There are certain rules for comparing numbers. Consider the following example.

Yesterday the thermometer showed 15˚ C, and today it shows 20˚ C. Today is warmer than yesterday. The number 15 is less than the number 20, we can write it like this: 15< 20. А, если мы представим эти числа на координатной прямой, то точка со значением 15 будет расположена левее точки со значением 20.

Now let's look at negative temperatures. Yesterday it was -12˚ C outside, and today -8˚ C. Today is warmer than yesterday. Therefore, they believe that the number -12 is less than the number -8. On a horizontal coordinate line, a point with a value of -12 is located to the left of a point with a value of -8. We can write it like this: -12< -8.

So, if you compare numbers using a horizontal coordinate line, the smaller of two numbers is the one whose image on the coordinate line is located to the left, and the larger is the one whose image is located to the right. For example, in our picture A > B and C, but B > C.

On the coordinate line, positive numbers are located to the right of zero, and negative numbers are located to the left of zero, every positive number Above zero, and any negative less than zero, and therefore every negative number is less than every positive number.

This means that the first thing you need to pay attention to when comparing numbers is the signs of the numbers being compared. A number with a minus (negative) is always less than a positive number.

If we compare two negative numbers, then we need to compare their moduli: the larger number will be the number whose modulus is smaller, and the smaller number will be the number whose modulus is smaller. For example, -7 and -5. The numbers being compared are negative. We compare their modules 5 and 7. 7 is greater than 5, which means -7 is less than -5. If you mark two negative numbers on a coordinate line, then the smaller number will be to the left, and the larger number will be located to the right. -7 is located to the left of -5, which means -7< -5.

Comparing fractions

From two fractions with same denominators the one with the smaller numerator is smaller, and the one with the larger numerator is larger.

You can only compare fractions with the same denominators.

Algorithm for comparing ordinary fractions

1) If a fraction has an integer part, we begin the comparison with it. The larger fraction will be the one whose whole part is larger. If the fractions do not have an integer part or they are equal, move on to the next point.

2) If fractions with different denominators it is necessary to bring them to a common denominator.

3) Compare the numerators of fractions. The larger fraction will be the one with the larger numerator.

Please note that the fraction with whole part There will always be more fractions without a whole part.

Comparison of decimals

Decimals can only be compared with the same number of digits (places) to the right of the decimal point.

Algorithm for comparing decimal fractions

1) Pay attention to the number of characters to the right of the decimal point. If the number of digits is the same, we can start comparing. If not, add the required number of zeros in one of the decimal fractions.

2) Compare decimal fractions from left to right: integers with integers, tenths with tenths, hundredths with hundredths, etc.

3) The larger fraction will be the one in which one of the parts is larger than the other fraction (we start the comparison with whole numbers: if the whole part of one fraction is larger, then the whole fraction is larger).

For example, let's compare decimal fractions:

1) Let's add in the first fraction required amount zeros to equalize the number of decimal places

57.300 and 57.321

2) We start comparing from left to right:

integers with integers: 57 = 57;

tenths with tenths: 3 = 3;

hundredths with hundredths: 0< 2.

Since the hundredths of the first decimal fraction turned out to be smaller, the whole fraction will be smaller:

57,300 < 57,321

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We continue to study rational numbers. In this lesson we will learn how to compare them.

From previous lessons we learned that the further to the right a number is located on the coordinate line, the larger it is. And accordingly, the further to the left the number is located on the coordinate line, the smaller it is.

For example, if you compare the numbers 4 and 1, you can immediately answer that 4 is more than 1. This is a completely logical statement and everyone will agree with it.

As proof, we can cite the coordinate line. It shows that the four lies to the right of the one

For this case, there is also a rule that can be used if desired. It looks like this:

Of the two positive numbers greater is the number whose modulus is greater.

To answer the question which number is greater and which is less, you first need to find the modules of these numbers, compare these modules, and then answer the question.

For example, compare the same numbers 4 and 1 using the above rule

Finding the modules of numbers:

|4| = 4

|1| = 1

Let's compare the found modules:

4 > 1

We answer the question:

4 > 1

For negative numbers there is another rule, it looks like this:

Of two negative numbers, the number whose modulus is smaller is greater.

For example, compare the numbers −3 and −1

Finding the modules of numbers

|−3| = 3

|−1| = 1

Let's compare the found modules:

3 > 1

We answer the question:

−3 < −1

The modulus of a number should not be confused with the number itself. Common mistake many newbies. For example, if the modulus of −3 is greater than the modulus of −1, this does not mean that −3 is greater than −1.

The number −3 is less than the number −1. This can be understood if we use the coordinate line

It can be seen that the number −3 lies further to the left than −1. And we know that the further to the left, the less.

If you compare a negative number with a positive one, the answer will suggest itself. Any negative number will be less than any positive number. For example, −4 is less than 2

It can be seen that −4 lies further to the left than 2. And we know that “the further to the left, the less.”

Here, first of all, you need to look at the signs of the numbers. A minus sign in front of a number indicates that the number is negative. If the number sign is missing, then the number is positive, but you can write it down for clarity. Recall that this is a plus sign

As an example, we looked at integers of the form −4, −3 −1, 2. Comparing such numbers, as well as depicting them on a coordinate line, is not difficult.

It is much more difficult to compare other kinds of numbers, such as fractions, mixed numbers, and decimals, some of which are negative. Here you will basically have to apply the rules, because it is not always possible to accurately depict such numbers on a coordinate line. In some cases, a number will be needed to make it easier to compare and understand.

Example 1. Compare rational numbers

So, you need to compare a negative number with a positive one. Any negative number is less than any positive number. Therefore, without wasting time, we answer that it is less than

Example 2.

You need to compare two negative numbers. Of two negative numbers, the one whose magnitude is smaller is greater.

Finding the modules of numbers:

Let's compare the found modules:

Example 3. Compare the numbers 2.34 and

You need to compare a positive number with a negative one. Any positive number is greater than any negative number. Therefore, without wasting time, we answer that 2.34 is more than

Example 4. Compare rational numbers and

Finding the modules of numbers:

We compare the found modules. But first, let’s bring them to a clear form to make it easier to compare, namely, we’ll convert them into improper fractions and bring them to a common denominator

According to the rule, of two negative numbers, the number whose absolute value is smaller is greater. This means rational is greater than , because the modulus of the number is less than the modulus of the number

Example 5.

You need to compare zero with a negative number. Zero is greater than any negative number, so without wasting time we answer that 0 is greater than

Example 6. Compare rational numbers 0 and

You need to compare zero with a positive number. Zero is less than any positive number, so without wasting time we answer that 0 is less than

Example 7. Compare rational numbers 4.53 and 4.403

You need to compare two positive numbers. Of two positive numbers, the number whose modulus is greater is greater.

Let's make the number of digits after the decimal point the same in both fractions. To do this, in the fraction 4.53 we add one zero at the end

Finding the modules of numbers

Let's compare the found modules:

According to the rule, of two positive numbers, the number whose absolute value is greater is greater. This means that the rational number 4.53 is greater than 4.403 because the modulus of 4.53 is greater than the modulus of 4.403

Example 8. Compare rational numbers and

You need to compare two negative numbers. Of two negative numbers, the number whose modulus is smaller is greater.

Finding the modules of numbers:

We compare the found modules. But first, let’s bring them to a clear form to make it easier to compare, namely, we’ll convert the mixed number into an improper fraction, then we’ll bring both fractions to a common denominator:

According to the rule, of two negative numbers, the number whose absolute value is smaller is greater. This means rational is greater than , because the modulus of the number is less than the modulus of the number

Comparing decimals is much easier than comparing fractions and mixed numbers. In some cases, by looking at the whole part of such a fraction, you can immediately answer the question of which fraction is larger and which is smaller.

To do this, you need to compare the modules of the entire parts. This will allow you to quickly answer the question in the task. After all, as you know, whole parts in decimal fractions have more weight than fractional parts.

Example 9. Compare rational numbers 15.4 and 2.1256

The modulus of the whole part of the fraction is 15.4 greater than the modulus of the whole part of the fraction 2.1256

therefore the fraction 15.4 is greater than the fraction 2.1256

15,4 > 2,1256

In other words, we didn’t have to waste time adding zeros to the fraction 15.4 and comparing the resulting fractions like ordinary numbers

154000 > 21256

The comparison rules remain the same. In our case, we compared positive numbers.

Example 10. Compare rational numbers −15.2 and −0.152

You need to compare two negative numbers. Of two negative numbers, the number whose modulus is smaller is greater. But we will compare only the modules of integer parts

We see that the modulus of the whole part of the fraction is −15.2 greater than the modulus of the whole part of the fraction −0.152.

This means rational −0.152 is greater than −15.2 because the modulus of the integer part of the number −0.152 is less than the modulus of the integer part of the number −15.2

−0,152 > −15,2

Example 11. Compare rational numbers −3.4 and −3.7

You need to compare two negative numbers. Of two negative numbers, the number whose modulus is smaller is greater. But we will compare only the modules of entire parts. But the problem is that the moduli of integers are equal:

In this case, you will have to use the old method: find the modules of rational numbers and compare these modules

Let's compare the found modules:

According to the rule, of two negative numbers, the number whose modulus is smaller is greater. This means rational −3.4 is greater than −3.7 because the modulus of the number −3.4 is less than the modulus of the number −3.7

−3,4 > −3,7

Example 12. Compare rational numbers 0,(3) and

You need to compare two positive numbers. Moreover, compare a periodic fraction with a simple fraction.

Let's convert the periodic fraction 0,(3) into an ordinary fraction and compare it with the fraction . After converting the periodic fraction 0,(3) into an ordinary fraction, it becomes the fraction

Finding the modules of numbers:

We compare the found modules. But first, let’s bring them to an understandable form to make it easier to compare, namely, let’s bring them to a common denominator:

According to the rule, of two positive numbers, the number whose absolute value is greater is greater. This means that a rational number is greater than 0,(3) because the modulus of the number is greater than the modulus of the number 0,(3)

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1. What numbers are missing? a) 497, 498, ..., 500; b) 902, 901, ..., 899. What does each digit mean in the numbers 902 and 498?
Name neighboring numbers for the number 498, the next number for the number 899, the previous one for the number 700.

2. Compare (>, 799 * 800 701 * 703
65 * 67 650 * 648
How to compare multi-digit numbers?

This article gives detailed review most important points concerning comparisons of rational numbers. If the signs of the numbers being compared are different, then you can immediately tell which number is larger and which is smaller, so at the very beginning we will look at the rule for comparing rational numbers with different signs. Next we will focus on comparing zero with another rational number. After this, we will dwell in detail on the comparison of positive rational numbers. Finally, let's move on to the rule for comparing negative rational numbers. We will dilute the theory with solutions to typical examples.

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Comparison of rational numbers with different signs

Easiest to do comparison of two rational numbers having different signs. In this case, the rule for comparing numbers with different signs is used: any positive number is greater than any negative, and any negative number is less than a positive one.

For example, of the two rational numbers 5/7 and −0.25, the larger number is 5/7, since it is positive, and the smaller number is −0.25, since it is negative. Another example: a negative rational number is less than a positive rational number 0.000(1) .

Comparing a rational number with zero

Very easy to do comparing zero with a rational number, different from zero. The rule is true: any positive number is greater than zero, and any negative number is less than zero.

Let's give a couple of examples of comparing a rational number with zero. The number 4/9 is greater than 0, since 4/9 is a positive number, on the other hand, 0 is less than 4/9. Another example: the number 0 is greater than the negative rational number −45.5, on the other hand, the number −45.5 is less than zero.

It also needs to be said about comparing zero to zero: zero is equal to zero, that is, 0=0 .

It should be noted here that the number zero can be written in a form other than 0. Indeed, the number 0 corresponds to any entry of the form 0/n, where n is any natural number, or entries 0.0, 0.00, ..., up to 0,(0) . That is, for example, when comparing two rational numbers, the entries of which are 0.00 and 0/3, we conclude that they are equal, since these entries correspond to the numbers 0 and 0, respectively.

Comparison of positive rational numbers

Comparison of positive rational numbers you should start by comparing their whole parts. In this case it is used next rule: greater is the number whose integer part is greater, and less is the number whose integer part is less.

Example.

Which rational number is 0.76 or greater?

Solution.

The rational numbers being compared are positive, and it is quite obvious that the integer part of the number 0.76, equal to zero, is less than the integer part of the number, equal to two (if necessary, see the comparison of integers). Therefore, , which means that of the two original numbers, the larger number is .

Answer:

Nuances in applying the above rule can only arise when one of the numbers being compared is a periodic decimal with a period of 9, which we mentioned in the section equal and unequal decimals.

Example.

Compare the rational numbers 15 and 14, (9) .

Solution.

A periodic fraction with a period of 9 of the form 14,(9) is just one of the forms of writing the number 15. That is, 15=14,(9) .

Answer:

The original rational numbers are equal.

If the integer parts of the rational numbers being compared are equal, the final result of the comparison will help to obtain a comparison of the fractional parts. The fractional part of a rational number can always be represented as an ordinary fraction m/n, and also as a finite or periodic decimal fraction. Thus, comparing the fractional parts of two positive rational numbers can always be reduced to comparing ordinary fractions or comparing decimals. As a result, of two positive rational numbers with equal integer parts, the greater is the one whose fractional part is larger, and the smaller is the one whose fractional part is smaller.

Example.

Compare the positive rational numbers 3,7 and .

Solution.

Obviously, the integer parts of the rational numbers being compared are equal to 3=3. Let's move on to comparing fractional parts, that is, to comparing the numbers 0.7 and 2/3.

We'll show you two ways.

In the first one, we will convert the decimal fraction into an ordinary fraction: 0.7 = 7/10. We come to a comparison of ordinary fractions 7/10 and 2/3. After reducing them to a common denominator 30, we obtain , which implies that and . Hence, .

In the second solution, we convert an ordinary fraction to a decimal, we have. So from comparing 0.7 and 2/3 we came to comparing the decimal fractions 0.7 and 0.(6), the result of which is: 0.7>0.(6). Therefore, and .

Obviously, both methods led us to the same result of comparing the original rational numbers.

Answer:

If both the integer and fractional parts of the positive rational numbers being compared are equal, then these numbers are equal.

Example.

Compare the numbers 4.5 and .

Solution.

Obviously, the integer parts of the numbers are equal. The fractional part of the number 4.5 is 0.5, converting this decimal fraction to an ordinary fraction gives 1/2. Thus, the fractional parts of the original numbers are also equal. Therefore, the original rational numbers are equal.

Answer:

Let's finish this point the following statement: if the records of the compared numbers completely coincide, then these numbers are equal. Indeed, in this case both the integer parts and the fractional parts of the numbers being compared are equal. For example, the rational numbers 5.698 and 5.698 are equal, and the numbers and are also equal.

Comparison of negative rational numbers

Comparison of negative rational numbers obeys the rule for comparing negative numbers: of two negative numbers, the greater is the one whose modulus is smaller, and the smaller is the one whose modulus is greater.

This rule reduces the comparison of negative rational numbers to the comparison of positive rational numbers discussed in the previous paragraph.

First level

Comparison of numbers. Comprehensive guide (2019)

When solving equations and inequalities, as well as problems with modules, you need to place the found roots on the number line. As you know, the roots found may 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 they are, look in the topic), or are complex mathematical expressions, then placing them on the number line is very problematic. Moreover, you cannot use calculators during the exam, and approximate calculations do not provide 100% guarantees that one number is less than another (what if there is a difference between the numbers being compared?).

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

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

How can they be compared, for example, with a number? This is the rub...)

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

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

Comparison of fractions

So, we need to compare two fractions: and.

There are several options on how to do this.

Option 1. Reduce fractions to a common denominator.

Let's write it in the form of an ordinary fraction:

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

Now we need to compare fractions:

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

1. just bring 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 with the larger numerator, that is, the first one.

2. “let’s discard” (consider that we have subtracted one from each fraction, and the ratio of the fractions to each other, accordingly, has not changed) and 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 subtracted one correctly? 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, bringing them to a common... numerator.

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

Yes Yes. This is not a typo. This method is rarely taught to anyone at school, 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 a fraction greatest?” Of course, you will say “when the numerator is as large as possible and the denominator as small as possible.”

For example, you can definitely say that it’s true? What if we need to compare the following fractions: ? I think you will also immediately put the sign correctly, 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 turn out to be very small, and accordingly: . As you can see, the denominators here are different, but the numerators are the same. However, in order to compare these two fractions, you don't have to look for 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 - compare and... We will compare and... Let us reduce these fractions not to a common denominator, but to a common numerator. To do this simply 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 (minuend) is greater than the second (subtrahend), and if negative, then vice versa.

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

As you already understand, we also convert to an ordinary fraction and get the same result - . Our expression takes the form:

Next, we will still have to resort to reduction to a common denominator. The question is: in the first way, converting fractions into improper ones, or in the second way, 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 reduced to a common denominator becomes much easier.

Let's bring it to a common denominator:

The main thing here is not to get confused about what number we subtracted from and where. Carefully look at the progress of the solution and do not accidentally confuse the signs. We subtracted the first number 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. Don’t rush to bring to a common denominator or subtract. Look: you can easily convert it to a decimal fraction. How long will it be? Right. What's more in the end?

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

Option 4: Comparing fractions using division.

Yes Yes. And this is also possible. The logic is simple: when we divide larger number for less, the answer we get is the number more than one, and if we divide the smaller number by the larger, then the answer falls on the interval from to.

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

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

Divide the first fraction by the second:

Let's shorten by and by.

The result obtained is less, which means the dividend is less than the divisor, that is:

We've sorted everything out possible options comparing fractions. How do you see them 5:

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

Ready to train? Compare fractions in the optimal way:

Let's compare the answers:

1. (- convert to decimal)
2. (divide one fraction by another and reduce by numerator and denominator)
3. (select the whole part and compare fractions based on the principle of the same numerator)
4. (divide one fraction by another and reduce by 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 up 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’s not difficult at all.

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

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

Let's open the brackets and compare what we get:

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

If, then of two degrees and the greater is the one whose index is less.

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

Let's introduce some natural number as the difference between and.

Logical, isn't it?

And now let us once again 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 interval from to, but the exponents are equal. Everything is very simple here.

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

Of course, you did the math quickly:

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 with both the left and right sides (if you multiply by, then you must multiply both).

In addition, there are cases when it is simply unprofitable to do any manipulations. 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:

1. - the same as
2. - the same as
3. - the same as
4. - the same as

3. Comparing numbers with roots

First, let's remember what roots are? Do you remember this recording?

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

Roots of odd degree exist for negative and positive numbers, and even roots- only for positive ones.

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

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

Let's say we need to compare:

To compare these two roots, you don’t need to do any calculations, just analyze the concept of “root” itself. Do you understand what I'm talking about? Yes, about this: otherwise it can be written as the third power of some number, equal to the radical expression.

What's more? or? Of course, you can compare this without any difficulty. The larger the number we raise to a power, the greater the value will be.

So. Let's derive a rule.

If the exponents of the roots are the same (in our case this is), then it is necessary to compare the radical expressions (and) - the larger the radical number, the more value roots at equal rates.

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

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

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 the root to a greater extent you will get a smaller number. Let's take for example:

Let us denote the value of the first root as, and the second - as, then:

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

If the radical 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 higher the indicator, the smaller this expression.

Try to compare the following roots:

Let's compare the results?

We sorted this out successfully :). Another question arises: what if we are all different? Both degree and radical expression? Not everything is so complicated, we just need to... “get rid” of the root. Yes Yes. Just get rid of it)

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

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

1. We look at the indicators of the roots - and. Their least common multiple is .
2. Let's raise both expressions to a power:
3. Let's transform the expression and open the brackets (more details in the chapter):
4. Let's count what we've done and put a sign:

4. Comparison of logarithms

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

1. What is the argument of a logarithm and what is its base?
2. What determines whether a function increases or decreases?

If you remember everything and have mastered it perfectly, let's get started!

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

• reduction to the same basis;
• reduction to the same argument;
• comparison with the third number.

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

Let's consider a comparison of 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:

1. The function, for, increases on the interval from, which means, by definition, then (“direct comparison”).
2. Example:- the grounds are the same, we compare the arguments accordingly: , therefore:
3. The function, for, decreases on the interval from, which means, by definition, then (“reverse comparison”). - the bases are the same, we compare the arguments accordingly: , however, the sign of the logarithms will be “reverse”, since the function is decreasing: .

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

1. The base is larger.
   • . In this case we use “reverse comparison”. For example: - the arguments are the same, and. Let’s compare the bases: however, the sign of the logarithms will be “reverse”:
2. The base a is in the gap.
   • . In this case we use “direct comparison”. For example:
   • . In this case we use “reverse comparison”. For example:

Let's write everything down in a general tabular form:

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

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

A little hint - for comparison, a 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 sorted this out. What sign will there be? 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? Why do we need a unit circle and how to find the value of trigonometric functions on it? If you don't 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 little. Let's draw a unit trigonometric circle and a triangle inscribed in it. Did you manage? Now mark on which side we plot the cosine and on which side the sine, using the sides of the triangle. (you, of course, remember that sine is the ratio of the opposite side to the hypotenuse, and cosine is the adjacent side?). Did you draw it? Great! The final touch is to put down where we will have it, where and so on. Did you put it down? Phew) Let's compare what happened to you and me.

Phew! Now let's start the comparison!

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

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

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

In a similar way, we compare the value of cosines. We only lower the perpendicular onto the axis... That's right, . Accordingly, we look at which point is to the right (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 a tangent is. So what is a tangent?) That's right, the ratio of sine to cosine.

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

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

Now analyze what you wrote. - we divide a large segment into a small one. The answer will contain a value that is definitely greater than one. Right?

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

So which trigonometric expression has the greater value?

Right:

As you now understand, comparing cotangents is the same thing, 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 number is greater: or? The answer is obvious. And now: or? Not so obvious anymore, right? So: or?

Often you need to know which numerical expression is greater. For example, in order to place the points on the axis in the correct order when solving an inequality.

Now I’ll teach you how to compare such numbers.

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

The essence of comparing numbers is this: 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 sides (and, of course, we can subtract too)
• “move everything to one side”, 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 need to make sure that both parts have the same sign; if both parts are positive, the sign does not change when raised to a power, but if they are negative, then it changes to the opposite.
• extract the root of the same degree from both parts. If we are extracting a root of an even degree, we must first make sure that both expressions are non-negative.
• any other equivalent transformations.

Important: it is advisable to make transformations such that the inequality sign does not change! That is, during transformations, it is undesirable to multiply by a negative number, and you cannot square it 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 it to get rid of the root:

Example.

Which is more: or?

Solution.

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

2. Multiplication by its conjugate.

Example.

Which is more: or?

Solution.

Let's 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 smaller than the left one:

3. Subtraction

Let's remember that.

Example.

Which is more: or?

Solution.

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

It can be seen that on the left side each term 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 what more...

The right side is larger.

Example.

Compare the numbers and...

Solution.

Let's remember the trigonometry formulas:

Let's check which quarters on trigonometric circle there are points and.

4. Division.

Here we also use a simple rule: .

At or, that is.

When the sign changes: .

Example.

Compare: .

Solution.

5. Compare the numbers with the third number

If and, then (law of transitivity).

Example.

Compare.

Solution.

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

It's obvious that.

On the other side, .

Example.

Which is more: or?

Solution.

Both numbers are larger, but smaller. Let's select 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 for different reasons and the same argument:

It can be explained this way: the larger the base, the lesser the degree it will have to be raised to get the same thing. If the base is smaller, then the opposite is true, since the corresponding function is monotonically decreasing.

Example.

Compare the numbers: and.

Solution.

According to the above rules:

And now the formula for the advanced.

The rule for comparing logarithms can be written more briefly:

Example.

Which is more: or?

Solution.

Example.

Compare which number is greater: .

Solution.

COMPARISON OF NUMBERS. BRIEFLY ABOUT THE MAIN THINGS

1. Exponentiation

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

2. Multiplication by its conjugate

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

3. Subtraction

4. Division

When or that is

When the sign changes:

5. Comparison with the third number

If and then

6. Comparison of logarithms

Basic Rules.