Comparing numbers by writing them. Comparison of numbers. Largest negative and smallest positive integer

The absolute value of a number

Modulus of a denote $|a|$. Vertical lines to the right and left of the number form the sign of the modulus.

For example, the modulus of any number (natural, integer, rational or irrational) is written as follows: $|5|$, $|-11|$, $|2,345|$, $|\sqrt(45)|$.

Definition 1

Modulus of a equals $a$ itself if $a$ is positive, $−a$ if $a$ is negative, or $0$ if $a=0$.

This definition of the modulus of a number can be written as follows:

$|a|= \begin(cases) a, & a > 0, \\ 0, & a=0,\\ -a, &a

You can use a shorter notation:

$|a|=\begin(cases) a, & a \geq 0 \\ -a, & a

Example 1

Calculate the modulus of the numbers $23$ and $-3.45$.

Solution.

Find the absolute value of $23$.

The number $23$ is positive, therefore, by definition, the modulus of a positive number is equal to this number:

Find the modulus of the number $–3.45$.

The number $–3.45$ is a negative number, therefore, according to the definition, the modulus of a negative number is equal to the number opposite to the given one:

Answer: $|23|=23$, $|-3,45|=3,45$.

Definition 2

The modulus of a number is the absolute value of a number.

Thus, the module of a number is the number under the sign of the module without taking into account its sign.

Modulus of a number as a distance

The geometric value of the modulus of the number: the modulus of a number is the distance.

Definition 3

Modulus of a is the distance from the reference point (zero) on the number line to the point that corresponds to the number $a$.

Example 2

For example, the modulus of $12$ is $12$, because the distance from the reference point to the point with coordinate $12$ is equal to twelve:

The point with coordinate $−8.46$ is located at a distance of $8.46$ from the origin, so $|-8.46|=8.46$.

Modulus of a number as an arithmetic square root

Definition 4

Modulus of a is the arithmetic square root of $a^2$:

$|a|=\sqrt(a^2)$.

Example 3

Calculate the modulus of the number $–14$ using the definition of the modulus of the number in terms of the square root.

Solution.

$|-14|=\sqrt(((-14)^2)=\sqrt((-14) \cdot (-14))=\sqrt(14 \cdot 14)=\sqrt((14)^2 )=14$.

Answer: $|-14|=14$.

Comparing negative numbers

Comparison of negative numbers is based on a comparison of the modules of these numbers.

Remark 1

Rule for comparing negative numbers:

If the modulus of one of the negative numbers is greater, then such a number is less;
if the modulus of one of the negative numbers is less, then such a number is large;
if the modules of the numbers are equal, then the negative numbers are equal.

Remark 2

On the number line, the smaller negative number is located to the left of the larger negative number.

Example 4

Compare negative numbers $−27$ and $−4$.

Solution.

According to the rule for comparing negative numbers, we first find the moduli of the numbers $–27$ and $–4$, and then compare the obtained positive numbers.

Thus, we get that $–27 |-4|$.

Answer: $–27

When comparing negative rational numbers, it is necessary to convert both numbers to the form of ordinary fractions or decimal fractions.

Comparing numbers is one of the easiest and most enjoyable topics in a mathematics course. However, it must be said that it is not so simple. For example, few people have difficulty comparing single or double digit positive numbers.

But numbers with a large number of signs already cause problems, often people get lost when comparing negative numbers and do not remember how to compare two numbers with different signs. We will try to answer all these questions.

Rules for Comparing Positive Numbers

Let's start with the simplest - with numbers that do not have any sign in front of them, that is, with positive ones.

First of all, it is worth remembering that all positive numbers are, by definition, greater than zero, even if we are talking about a fractional number without an integer. For example, the decimal fraction 0.2 will be greater than zero, since on the coordinate line the point corresponding to it is still two small divisions away from zero.
If we are talking about comparing two positive numbers with a large number of characters, then you need to compare each of the digits. For example, 32 and 33. The tens digit for these numbers is the same, but the number 33 is larger, because in the units digit "3" is greater than "2".
How do you compare two decimals? Here you need to look first of all at the integer part - for example, a fraction of 3.5 will be less than 4.6. What if the integer part is the same, but the decimal places are different? In this case, the rule for integers applies - you need to compare signs by digits until you find larger and smaller tenths, hundredths, thousandths. For example, 4.86 is greater than 4.75 because eight tenths is greater than seven.

Comparing negative numbers

If we have some numbers -a and -c in the problem, and we need to determine which of them is greater, then the universal rule applies. First, the modules of these numbers are written out - |a| and |c| - and are compared with each other. The number whose modulus is greater will be smaller in comparison of negative numbers, and vice versa - the larger number will be the one whose modulus is less.

What if you need to compare a negative and a positive number?

Only one rule works here, and it is elementary. Positive numbers are always greater than numbers with a minus sign - whatever they are. For example, the number "1" will always be greater than the number "-1458" simply because the unit is to the right of zero on the coordinate line.

You also need to remember that any negative number is always less than zero.

When solving equations and inequalities, as well as problems with modules, it is required to locate the found roots on the real line. As you know, the found roots can be different. They can be like this:, or they can be like this:,.

Accordingly, if the numbers are not rational but irrational (if you forgot what it is, look in the topic), or are complex mathematical expressions, then placing them on the number line is very problematic. Moreover, calculators cannot be used in the exam, and an approximate calculation does not give 100% guarantees that one number is less than another (what if there is a difference between the compared numbers?).

Of course, you know that positive numbers are always greater than negative ones, and that if we represent a number axis, then when compared, the largest numbers will be to the right than the smallest: ; ; etc.

But is it always so easy? Where on the number line we mark .

How to compare them, for example, with a number? That's where the rub is...)

First, let's talk about in general terms how and what to compare.

Important: it is desirable to make transformations in such a way that the inequality sign does not change! That is, in the course of transformations, it is undesirable to multiply by a negative number, and it is forbidden square if one of the parts is negative.

Fraction Comparison

So, we need to compare two fractions: and.

There are several options for how to do this.

Option 1. Bring fractions to a common denominator.

Let's write it as an ordinary fraction:

- (as you can see, I also reduced by the numerator and denominator).

Now we need to compare fractions:

Now we can continue to compare also in two ways. We can:

just reduce everything to a common denominator, presenting both fractions as improper (the numerator is greater than the denominator):
Which number is greater? That's right, the one whose numerator is greater, that is, the first.
“discard” (assume that we subtracted one from each fraction, and the ratio of fractions to each other, respectively, has not changed) and we will compare the fractions:
We also bring them to a common denominator:
We got exactly the same result as in the previous case - the first number is greater than the second:
Let's also check whether we have correctly subtracted one? Let's calculate the difference in the numerator in the first calculation and the second:
1)
2)

So, we looked at how to compare fractions, bringing them to a common denominator. Let's move on to another method - comparing fractions by bringing them to a common ... numerator.

Option 2. Comparing fractions by reducing to a common numerator.

Yes Yes. This is not a typo. At school, this method is rarely taught to anyone, but very often it is very convenient. So that you quickly understand its essence, I will ask you only one question - “in what cases is the value of the fraction the largest?” Of course, you will say "when the numerator is as large as possible, and the denominator is as small as possible."

For example, you will definitely say that True? And if we need to compare such fractions: I think you, too, will immediately correctly put the sign, because in the first case they are divided into parts, and in the second into whole ones, which means that in the second case the pieces are very small, and accordingly:. As you can see, the denominators are different here, but the numerators are the same. However, in order to compare these two fractions, you do not need to find a common denominator. Although ... find it and see if the comparison sign is still wrong?

But the sign is the same.

Let's return to our original task - to compare and. We will compare and We bring these fractions not to a common denominator, but to a common numerator. For this it's simple numerator and denominator multiply the first fraction by. We get:

And. Which fraction is larger? That's right, the first one.

Option 3. Comparing fractions using subtraction.

How to compare fractions using subtraction? Yes, very simple. We subtract another from one fraction. If the result is positive, then the first fraction (reduced) is greater than the second (subtracted), and if negative, then vice versa.

In our case, let's try to subtract the first fraction from the second: .

As you already understood, we also translate into an ordinary fraction and get the same result -. Our expression becomes:

Further, we still have to resort to reduction to a common denominator. The question is how: in the first way, converting fractions into improper ones, or in the second, as if “removing” the unit? By the way, this action has a completely mathematical justification. Look:

I like the second option better, since multiplying in the numerator when reducing to a common denominator becomes many times easier.

We bring to a common denominator:

The main thing here is not to get confused about what number and where we subtracted from. Carefully look at the course of the solution and do not accidentally confuse the signs. We subtracted the first from the second number and got a negative answer, so? .. That's right, the first number is greater than the second.

Got it? Try comparing fractions:

Stop, stop. Do not rush to bring to a common denominator or subtract. Look: it can be easily converted to a decimal fraction. How much will it be? Right. What ends up being more?

This is another option - comparing fractions by reducing to a decimal.

Option 4. Comparing fractions using division.

Yes Yes. And so it is also possible. The logic is simple: when we divide a larger number by a smaller one, we get a number greater than one in the answer, and if we divide a smaller number by a larger one, then the answer falls on the interval from to.

To remember this rule, take for comparison any two prime numbers, for example, and. Do you know what's more? Now let's divide by. Our answer is . Accordingly, the theory is correct. If we divide by, what we get is less than one, which in turn confirms what is actually less.

Let's try to apply this rule on ordinary fractions. Compare:

Divide the first fraction by the second:

Let's shorten by and by.

The result is less, so the dividend is less than the divisor, that is:

We have analyzed all possible options for comparing fractions. As you can see there are 5 of them:

reduction to a common denominator;
reduction to a common numerator;
reduction to the form of a decimal fraction;
subtraction;
division.

Ready to workout? Compare fractions in the best way:

Let's compare the answers:

(- convert to decimal)
(divide one fraction by another and reduce by the numerator and denominator)
(select the whole part and compare fractions according to the principle of the same numerator)
(divide one fraction by another and reduce by the numerator and denominator).

2. Comparison of degrees

Now imagine that we need to compare not just numbers, but expressions where there is a degree ().

Of course, you can easily put a sign:

After all, if we replace the degree with multiplication, we get:

From this small and primitive example, the rule follows:

Now try to compare the following: . You can also easily put a sign:

Because if we replace exponentiation with multiplication...

In general, you understand everything, and it is not difficult at all.

Difficulties arise only when, when compared, the degrees have different bases and indicators. In this case, it is necessary to try to bring to a common basis. For example:

Of course, you know that this, accordingly, the expression takes the form:

Let's open the brackets and compare what happens:

A somewhat special case is when the base of the degree () is less than one.

If, then of two degrees or more, the one whose indicator is less.

Let's try to prove this rule. Let be.

We introduce some natural number as the difference between and.

Logical, isn't it?

Now let's pay attention to the condition - .

Respectively: . Hence, .

For example:

As you understand, we considered the case when the bases of the powers are equal. Now let's see when the base is in the range from to, but the exponents are equal. Everything is very simple here.

Let's remember how to compare this with an example:

Of course, you quickly calculated:

Therefore, when you come across similar problems for comparison, keep in mind some simple similar example that you can quickly calculate, and based on this example, put down signs in a more complex one.

When performing transformations, remember that if you multiply, add, subtract or divide, then all actions must be done on both the left and right sides (if you multiply by, then you need to multiply both).

In addition, there are times when doing any manipulations is simply unprofitable. For example, you need to compare. In this case, it is not so difficult to raise to a power, and arrange the sign based on this:

Let's practice. Compare degrees:

Ready to compare answers? Here's what I got:

- the same as
- the same as
- the same as
- the same as

3. Comparison of numbers with a root

Let's start with what are roots? Do you remember this entry?

The root of a real number is a number for which equality holds.

Roots odd degree exist for negative and positive numbers, and even roots- Only for positive.

The value of the root is often an infinite decimal, which makes it difficult to accurately calculate it, so it is important to be able to compare roots.

If you forgot what it is and what it is eaten with -. If you remember everything, let's learn to compare the roots step by step.

Let's say we need to compare:

To compare these two roots, you do not need to do any calculations, just analyze the very concept of "root". Got what I'm talking about? Yes, about this: otherwise it can be written as the third power of some number, equal to the root expression.

What more? or? This, of course, you can compare without any difficulty. The larger the number we raise to a power, the larger the value will be.

So. Let's get the rule.

If the exponents of the roots are the same (in our case, this is), then it is necessary to compare the root expressions (and) - the larger the root number, the greater the value of the root with equal indicators.

Difficult to remember? Then just keep an example in mind and. That more?

The exponents of the roots are the same, since the root is square. The root expression of one number () is greater than another (), which means that the rule is really true.

But what if the radical expressions are the same, but the degrees of the roots are different? For example: .

It is also quite clear that when extracting a root of a greater degree, a smaller number will be obtained. Let's take for example:

Denote the value of the first root as, and the second - as, then:

You can easily see that there should be more in these equations, therefore:

If the root expressions are the same(in our case), and the exponents of the roots are different(in our case, this is and), then it is necessary to compare the exponents(And) - the larger the exponent, the smaller the given expression.

Try comparing the following roots:

Let's compare the results?

We have successfully dealt with this :). Another question arises: what if we are all different? And the degree, and the radical expression? Not everything is so difficult, we just need to ... "get rid" of the root. Yes Yes. Get rid of it.)

If we have different degrees and radical expressions, it is necessary to find the least common multiple (read the section about) for the root exponents and raise both expressions to a power equal to the least common multiple.

That we are all in words and in words. Here's an example:

We look at the indicators of the roots - and. Their least common multiple is .
Let's raise both expressions to a power:
Let's transform the expression and expand the brackets (more details in the chapter):
Let's consider what we have done, and put a sign:

4. Comparison of logarithms

So, slowly but surely, we approached the question of how to compare logarithms. If you don’t remember what kind of animal this is, I advise you to read the theory from the section first. Read? Then answer some important questions:

What is the argument of the logarithm and what is its base?
What determines whether a function is increasing or decreasing?

If you remember everything and learned it well - let's get started!

In order to compare logarithms with each other, you need to know only 3 tricks:

reduction to the same base;
casting to the same argument;
comparison with the third number.

First, pay attention to the base of the logarithm. You remember that if it is less, then the function decreases, and if it is greater, then it increases. This is what our judgments will be based on.

Consider comparing logarithms that have already been reduced to the same base or argument.

To begin with, let's simplify the problem: let in the compared logarithms equal grounds. Then:

The function, when increases on the interval from, means, by definition, then (“direct comparison”).
Example:- the bases are the same, respectively, we compare the arguments: , therefore:
The function, at, decreases on the interval from, which means, by definition, then (“reverse comparison”). - the bases are the same, respectively, we compare the arguments: , however, the sign of the logarithms will be “reverse”, since the function decreases: .

Now consider the cases where the bases are different, but the arguments are the same.

The base is bigger.
- . In this case, we use "reverse comparison". For example: - the arguments are the same, and. We compare the bases: however, the sign of the logarithms will be “reverse”:
Base a is in between.
- . In this case, we use "direct comparison". For example:
- . In this case, we use "reverse comparison". For example:

Let's write everything in a general tabular form:

, wherein	, wherein

Accordingly, as you already understood, when comparing logarithms, we need to bring to the same base, or argument, We come to the same base using the formula for moving from one base to another.

You can also compare logarithms with a third number and, based on this, infer what is less and what is more. For example, think about how to compare these two logarithms?

A little hint - for comparison, the logarithm will help you a lot, the argument of which will be equal.

Thought? Let's decide together.

We can easily compare these two logarithms with you:

Don't know how? See above. We just took it apart. What sign will be there? Right:

Agree?

Let's compare with each other:

You should get the following:

Now combine all our conclusions into one. Happened?

5. Comparison of trigonometric expressions.

What is sine, cosine, tangent, cotangent? What is the unit circle for and how to find the value of trigonometric functions on it? If you do not know the answers to these questions, I highly recommend that you read the theory on this topic. And if you know, then comparing trigonometric expressions with each other is not difficult for you!

Let's refresh our memory a bit. Let's draw a unit trigonometric circle and a triangle inscribed in it. Did you manage? Now mark on which side we have the cosine, and on which sine, using the sides of the triangle. (Of course, you remember that the sine is the ratio of the opposite side to the hypotenuse, and the cosine of the adjacent one?). Did you draw? Great! The final touch - put down where we will have it, where and so on. Put down? Phew) Compare what happened with me and you.

Phew! Now let's start the comparison!

Suppose we need to compare and . Draw these angles using the hints in the boxes (where we have marked where), laying out the points on the unit circle. Did you manage? Here's what I got.

Now let's lower the perpendicular from the points we marked on the circle to the axis ... Which one? Which axis shows the value of the sines? Right, . Here is what you should get:

Looking at this figure, which is bigger: or? Of course, because the point is above the point.

Similarly, we compare the value of cosines. We only lower the perpendicular onto the axis ... Right, . Accordingly, we look at which point is to the right (well, or higher, as in the case of sines), then the value is greater.

You probably already know how to compare tangents, right? All you need to know is what is tangent. So what is tangent?) That's right, the ratio of sine to cosine.

To compare the tangents, we also draw an angle, as in the previous case. Let's say we need to compare:

Did you draw? Now we also mark the values of the sine on the coordinate axis. Noted? And now indicate the values of the cosine on the coordinate line. Happened? Let's compare:

Now analyze what you have written. - we divide a large segment into a small one. The answer will be a value that is exactly greater than one. Right?

And when we divide the small one by the big one. The answer will be a number that is exactly less than one.

So the value of which trigonometric expression is greater?

Right:

As you now understand, the comparison of cotangents is the same, only in reverse: we look at how the segments that define cosine and sine relate to each other.

Try to compare the following trigonometric expressions yourself:

Examples.

Answers.

COMPARISON OF NUMBERS. AVERAGE LEVEL.

Which of the numbers is greater: or? The answer is obvious. And now: or? Not so obvious anymore, right? And so: or?

Often you need to know which of the numeric expressions is greater. For example, when solving an inequality, put points on the axis in the correct order.

Now I will teach you to compare such numbers.

If you need to compare numbers and, put a sign between them (derived from the Latin word Versus or abbreviated vs. - against):. This sign replaces the unknown inequality sign (). Further, we will perform identical transformations until it becomes clear which sign should be put between the numbers.

The essence of comparing numbers is as follows: we treat the sign as if it were some kind of inequality sign. And with the expression, we can do everything we usually do with inequalities:

add any number to both parts (and subtract, of course, we can also)
“move everything in one direction”, that is, subtract one of the compared expressions from both parts. In place of the subtracted expression will remain: .
multiply or divide by the same number. If this number is negative, the inequality sign is reversed: .
Raise both sides to the same power. If this power is even, you must make sure that both parts have the same sign; if both parts are positive, the sign does not change when raised to a power, and if they are negative, then it changes to the opposite.
take the root of the same degree from both parts. If we extract the root of an even degree, you must first make sure that both expressions are non-negative.
any other equivalent transformations.

Important: it is desirable to make transformations in such a way that the inequality sign does not change! That is, in the course of transformations, it is undesirable to multiply by a negative number, and it is impossible to square if one of the parts is negative.

Let's look at a few typical situations.

1. Exponentiation.

Example.

Which is more: or?

Solution.

Since both sides of the inequality are positive, we can square to get rid of the root:

Example.

Which is more: or?

Solution.

Here, too, we can square, but this will only help us get rid of the square root. Here it is necessary to raise to such a degree that both roots disappear. This means that the exponent of this degree must be divisible by both (the degree of the first root) and by. This number is, so we raise it to the th power:

2. Multiplication by the conjugate.

Example.

Which is more: or?

Solution.

Multiply and divide each difference by the conjugate sum:

Obviously, the denominator on the right side is greater than the denominator on the left. Therefore, the right fraction is less than the left:

3. Subtraction

Let's remember that.

Example.

Which is more: or?

Solution.

Of course, we could square everything, regroup, and square again. But you can do something smarter:

It can be seen that each term on the left side is less than each term on the right side.

Accordingly, the sum of all terms on the left side is less than the sum of all terms on the right side.

But be careful! We were asked more...

The right side is larger.

Example.

Compare numbers and.

Solution.

Remember the trigonometry formulas:

Let us check in which quarters the points and lie on the trigonometric circle.

4. Division.

Here we also use a simple rule: .

With or, that is.

When the sign changes: .

Example.

Make a comparison: .

Solution.

5. Compare the numbers with the third number

If and, then (law of transitivity).

Example.

Compare.

Solution.

Let's compare the numbers not with each other, but with the number.

It's obvious that.

On the other side, .

Example.

Which is more: or?

Solution.

Both numbers are larger but smaller. Choose a number such that it is greater than one but less than the other. For example, . Let's check:

6. What to do with logarithms?

Nothing special. How to get rid of logarithms is described in detail in the topic. The basic rules are:

\[(\log _a)x \vee b(\rm( )) \Leftrightarrow (\rm( ))\left[ (\begin(array)(*(20)(l))(x \vee (a^ b)\;(\rm(at))\;a > 1)\\(x \wedge (a^b)\;(\rm(at))\;0< a < 1}\end{array}} \right.\] или \[{\log _a}x \vee {\log _a}y{\rm{ }} \Leftrightarrow {\rm{ }}\left[ {\begin{array}{*{20}{l}}{x \vee y\;{\rm{при}}\;a >1)\\(x \wedge y\;(\rm(at))\;0< a < 1}\end{array}} \right.\]

We can also add a rule about logarithms with different bases and the same argument:

It can be explained as follows: the larger the base, the lesser degree it will have to be erected to get the same one. If the base is smaller, then the opposite is true, since the corresponding function is monotonically decreasing.

Example.

Compare numbers: i.

Solution.

According to the above rules:

And now the advanced formula.

The rule for comparing logarithms can also be written shorter:

Example.

Which is more: or?

Solution.

Example.

Compare which of the numbers is greater: .

Solution.

COMPARISON OF NUMBERS. BRIEFLY ABOUT THE MAIN

1. Exponentiation

If both sides of the inequality are positive, they can be squared to get rid of the root

2. Multiplication by the conjugate

A conjugate is a multiplier that complements the expression to the formula for the difference of squares: - conjugate for and vice versa, because .

3. Subtraction

4. Division

At or that is

When the sign changes:

5. Comparison with the third number

If and then

6. Comparison of logarithms

Basic Rules:

Logarithms with different bases and the same argument.

But is it always so easy? Where on the number line we mark .

How to compare them, for example, with a number? That's where the rub is...)

To begin with, let's talk in general terms about how and what to compare.

Fraction Comparison

So, we need to compare two fractions: and.

There are several options for how to do this.

Option 1. Bring fractions to a common denominator.

Let's write it as an ordinary fraction:

- (as you can see, I also reduced by the numerator and denominator).

Now we need to compare fractions:

Now we can continue to compare also in two ways. We can:

just reduce everything to a common denominator, presenting both fractions as improper (the numerator is greater than the denominator):
Which number is greater? That's right, the one whose numerator is greater, that is, the first.
“discard” (assume that we subtracted one from each fraction, and the ratio of fractions to each other, respectively, has not changed) and we will compare the fractions:
We also bring them to a common denominator:
We got exactly the same result as in the previous case - the first number is greater than the second:
Let's also check whether we have correctly subtracted one? Let's calculate the difference in the numerator in the first calculation and the second:
1)
2)

So, we looked at how to compare fractions, bringing them to a common denominator. Let's move on to another method - comparing fractions by bringing them to a common ... numerator.

Option 2. Comparing fractions by reducing to a common numerator.

But the sign is the same.

And. Which fraction is larger? That's right, the first one.

Option 3. Comparing fractions using subtraction.

In our case, let's try to subtract the first fraction from the second: .

As you already understood, we also translate into an ordinary fraction and get the same result -. Our expression becomes:

I like the second option better, since multiplying in the numerator when reducing to a common denominator becomes many times easier.

We bring to a common denominator:

Got it? Try comparing fractions:

Stop, stop. Do not rush to bring to a common denominator or subtract. Look: it can be easily converted to a decimal fraction. How much will it be? Right. What ends up being more?

This is another option - comparing fractions by reducing to a decimal.

Option 4. Comparing fractions using division.

Let's try to apply this rule on ordinary fractions. Compare:

Divide the first fraction by the second:

Let's shorten by and by.

The result is less, so the dividend is less than the divisor, that is:

We have analyzed all possible options for comparing fractions. As you can see there are 5 of them:

reduction to a common denominator;
reduction to a common numerator;
reduction to the form of a decimal fraction;
subtraction;
division.

Ready to workout? Compare fractions in the best way:

Let's compare the answers:

(- convert to decimal)
(divide one fraction by another and reduce by the numerator and denominator)
(select the whole part and compare fractions according to the principle of the same numerator)
(divide one fraction by another and reduce by the numerator and denominator).

2. Comparison of degrees

Now imagine that we need to compare not just numbers, but expressions where there is a degree ().

Of course, you can easily put a sign:

After all, if we replace the degree with multiplication, we get:

From this small and primitive example, the rule follows:

Now try to compare the following: . You can also easily put a sign:

Because if we replace exponentiation with multiplication...

In general, you understand everything, and it is not difficult at all.

Difficulties arise only when, when compared, the degrees have different bases and indicators. In this case, it is necessary to try to bring to a common basis. For example:

Of course, you know that this, accordingly, the expression takes the form:

Let's open the brackets and compare what happens:

A somewhat special case is when the base of the degree () is less than one.

If, then of two degrees or more, the one whose indicator is less.

Let's try to prove this rule. Let be.

We introduce some natural number as the difference between and.

Logical, isn't it?

Now let's pay attention to the condition - .

Respectively: . Hence, .

For example:

As you understand, we considered the case when the bases of the powers are equal. Now let's see when the base is in the range from to, but the exponents are equal. Everything is very simple here.

Let's remember how to compare this with an example:

Of course, you quickly calculated:

Let's practice. Compare degrees:

Ready to compare answers? Here's what I got:

- the same as
- the same as
- the same as
- the same as

3. Comparison of numbers with a root

Let's start with what are roots? Do you remember this entry?

The root of a real number is a number for which equality holds.

Roots odd degree exist for negative and positive numbers, and even roots- Only for positive.

The value of the root is often an infinite decimal, which makes it difficult to accurately calculate it, so it is important to be able to compare roots.

If you forgot what it is and what it is eaten with -. If you remember everything, let's learn to compare the roots step by step.

Let's say we need to compare:

What more? or? This, of course, you can compare without any difficulty. The larger the number we raise to a power, the larger the value will be.

So. Let's get the rule.

Difficult to remember? Then just keep an example in mind and. That more?

The exponents of the roots are the same, since the root is square. The root expression of one number () is greater than another (), which means that the rule is really true.

But what if the radical expressions are the same, but the degrees of the roots are different? For example: .

It is also quite clear that when extracting a root of a greater degree, a smaller number will be obtained. Let's take for example:

Denote the value of the first root as, and the second - as, then:

You can easily see that there should be more in these equations, therefore:

Try comparing the following roots:

Let's compare the results?

That we are all in words and in words. Here's an example:

We look at the indicators of the roots - and. Their least common multiple is .
Let's raise both expressions to a power:
Let's transform the expression and expand the brackets (more details in the chapter):
Let's consider what we have done, and put a sign:

4. Comparison of logarithms

What is the argument of the logarithm and what is its base?
What determines whether a function is increasing or decreasing?

If you remember everything and learned it well - let's get started!

In order to compare logarithms with each other, you need to know only 3 tricks:

reduction to the same base;
casting to the same argument;
comparison with the third number.

First, pay attention to the base of the logarithm. You remember that if it is less, then the function decreases, and if it is greater, then it increases. This is what our judgments will be based on.

Consider comparing logarithms that have already been reduced to the same base or argument.

To begin with, let's simplify the problem: let in the compared logarithms equal grounds. Then:

The function, when increases on the interval from, means, by definition, then (“direct comparison”).
Example:- the bases are the same, respectively, we compare the arguments: , therefore:
The function, at, decreases on the interval from, which means, by definition, then (“reverse comparison”). - the bases are the same, respectively, we compare the arguments: , however, the sign of the logarithms will be “reverse”, since the function decreases: .

Now consider the cases where the bases are different, but the arguments are the same.

The base is bigger.
- . In this case, we use "reverse comparison". For example: - the arguments are the same, and. We compare the bases: however, the sign of the logarithms will be “reverse”:
Base a is in between.
- . In this case, we use "direct comparison". For example:
- . In this case, we use "reverse comparison". For example:

Let's write everything in a general tabular form:

, wherein	, wherein