

It turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal, or both ordinary. Let's get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of degrees:

ADVANCED LEVEL

Determination of the degree

A degree is an expression of the form:, where:

— base of degree;
- exponent.

Degree with natural exponent (n = 1, 2, 3, ...)

Raising a number to a natural power n means multiplying the number by itself times:

Integer degree (0, ± 1, ± 2, ...)

If the exponent is whole positive number:

Erection to zero degree:

The expression is indefinite, because, on the one hand, to any degree - this, and on the other - any number to the th degree - this.

If the exponent is whole negative number:

(because you cannot divide by).

Once again about zeros: expression is undefined in case. If, then.

Examples:

Rational grade

- natural number;
- an integer;

Examples:

Power properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression, we get the following product:

But by definition, it is the degree of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule necessarily must have the same bases. Therefore, we combine the degrees with the base, but remains a separate factor:

One more important note: this rule is - for the product of degrees only!

By no means should I write that.

Just as with the previous property, let us turn to the definition of the degree:

Let's rearrange this piece like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In essence, this can be called "bracketing the indicator". But you should never do this in total:!

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

A degree with a negative base.

Up to this point, we have only discussed how it should be index degree. But what should be the foundation? In degrees with natural indicator the basis can be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, will the number be positive or negative? A? ?

With the first, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But negative is a little more interesting. After all, we remember a simple rule from the 6th grade: "minus by minus gives a plus." That is, or. But if we multiply by (), we get -.

And so on to infinity: with each subsequent multiplication, the sign will change. One can formulate such simple rules:

even degree, - number positive.
Negative number raised to odd degree, - number negative.
A positive number to any degree is a positive number.
Zero to any power is equal to zero.

Decide on your own which sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We just look at the base and exponent and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, unless the base is zero. The foundation is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and, divide them into each other, divide them into pairs and get:

Before examining the last rule, let's solve a few examples.

Calculate the values of the expressions:

Solutions :

If we ignore the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

Let's take a close look at the denominator. It looks a lot like one of the multipliers in the numerator, but what's wrong? Wrong order of terms. If they were reversed, Rule 3 could be applied. But how to do it? It turns out to be very easy: here the even degree of the denominator helps us.

If you multiply it by, nothing changes, right? But now it turns out the following:

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change at the same time! It cannot be replaced with by changing only one disadvantage that we do not want!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing more than a definition of an operation multiplication: there were only multipliers. That is, it is, by definition, the degree of a number with an exponent:

Example:

Irrational grade

In addition to the information about the degrees for the intermediate level, we will analyze the degree with an irrational exponent. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are whole numbers (that is, irrational numbers are all real numbers except rational).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely a number; a degree with a negative integer exponent is as if some kind of "reverse process" took place, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians created to extend the concept of a degree to the entire space of numbers.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number. But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the institute.

So what do we do when we see an irrational exponent? We are trying with all our might to get rid of it! :)

For example:

Decide for yourself:

Answers:

We recall the formula for the difference of squares. Answer: .
We bring fractions to the same form: either both decimal places, or both ordinary ones. We get, for example:.
Nothing special, we apply the usual degree properties:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree is called an expression of the form:, where:

Integer degree

degree, the exponent of which is a natural number (i.e. whole and positive).

Rational grade

degree, the exponent of which is negative and fractional numbers.

Irrational grade

degree, the exponent of which is an infinite decimal fraction or root.

Power properties

Features of degrees.

Negative number raised to even degree, - number positive.
Negative number raised to odd degree, - number negative.
A positive number to any degree is a positive number.
Zero is equal to any power.
Any number to the zero degree is equal to.

NOW YOUR WORD ...

How do you like the article? Write down in the comments like whether you like it or not.

Tell us about your experience with degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". This is how it sounds:

Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.

This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly catch up with the turtle."

How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:

During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia Zeno tells about a flying arrow:

The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What do I want to turn Special attention, so it is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, 4 July 2018

The distinction between set and multiset is very well documented in Wikipedia. We look.

As you can see, "there cannot be two identical elements in a set", but if there are identical elements in a set, such a set is called a "multiset". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "completely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.

No matter how mathematicians hide behind the phrase "chur, I'm in the house", or rather "mathematics studies abstract concepts," there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to the mathematicians themselves.

We studied mathematics very well and now we are sitting at the checkout, giving out salaries. Here comes a mathematician to us for his money. We count the entire amount to him and lay out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and hand the mathematician his “mathematical set of salary”. Let us explain the mathematics that he will receive the rest of the bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different banknote numbers on bills of the same denomination, which means that they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms in each coin is unique ...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science did not lie anywhere near here.

Look here. We select football stadiums with the same pitch. The area of the fields is the same, which means we have got a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How is it correct? And here the mathematician-shaman-shuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the multiset. In any case, he will convince us that he is right.

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

Sunday, 18 March 2018

The sum of the digits of the number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that is why they are shamans in order to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Need proof? Open Wikipedia and try to find the Sum of Digits of a Number page. It doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers and in the language of mathematics the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans - it is elementary.

Let's see what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What should be done in order to find the sum of the digits of this number? Let's go through all the steps in order.

1. We write down the number on a piece of paper. What have we done? We have converted the number to the graphic symbol of the number. This is not a mathematical operation.

2. We cut one resulting picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number 12345, I don't want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not look at every step under a microscope, we have already done that. Let's see the result.

As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you would get completely different results when determining the area of a rectangle in meters and centimeters.

Zero in all number systems looks the same and has no sum of digits. This is another argument for the fact that. A question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists - no. Reality is not all about numbers.

The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, it means it has nothing to do with mathematics.

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

Sign on the door Opens the door and says:

Ouch! Isn't this a women's toilet?
- Young woman! This is a laboratory for the study of the indiscriminate holiness of souls during the ascension to heaven! Halo on top and arrow pointing up. What other toilet?

Female ... The nimbus above and the down arrow is male.

If a piece of design art like this flashes before your eyes several times a day,

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

Personally, I make an effort on myself so that in a pooping person (one picture), I can see minus four degrees (a composition of several pictures: a minus sign, number four, the designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a stereotype of perception of graphic images. And mathematicians constantly teach us this. Here's an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive the number and the letter as one graphic symbol.

In this article we will figure out what is degree of... Here we will give definitions of the degree of a number, while considering in detail all possible exponents, starting with a natural exponent and ending with an irrational one. In the material you will find a lot of examples of degrees, covering all the subtleties that arise.

Page navigation.

Degree with natural exponent, square of number, cube of number

Let's start with. Looking ahead, we say that the definition of the degree of a number a with natural exponent n is given for a, which we will call basis degree, and n, which we will call exponent... We also note that the degree with a natural exponent is determined through the product, so to understand the material below, you need to have an idea of the multiplication of numbers.

Definition.

Power of number a with natural exponent n is an expression of the form a n, the value of which is equal to the product of n factors, each of which is equal to a, that is,.
In particular, the power of a number a with exponent 1 is the number a itself, that is, a 1 = a.

It should be said right away about the rules for reading degrees. Universal way read record a n is: "a to the power of n". In some cases, the following options are also acceptable: "a to the n-th power" and "n-th power of the number a". For example, let's take the power of 8 12, which is “eight to the power of twelve”, or “eight to the twelfth power”, or “the twelfth power of eight”.

The second degree of a number, as well as the third degree of a number, have their own names. The second power of a number is called by the square of the number for example, 7 2 reads “seven squared” or “the square of the number seven”. The third power of a number is called cube numbers for example, 5 3 can be read as “cube of five” or “cube of number 5”.

It's time to lead examples of degrees with natural values... Let's start with the exponent 5 7, here 5 is the base of the exponent and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9.

Note that in the last example, the base of the 4.32 degree is written in parentheses: to avoid confusion, we will put in parentheses all bases of the degree that are different from natural numbers. As an example, we give the following degrees with natural indicators , their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity in this moment, we will show the difference between the entries of the form (−2) 3 and −2 3. The expression (−2) 3 is the power of −2 with a natural exponent of 3, and the expression −2 3 (it can be written as - (2 3)) corresponds to the number, the value of the power 2 3.

Note that there is a notation for the degree of a number a with exponent n of the form a ^ n. Moreover, if n is a multivalued natural number, then the exponent is taken in parentheses. For example, 4 ^ 9 is another notation for the power of 4 9. And here are some more examples of writing degrees using the "^" symbol: 14 ^ (21), (−2,1) ^ (155). In what follows, we will mainly use the notation for the degree of the form a n.

One of the tasks, the inverse of raising to a power with a natural exponent, is the problem of finding the base of a degree from a known value of the degree and a known exponent. This task leads to.

It is known that the set of rational numbers consists of integers and fractional numbers, and each a fractional number can be presented as positive or negative common fraction... We defined the degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of a degree with a rational exponent, we need to give a sense to the degree of a number a with a fractional exponent m / n, where m is an integer and n is a natural number. Let's do it.

Consider a degree with a fractional exponent of the form. For the property of degree to degree to be valid, the equality ... If we take into account the obtained equality and the way we determined it, then it is logical to accept, provided that for the given m, n and a, the expression makes sense.

It is easy to verify that for all properties of a degree with an integer exponent (this is done in the section on properties of a degree with a rational exponent).

The above reasoning allows us to do the following. output: if for given m, n and a the expression makes sense, then the power of the number a with fractional exponent m / n is called the nth root of a to the power of m.

This statement brings us very close to determining the degree with a fractional exponent. It remains only to describe for which m, n and a the expression makes sense. There are two main approaches depending on the constraints on m, n and a.

The easiest way is to restrict a by assuming a≥0 for positive m and a> 0 for negative m (since for m≤0 the degree 0 m is not defined). Then we get the following definition of a fractional exponent.

Definition.

The power of a positive number a with a fractional exponent m / n, where m is an integer and n is a natural number, is called the nth root of the number a to the power of m, that is,.

A fractional power of zero is also determined with the only proviso that the indicator must be positive.

Definition.

Power of zero with positive fractional exponent m / n, where m is a positive integer and n is a natural number, is defined as .
When the degree is not determined, that is, the degree of a number zero with a fractional negative exponent does not make sense.

It should be noted that with such a definition of a degree with a fractional exponent, there is one nuance: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0. For example, it makes sense to write or, and the definition given above forces us to say that degrees with a fractional exponent of the form do not make sense, since the base should not be negative.

Another approach to determining the exponent with a fractional exponent m / n is to consider separately the odd and even exponents of the root. This approach requires an additional condition: the degree of the number a, the indicator of which is, is considered the power of the number a, the indicator of which is the corresponding irreducible fraction (the importance of this condition will be explained below). That is, if m / n is an irreducible fraction, then for any natural number k, the degree is preliminarily replaced by.

For even n and positive m, the expression makes sense for any non-negative a (an even root of a negative number does not make sense), for negative m, the number a must still be nonzero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be nonzero (so that there is no division by zero).

The above reasoning leads us to such a definition of the degree with a fractional exponent.

Definition.

Let m / n be an irreducible fraction, m an integer, and n a natural number. For any cancellable fraction, the exponent is replaced by. The power of a number with an irreducible fractional exponent m / n is for

Let us explain why a degree with a reducible fractional exponent is previously replaced by a degree with an irreducible exponent. If we simply defined the degree as, and did not make a reservation about the irreducibility of the fraction m / n, then we would be faced with situations similar to the following: since 6/10 = 3/5, then the equality should hold , but , a .

In one of the previous articles, we already mentioned the degree of number. Today we will try to orient ourselves in the process of finding its meaning. Scientifically speaking, we will be figuring out how to properly raise to a power. We will figure out how this process is carried out, at the same time we will touch upon all possible indicators of degree: natural, irrational, rational, whole.

So, let's take a closer look at the solutions of the examples and find out what it means:

Definition of the concept.
Raising to negative Art.
Whole indicator.
Raising a number to an irrational power.

Here is a definition that accurately reflects the meaning: "Exponentiation is the definition of the meaning of the power of a number."

Accordingly, raising the number a to Art. r and the process of finding the value of the exponent a with exponent r are identical concepts. For example, if the task is to calculate the value of the power (0.6) 6 ″, then it can be simplified to the expression “Raise the number 0.6 to the power of 6”.

After that, you can proceed directly to the construction rules.

Negative exponentiation

For clarity, you should pay attention to the following chain of expressions:

110 = 0.1 = 1 * 10 in minus 1 st.,

1100 = 0.01 = 1 * 10 in minus 2 steps.,

11000 = 0.0001 = 1 * 10 minus 3 st.,

110000 = 0.00001 = 1 * 10 in minus 4 degrees.

Thanks to these examples, you can clearly see the ability to instantly calculate 10 to any minus power. For this purpose, it is quite corny to shift the decimal component:

10 to -1 degrees - before the unit 1 zero;
at -3 - three zeros before one;
in -9 is 9 zeros and so on.

It is just as easy to understand according to this scheme, how much will be 10 to minus 5 tbsp. -

1100000=0,000001=(1*10)-5.

How to raise a natural number

Recalling the definition, we take into account that the natural number a in Art. n is equal to the product of n factors, each of which is equal to a. Let's illustrate: (a * a * ... a) n, where n is the number of numbers that are multiplied. Accordingly, in order to raise a to n, it is necessary to calculate the product of the following form: a * a * ... and divide by n times.

From this it becomes obvious that erection in natural art. relies on the ability to multiply(This material is covered in the section on multiplying real numbers). Let's take a look at the problem:

Erect -2 in the 4th st.

We are dealing with a natural indicator. Accordingly, the course of the decision will be as follows: (-2) in art. 4 = (-2) * (- 2) * (- 2) * (- 2). Now it remains only to carry out the multiplication of whole numbers: (- 2) * (- 2) * (- 2) * (- 2). We get 16.

Answer to the problem:

(-2) in art. 4 = 16.

Example:

Calculate the value: three point two sevens squared.

This example is equal to the following product: three point two sevenths multiplied by three point two sevenths. Remembering how the multiplication of mixed numbers is carried out, we complete the construction:

3 point 2 sevenths multiply by themselves;
equals 23 sevenths multiplied by 23 sevenths;
equal to 529 forty-ninth;
abbreviate and get 10 thirty-nine forty-nine.

Answer: 10 39/49

With regard to the issue of raising to an irrational indicator, it should be noted that calculations begin to be carried out after the completion of the preliminary rounding of the basis of the degree to a certain category that would allow obtaining a value with a given accuracy. For example, we need to square the number P (pi).

We start by rounding P to hundredths and get:

P squared = (3.14) 2 = 9.8596. However, if we reduce P to ten thousandths, we get P = 3.14159. Then squaring gets a completely different number: 9.8695877281.

It should be noted here that in many problems there is no need to raise irrational numbers to a power. As a rule, the answer is written either in the form of a degree, for example, the root of 6 to the power of 3, or, if the expression allows, its transformation is carried out: the root of 5 to the 7th degree = 125 root of 5.

How to raise a number to a whole power

This algebraic manipulation is appropriate take into account for the following cases:

for whole numbers;
for a zero indicator;
for a whole positive indicator.

Since practically all positive integers coincide with the mass of natural numbers, then setting to a positive integer power is the same process as setting in Art. natural. We described this process in the previous paragraph.

Now let's talk about calculating Art. null. We have already found out above that the zero degree of the number a can be determined for any nonzero a (real), while a in Art. 0 will equal 1.

Accordingly, raising any real number to zero st. will give one.

For example, 10 in st. 0 = 1, (-3.65) 0 = 1, and 0 in st. 0 cannot be determined.

In order to complete the raising to an integer power, it remains to decide on the options for integer negative values. We remember that Art. from a with integer exponent -z will be defined as a fraction. The denominator of the fraction is Art. with a positive integer value, the value of which we have already learned to find. Now it only remains to consider an example of construction.

Example:

Calculate the value of the number 2 in a cube with a negative integer exponent.

Solution process:

According to the definition of a degree with a negative indicator, we denote: two at minus 3 tbsp. equals one to two in the third degree.

The denominator is calculated simply: two cubed;

3 = 2*2*2=8.

Answer: two in minus 3rd tbsp. = one eighth.

First level

The degree and its properties. Comprehensive guide (2019)

Why are degrees needed? Where will they be useful to you? Why do you need to take the time to study them?

To find out everything about degrees, what they are for, how to use your knowledge in everyday life, read this article.

And, of course, knowledge of degrees will bring you closer to successfully passing the OGE or USE and to entering the university of your dreams.

Let "s go ... (Let's go!)

Important note! If instead of formulas you see gibberish, clear the cache. To do this, press CTRL + F5 (on Windows) or Cmd + R (on Mac).

FIRST LEVEL

Exponentiation is the same mathematical operation as addition, subtraction, multiplication, or division.

Now I will explain everything in human language using very simple examples. Pay attention. The examples are elementary, but they explain important things.

Let's start with addition.

There is nothing to explain. You already know everything: there are eight of us. Each has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

So, to count faster, easier and without errors, you just need to remember multiplication table... You can, of course, do everything slower, harder and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful:

What other clever counting tricks have lazy mathematicians come up with? Right - raising a number to a power.

Raising a number to a power

All you need to do is remember what is highlighted in the table of powers of numbers... Believe me, this will make your life much easier.

By the way, why is the second degree called square numbers, and the third - cube? What does it mean? That's a very good question. Now you will have both squares and cubes.

Life example # 1

Let's start with a square or the second power of a number.

Real life example # 2

Here's a task for you, count how many squares are on the chessboard using the square of the number ... On one side of the cells and on the other, too. To count their number, you need to multiply eight by eight or ... if you notice that the chessboard is a square with a side, then you can square eight. You will get cells. () So?

Life example no. 3

Now the cube or the third power of the number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Surprisingly, right?) Draw a pool: the bottom is a meter in size and a meter deep and try to calculate how many cubic meters by meter will enter your pool.

Well, to finally convince you that the degrees were invented by idlers and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Life example no. 4

Real life example no. 5

Now you know that by raising a number to a power, you will greatly facilitate your life. Let's take a look at what you can do with degrees and what you need to know about them.

Terms and concepts ... so as not to get confused

Well, at the same time that such degree basis? Even simpler - this is the number that is below, at the base.

Here's a drawing to be sure.

Well, in general terms, in order to generalize and remember better ... A degree with a base "" and an indicator "" is read as "in degree" and is written as follows:

Degree of number with natural exponent

You probably guessed by now: because the exponent is a natural number. Yes, but what is natural number? Elementary! Natural numbers are those that are used in counting when listing objects: one, two, three ... When we count objects, we do not say: "minus five", "minus six", "minus seven". We also do not say: "one third", or "zero point, five tenths." These are not natural numbers. What numbers do you think they are?

There are also irrational numbers. What are these numbers? In short, an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Summary:

Let us define the concept of a degree, the exponent of which is a natural number (that is, an integer and positive).

Any number in the first power is equal to itself:
To square a number is to multiply it by itself:
To cube a number is to multiply it by itself three times:

Definition. Raising a number to a natural power means multiplying the number by itself times:
.

Power properties

Where did these properties come from? I will show you now.

Let's see: what is and ?

A-priory:

How many factors are there in total?

It's very simple: we added multipliers to the multipliers, and the total is multipliers.

But by definition, it is the degree of a number with an exponent, that is, as required to prove.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule necessarily must have the same bases!
Therefore, we combine the degrees with the base, but remains a separate factor:

just for the product of degrees!

In no case can you write that.

2.that is -th power of a number

Just as with the previous property, let us turn to the definition of the degree:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In essence, this can be called "bracketing the indicator". But you should never do this in total:

Let's remember the abbreviated multiplication formulas: how many times did we want to write?

But this is not true, after all.

Degree with negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the foundation?

In degrees with natural rate the basis can be any number... Indeed, we can multiply any numbers by each other, be they positive, negative, or even.

Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, will the number be positive or negative? A? ? With the first, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But negative is a little more interesting. After all, we remember a simple rule from the 6th grade: "minus by minus gives a plus." That is, or. But if we multiply by, it works.

Decide on your own which sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Did you manage?

Here are the answers: In the first four examples, hopefully everything is clear? We just look at the base and exponent and apply the appropriate rule.

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive.

Well, unless the base is zero. The foundation is not equal, is it? Obviously not, since (because).

Example 6) is no longer so easy!

6 examples to train

Parsing the solution 6 examples

If we ignore the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

But how to do that? It turns out to be very easy: here the even degree of the denominator helps us.

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

Whole we call the natural numbers opposite to them (that is, taken with the sign "") and the number.

positive integer, but it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at some new cases. Let's start with an indicator equal to.

Any number in the zero degree is equal to one:

As always, let us ask ourselves the question: why is this so?

Consider a degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and got the same as it was -. And what number should you multiply so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number in the zero degree is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

From here it is already easy to express what you are looking for:

Now we will extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number in the negative power is inverse to the same number in the positive power. But at the same time the base cannot be null:(because you cannot divide by).

Let's summarize:

I. Expression not specified in case. If, then.

II. Any number to the zero degree is equal to one:.

III. A number that is not equal to zero is in negative power inverse to the same number in a positive power:.

Tasks for independent solution:

Well, and, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

Let's continue to expand the circle of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is Fractional degree, consider the fraction:

Let's raise both sides of the equation to the power:

Now let's remember the rule about "Degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the th root.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse operation of the exponentiation:.

It turns out that. Obviously, this particular case can be extended:.

Now we add the numerator: what is it? The answer is easily obtained using the degree-to-degree rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, you cannot extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But this is where the problem arises.

The number can be represented as other, cancellable fractions, for example, or.

And it turns out that it does exist, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write. But if we write down the indicator in a different way, and again we get a nuisance: (that is, we got a completely different result!).

To avoid such paradoxes, we consider only positive radix with fractional exponent.

So if:

- natural number;
- an integer;