What is greater is the number with a minus or natural. Negative numbers. What to do with logarithms
There are many types of numbers, one of them is integers. Integers appeared in order to make it easier to count not only in a positive direction, but also in a negative one.
Consider an example:
During the day it was 3 degrees outside. By evening the temperature dropped by 3 degrees.
3-3=0
It was 0 degrees outside. And at night the temperature dropped by 4 degrees and began to show on the thermometer -4 degrees.
0-4=-4
A series of integers.
We cannot describe such a problem with natural numbers; we will consider this problem on a coordinate line.
We have a series of numbers:
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
This series of numbers is called next to whole numbers.
Integer positive numbers. Whole negative numbers.
A series of integers consists of positive and negative numbers. To the right of zero are natural numbers, or they are also called whole positive numbers. And to the left of zero go whole negative numbers.
Zero is neither positive nor negative. It is the boundary between positive and negative numbers.
is a set of numbers consisting of natural numbers, negative integers and zero.
A series of integers in positive and negative directions is endless multitude.
If we take any two integers, then the numbers between these integers will be called end set.
For instance:
Let's take integers from -2 to 4. All numbers between these numbers are included in the finite set. Our finite set of numbers looks like this:
-2, -1, 0, 1, 2, 3, 4.
Natural numbers are denoted by the Latin letter N.
Integers are denoted by the Latin letter Z. The whole set of natural numbers and integers can be depicted in the figure. 
Nonpositive integers in other words, they are negative integers.
Non-negative integers are positive integers.
Negative and imaginary numbers
We now venture into algebra. The use of negative and imaginary numbers in algebra confirms the four-part nature of analysis and provides an additional chance to use three-part analysis. In this case, we must again warn that we intend to use the concepts of algebra for purposes far beyond the usual application of these concepts, since some of the discoveries of algebra make a significant contribution to our study.
The evolution of mathematics has gone by leaps and bounds after the discovery of the possibility of using negative numbers ( negative quantities). If we represent positive numbers as a series going to the right of zero, then there will be negative numbers to the left of zero.
etc... -3, -2, -1, 0, +1, +2, +3… etc.
With this graph, we can think of addition as moving to the right, and subtracting as moving to the left. It becomes possible to subtract a larger number from a smaller one; for example, if we subtract 3 from 1, we get -2, which is a real (though negative) number.
The next important concept is imaginary numbers. They were not discovered, but rather accidentally discovered. Mathematicians came to the conclusion that numbers have roots, that is, numbers that, when multiplied by themselves, give the desired number. The discovery of negative numbers and matching them with roots caused a panic in scientific circles. What should be the numbers, multiplication of which would give the number -1? For a while there was no answer. The square root of a negative number was impossible to calculate. That is why it was called imaginary. But when Gauss, nicknamed "the prince of mathematicians," discovered a method for representing imaginary numbers, an opportunity was soon found for their application. Today they are used on a par with real numbers. The imaginary number representation method uses the Argand diagram, which represents the wholeness as a circle, and the roots of this wholeness as segments of the circle.
Recall that a series of negative and positive numbers diverge in opposite directions from one point - zero. Thus, the square roots of integers, +1 or -1, can also be expressed as opposite ends of a line, with zero at the center. This line can also be represented as an angle of 180 0 , or a diameter.
Gauss developed the original suggestion and drew the square root of -1 as half the distance between +1 and -1, or as the 90° angle between the line from -1 to +1. Therefore, if the division of the whole into plus and minus is the diameter, or 180 0, then the second division leads to the appearance of another axis that divides this diameter in half, i.e., by an angle of 90 0 .
Thus, we get two axes - a horizontal one, representing the infinities of positive and negative numbers, and a vertical one, representing the infinities of imaginary positive and negative numbers. It turns out the usual coordinate axis, where the number described by this scheme and the axes is a number that has real and imaginary parts.
Using the Argand diagram (this circle with an integer radius (radius +1) on complex system coordinates), the following roots of the whole (cube roots, roots in the fourth, fifth powers, etc.) we find by simply dividing the circle into three, five, etc. equal parts. Finding a whole root turns into a process of inscribing polygons in a circle: a triangle for a cube root, a pentagon for a fifth power root, and so on. The roots become points on the circle; their values have real and imaginary parts, and they are calculated, respectively, along the horizontal or vertical coordinate axes. This means that they are measured in terms of square roots and roots to the fourth power.
From this powerful logical simplification, it becomes clear that analysis is a fourfold process. Any situation can be considered in terms of four factors or aspects. This not only confirms Aristotle's idea of four categories, but also explains why quadratic equations(in other words, "four-sided") are so popular in mathematics.
But the conclusion about the nature of analysis as a four-fold essentially implies its work in both directions. Analysis shows both the inclusiveness of the fourfold and its limitations. And also the fact that sometimes the essence of experience does not lend itself to any analysis.
Being "inside" the geometric method, we have shown that these non-analytical factors include triplicity, fiveness, sevenness. Although we are able to give an analytical description of them, it is not capable of revealing their true nature.
Technological map of lesson No. 35FULL NAME. teachers: Ivanova Olga Anatolievna
Thing: Mathematics
Class: 6 A
Name of the educational and methodical set (EMC): Mathematics. Textbook for grade 6 / Nikolsky S.M., Potapov M.K.
Lesson topic: negative integers
Lesson type: Lesson of primary presentation of new knowledge
The place of the lesson in the system of lessons: Lesson 1 in the topic "Integers"
Lesson Objectives:
Tutorial: learn how to find the temperature difference using thermometer readings, get acquainted with the rule for subtracting numbers using a series of integers;
Developing: develop analytical thinking, highlight the main thing and generalize
Educational: foster a sense of mutual cooperation, the ability to listen
Didactic task of the lesson: introduce the concept of negative, positive numbers, a series of integers; learn the rules for subtracting numbers using a thermometer and a series of whole numbers
Planned results
Subject Results: know and understand the meaning of concepts : positive number, negative number , a series of integers, be able to subtract numbers using a series of integers, apply the acquired knowledge in other lessons.
Metasubject results:
Cognitive: the ability to understand the learning task of the lesson, identify and formulate cognitive goals, build a logical chain of reasoning.
Regulatory: control and evaluate their own activities and the activities of partners, plan and adjust their activities;
Communicative: be able to fully and clearly express their thoughts, listen to the interlocutor and conduct a dialogue.
Personal: be motivated to learning activities accept and master the social role of the student, use the acquired knowledge of educational cooperation with adults and peers in different situations.
Basic concepts: negative numbers, positive numbers, series of integers
Intersubject communications: physics
Resources:http :// www . uroki . net ; http :// www . zavuch . info
Forms of work: frontal conversation, work in pairs, individual work.
Lesson stages
Teacher activity
Student activities
time
Formed UUD
1.
Organizational stage
Greeting students. Readiness for the lesson.
Check if everything is ok? Books, pens and notebooks? The bell rang now: the lesson begins!
Work diligently at the lesson, and success awaits you for sure!
Preparing to start the lesson
Personal: have a positive attitude towards learning cognitive activity willing to acquire new knowledge, skills, improve existing ones.
Cognitive: are aware of the educational and cognitive task.
Regulatory: plan in cooperation with the teacher, classmates independently the necessary actions.
Communicative: listen and hear each other.
2.
Knowledge update
Guys, what is the most important skill in math? Let's check how you can count: we will conduct a mathematical warm-up.
Examples are written on the board, we solve them orally and say the answer.
Guys, what can you say about the numbers written in the first and second columns? What are they?
What mathematical operations with numbers did you perform?
Suggest answers (count)
Oral work with examples on the board.
Answer questions (natural, fractional)
(addition, subtraction, multiplication, division)
Evaluation of your activities
Personal: show steady cognitive interest to verbal counting.
Cognitive: perform educational and cognitive actions in a mental form; carry out to solve learning objectives operations of analysis, synthesis, comparison, qualification.
Regulatory: Accept and save the learning task.
Communicative: express and justify their point of view.
3.
goal setting
Organization of work with handouts.
Guys, pay attention to the sheets with task 1
The demonstration thermometer shows the solution to the problem.
Guys, what new concept are we facing? How do we record thermometer readings? What does the entry -3 mean 0 WITH.
From what point do we measure temperature? What do we call temperatures above 0? Below 0? What role does 0 play?
What is the topic of the lesson?
The teacher corrects the students' answers and voices the topic of the lesson. Lesson topic: negative integers.
Together with students:
formulates the purpose of educational activity;
builds a project (algorithm) for getting out of a problem situation.
Organizes and supplements joint educational activities
Read the problem and offer solutions.
Answer questions
Student responses
Temperature in the evening -3 0 WITH
Put a minus before 3.
3 0 From frost.
We count from 0. Plus (positive), minus (negative). the border
Negative temperatures (numbers)
Students write the topic in their notebook.
Formulate the purpose of learning activities in a dialogue with the teacher.
Personal: conduct a dialogue on the basis of equal relations and mutual respect and acceptance.
cognitive: extract the necessary information from the explanation, statements of classmates, systematize knowledge.
Regulatory: plan the necessary actions.
Communicative: build monologue statements, carry out joint activities.
4
Organization of work with the textbook
206 in notebooks
Check each other's answers
Task 2
solve examples using thermometer:
10 0 C -5 0 С=+5 0 WITH
15 0 C -15 0 С=+0 0 WITH
0 0 C -10 0 С=-10 0 WITH
10 0 С – 15 0 С=-5 0 С
15 0 S-20 0 С=-5 0 WITH
Guys, imagine that you and I placed the thermometer horizontally and got the following record
What do we call the numbers to the right of 0? To the left of 0?
Formulate the definition of a positive and negative number
Work orally and in notebooks.
Mutual check
Work in pairs; checking the solution at the blackboard with an explanation with a thermometer
Performance evaluation
Positive, negative.
Formulate a definition
Personal: solve problems in a constructive way.
cognitive: read and listen, extracting the necessary information.
Regulatory: control learning activities, notice the mistakes made; they are aware of the rule of control and successfully use it in solving a learning problem.
Communicative: work together in pairs.
4.
Physical education minute
Now imagine that zero is your arms folded at your chest, then left hand will show the location of what numbers? Right?
Show me where relative to zero is the number 5? -7? -10? one hundred? 15? -twenty?
Let's warm up
Answer questions, show the arrangement of numbers
They are distracted from educational activities, warming up.
personal: about awareness of the value of health
cognitive: Establish causal relationships between their health and exercise.
Regulatory: adequately independently assess the correctness of the performance of the action and make the necessary adjustments to the performance both at the end of the action and in the course of implementation.
5.
Primary perception and assimilation of the material
Guys, let's get back to the record.
7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
What does this entry mean?
What numbers make up a series of integers?
The tutorial will help you find the answer
How can a series of integers help us when subtracting numbers?
Try using a series of integers to complete task 3
Self-execution of the exercise
Completing task 3
Let's check what results you got.
Work with the textbook, search for the answer to the question. (series of integers)
The series of integers consists of natural numbers, negative integers and zero.
When subtracting, we will move to the left along the row
Completing assignments in notebooks
Verification with verbal commentary
Decision discussion
Performance evaluation
Personal: show the need for self-expression and self-realization.
cognitive: search for the necessary information (from the materials of the textbook and the teacher's story, by recalling it in memory).
Regulatory: independently control their time allotted for solving a specific task, and manage it.
Communicative: display in inner speech the content of the actions performed.
6.
Reflection
What new concept did we meet in today's lesson?
What have we learned in today's lesson?
What was the most difficult?
Summarizes the lesson. Evaluates the work of the class and individual students.
Give an adequate assessment of their activities.
Personal: understand the value of knowledge for a person.
cognitive: acquire the ability to use knowledge and skills in practical activities and Everyday life; establish the relationship between the amount of knowledge acquired in the lesson, skills and operational, research, analytical skills as integrated, complex skills.
Regulatory: evaluate their work; correct and explain their mistakes.
Communicative: formulate their own thoughts, express and justify their point of view.
7
Gives homework.
425, 426, 434 * in
Students write down their homework
Formulas in Excel will help to calculate not only positive, but also negative numbers. In what ways you can write a number with a minus, see the article "How to enter a negative number in Excel".
To find sum of negative numbers in excel
, need SUMIF function in Excel
.
For example, we have such a table.
In cell A7 we set the formula. To do this, go to the tab of the Excel table "Formulas", select "Math" and select the Excel function "SUMIF". 
Fill in the lines in the window that appears:
"Range" - indicate all the cells of the column or row in which we add the numbers. About the range in the table, see the article "What is a range in Excel" .
"Criterion" - here we write "<0» .
We press the "OK" button.
It turned out like this.
See the formula in the formula bar.How to set the "greater than" or "less than" sign in a formula, see the article "Where is the button on the keyboard» .
Sum only positive numbers in Excel.
It is necessary to write the formula in the same way, only in the line of the "Criterion" function window write "> 0" It turned out like this. 
The "SUMIF" function in Excel can count the values of the cells not all in a row, but selectively according to the condition that we write in the formula. This function is useful for calculating data for a certain date or a particular customer's order, student totals, etc. Read more about how to use this function.
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 instance:
Of course, you know that this, accordingly, the expression takes the form:
Let's open the brackets and compare what happens:
Several a special case 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.
Let us introduce some natural number as the difference between and.
Logical, isn't it?
Now let's pay attention to the condition - .
Respectively: . Hence, .
For instance:
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? That's what I did:
- - 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 instance: .
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 instance:
- . In this case, we use "reverse comparison". For instance:
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:
I 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? Fine! 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? That's what I did.
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 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 instance, . 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 grounds and the same argument:
It can be explained as follows: the larger the base, the less it will have to be raised in order 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
Fundamental rules:
Logarithms with different bases and the same argument:
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
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But think for yourself...
What does it take to be sure to be better than others on the exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
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