How to read math
Mathematics is “a language that neither read nor understand is possible without initiation” (Edward Rothstein, “Emblems of the Mind”)
Reading protocol - a set of strategies that the reader must use to get all the benefits of reading the text. A set of strategies for poetry is different from fiction, and strategies for reading fiction are different from scientific articles. It would be ridiculous to read a fiction book and wonder what sources allowed the author to say that the main character is tanned blond; but it would be wrong to read scientific literature and not ask such a question. This reading protocol is extended to viewing and listening protocols in art and music. In fact, most introductory courses in literature, music, and art are devoted to the study of these protocols.
For mathematics there is a special reading protocol. As we learn to read literature, so must we learn to read math. Schoolchildren should study reading protocol for mathematics in the same way that they learn the rules of reading a novel or a poem, learn to understand music and painting. Edward Rothstein's remarkable Emblems of the Mind reveals the relationship between mathematics and music, implicitly affecting reading protocols for mathematics.
When we read a novel, we are absorbed by the plot and characters. We are trying to follow different storylines and how each of them affects the development of the characters. The characters themselves must become real people for us - both those who delight us and those we despise. We do not stop at every word, but present the words as strokes in a picture. Even if the specific word is unknown, the overall picture is still clear. We rarely interrupt to reflect on a particular phrase or sentence. Instead, we allow the work to captivate us in its stream and quickly get through to the end. This is a useful and relaxing exercise, it gives food for thought.
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Writers often characterize characters, involving them in carefully selected action episodes, rather than describing carefully selected adjectives. They depict one aspect, then another, then again the first in a new light, and so on. As the overall picture grows, it becomes more and more clear. This is the way to convey complex thoughts that are not precisely defined.
Mathematical ideas are inherently accurate and well defined, so you can give a clear and very short description. Both the mathematical article and the artistic novel tell a certain story and develop complex ideas, but the mathematical article does it with far fewer words and symbols than in the book. The beauty of the book is in the aesthetic way of using language to awaken emotions and to represent things that are not amenable to precise definition. The beauty of a mathematical article is in the elegance and effectiveness of the laconic expression of clear ideas of great complexity.
What are some common mistakes made by people trying to read math? How can these errors be corrected?
“Reading mathematics is generally not linear ... To understand the text, you need cross-references, fast scanning, pauses and repeated reading” (from the same place)
Do not think that understanding each phrase individually will allow you to understand the whole idea. It's like trying to see a portrait, looking at each square centimeter of the painting. You will see the details, texture and color, but completely miss the portrait. A math article tells a story. Try to understand what this story is before going into details. You can take a closer look later when you build an understanding framework. Do it the same way you could re-read an art book.
“The three-line proof of the modest theorem is the quintessence of years of work. Reading mathematics ... provides for a return to the thought process that occurred during its recording "(from the same place)
Examine examples for patterns. Try special cases.
A mathematical article usually tells only a small part of a large and extensive history. As a rule, the author wanders for months in the dark, trying to discover something new. In the end, he organizes his discovery in the form of an article where he hides all the mistakes (and their causes) and presents a complete idea in the form of a clean neat stream. To truly understand the idea and recreate what the author has hidden, read between the lines.
Mathematicians are very laconic, but much is hidden behind these few words. The reader must participate. At each stage, he / she has to decide how clear the presented idea is. Ask yourself the following questions:
If you spend too little effort on such participation, then it's like reading a book without concentration. Half an hour later you suddenly realize that you just turned the pages, thinking about yours, and did not remember anything that you read.
Reading mathematics too quickly leads to frustration. Half an hour of focusing on the art book will allow the average reader to read 20-60 pages with a full understanding of the text, depending on the book and reading experience. The same half an hour over a mathematical article will give 0-10 lines, depending on the article and the experience of reading mathematics. Work and time can not be replaced. You can speed up mathematical reading with practice, but be careful. As with any skill, trying to do too much too quickly can throw you back and eliminate motivation. Imagine that you exercise vigorously for an hour, if you haven’t been doing two years before. Maybe you will go through the first stage, but it is unlikely to continue further. The disappointment of constant observation of how experienced guys tirelessly perform twice as much as yours, while your whole body aches the next day because of fatigue is too hard to accept.
For example, take this theorem from the treatise “The Case of the Computer” by Levi ben Gershom, written in 1321.
"When you add consecutive numbers, starting with 1, and the number of items is odd, the result is equal to the product of the average and the last." For modern mathematics it will be natural to write the theorem as follows:
The reader will need about the same time to unravel this small formula as he needs to understand two lines of the text version of the theorem. Example of Levy's theorem:
The best way to understand what you read is to make the idea your own. It means tracing the idea to its roots and rediscovering it yourself. Mathematicians often say that in order to understand something you must first read it, then write it in your own words, and then teach someone else. Each has different tools and a different level of "assimilation" of complex ideas. You need to adapt the idea to your own vision and experience.
“The meaning will seldom be completely understood, because each symbol or word already constitutes an extraordinary condensation of concept and reference” (from the same place)
A well-written mathematical text will accurately use the word in only one sense, making a distinction, say, between a combination and a permutation (or arrangement ). A clear mathematical definition may imply that a “yellow mad dog” and a “mad yellow dog” are a different arrangement of words, but the same combination. Most English-speaking readers will not agree with this. Exceptional accuracy is completely alien to most fiction and poetry, where the use of various words, synonyms and changeable descriptions is considered good form .
The reader is expected to understand that the absolute value is not some arbitrary value that turns out to be absolute, but the function has nothing to do with anything functional.
A particularly notorious example is the use of the phrase “from this it easily follows that ...” and similar constructions. They mean something like the following:
Now we can verify the validity of the following statement. Verification will require a certain amount of essentially mechanical, although perhaps difficult, work. As an author, I could spend it, but it will take a lot of space and, probably, will not lead to good results, since it is in your best interest to make calculations yourself and clarify what is happening here. I promise that no new ideas are involved. Although, of course, you may need to think a bit to find the right combination of good ideas to apply.
In other words, this phrase, when properly applied, is a signal to the author that there is something tedious and even difficult here, but not suggesting bright insights. Then the reader is free to decide how much he wants to understand all the details himself or if the author’s assurances are enough for him - and then we can say: “Okay, I believe you for the word.”
Now, regardless of your opinion on the appropriateness of the use of such a phrase in a particular situation and on the correctness of its use by the author, you should understand what it really means. The phrase "From this it easily follows that ..." does not mean
If you don't see it right away, you're a fool.
And she does not mean
This should not take more than two minutes.
But a person unfamiliar with a mathematical vocabulary may misunderstand it and will face disappointment. This is in addition to the question that the “tedious task” for one person may be the most difficult problem for another. Therefore, the author must correctly assess their audience.
Texts are written with the expectation of readers of a certain level. Make sure that you are such a reader or you want to do everything to join them.
From Thomas Eliot, "Song of Simeon":
For example, Eliot's poem largely implies that readers either know who Simeon is or want to find out. It also assumes that the reader will be either an experienced poetry reader, or one who wants to gain such experience. The author assumes that readers either know or understand the hints here. It goes beyond simple things like who Simeon was. For example, why are Roman hyacinths? Why is it important?
Eliot assumes that the reader will read slowly and pay attention to the pictures: the author compares dust and memory, compares old age with winter, compares the expectation of death with a fluff on the back of the palm, etc. He assumes that the reader will take it as poetry; in a sense, he assumes that the reader is familiar with the whole poetic tradition. He should notice that the odd lines rhyme and the rest do not, etc.
Most importantly, he assumes that the reader will connect to reading not only his mind, but also his emotions and imagination, which will draw the image of an old man, tired of life but forced to cling to it, waiting for some important event.
Most mathematical books are also written with some assumptions about the audience: that they know certain things, that they have reached a certain level of “mathematical maturity”, and so on. Before you begin to read, make sure that the author expects from you.
To experiment with the principles presented here, I included a small mathematical fragment, often called the birthday paradox. The first part is an exact mathematical article describing the problem and its solution. The second part is the reader's fictional attempt to understand the article using the appropriate reading protocol. The subject of the article is probability. It is accessible to the bright and motivated reader, even without a mathematical education.
A professor in a class of 30 random students often suggests that at least two people in a class have the same birthday (month and day, not necessarily a year). Will you accept the argument? And if there are fewer students in the class? Would you accept the offer to argue?
Suppose birthdays n people are evenly distributed over 365 days of the year (for simplicity, we do not take into account leap years). Let us prove that the probability of the same birthday (month and day) in at least two of them is:
What is the probability that among 30 random people in the room there will be at least two people with the same birthday? For n=30 this probability will be 71%, that is, the professor will win the dispute in 71 cases out of 100 in the long run. As it turns out, with 23 students in a class, the probability is about 50%.
Here is the proof: Let P(n) is the desired probability. Let be Q(n)=1−P(n) it is likely that everyone has different birthdays. Now find Q(n) by calculating the number of sets from n birthdays without duplicates and dividing it by the total number of possible sets of n birthdays Then find P(n) .
Total number of sets from n birthdays without duplicates:
This is because for the first DR there are probably 365 variants, for the second - 364 and so on for n person. Total amount n birthdays without any restrictions is simply 365n because for each of n Birthdays have 365 options. In this way, Q(n) equals
Solution for P(n) gives P(n)=1−q(n) and therefore our result.
In this section, the naive reader tries to understand the meaning of the last few paragraphs. Readers' replicas are a metaphorical expression of his thoughts out loud, and Professional's comments are a research paper that the reader must do. The relevant protocols are in bold and inserted in the appropriate places of the story.
It looks like the Reader seems to grasp things very quickly. But be sure that in reality there is a lot of time between the reader’s comments and that I have left out many of his comments that lead to a dead end. You can understand what he is experiencing only if you read between the lines and imagine his way of thinking. Thinking about it is part of your own efforts.
Reader (W): I don’t know anything about probabilities, can I figure this out?
Professional (P): Let's try. You may have to make big digressions at every step.
D: What does the phrase "30 random (chaotic) students" mean?
"When I take a word, it means what I want, neither more nor
P: Good question. The phrase does not mean that we have 30 people who are crazed or sick in the head. It means that we must assume that the birthdays of these 30 people are independent of each other and that every birthday is equally likely for each person. A little further, the author describes this more formally: “Suppose that birthdays n people are evenly distributed over 365 days of the year. "
Ch: Isn't it obvious? Why specifically to stipulate?
P: Yes, the assumption seems obvious. The author simply sets the basic condition. This sentence ensures that everything is fine, and the solution does not imply any imaginary bizarre science fiction.
Q: What do you mean?
P: For example, the author is not looking for a solution like this: everyone lives in the Land of Independence and was born on the 4th of July, so the chances of coincidence of the birthdays of two or more people are 100%. Mathematicians do not like solutions of this kind. By the way, the assumption also implies that we do not take into account leap years. That is, no one in this task was born February 29th. Read on.
D: I do not understand this long formula, What is n ?
P: The author solves the problem for an arbitrary number of people, not just for 30. From this point on, the author names the number of people. n .
Ch: I still don't understand. What reply?
P: Well, if you need an answer only for 30, just install n=30 .
Q: Okay, but it's hard to count. Where is my calculator? Let's see: 365 × 364 × 363 × ... × 336. This is tedious, and the result does not even fit on the screen. It is written here:
Even if knowing the formula, I cannot calculate the result, how can I understand where the formula came from?
P: You are right that this is not an accurate result, but if you continue further and divide, the answer will not be too far from the exact one.
Ch: All this is somehow uncomfortable. I would like to calculate the exact value. Is there another way to make calculations?
P: How many factors do you have in the numerator? How much in the denominator?
D: Do you mean 365 as the first factor, 364 as the second? Then it turns out 30 factors at the top. Also 30 multipliers below (30 copies 365).
P: Now can you calculate the result?
H: Oh, I see. I can pair each numerator from the numerator with each denominator multiplier, so it will turn out 365/365 for the first multiplier, then multiply by 364/365 and so on for all 30 multipliers. So the result will always fit in my calculator. (In a few minutes) ... So, I received .29368 if to round up to five characters.
P: What does this number mean?
Ch: I forgot what I was doing. So let's see. I was looking for an answer for n=30 . The number 0,29368 is all calculations, except subtraction from one. If you continue, you get 0.70632. Now what does this mean?
P: It is useful to learn more about probabilities here, but simply this means that in a group of 30 people two or more birthdays will coincide in 70,632 out of 100,000 cases, that is, in about 71% of cases.
H: Interesting. I would not have guessed. Would you like to say that in my class of 30 students the probability is quite high that at least two people will have the same birthday?
P: Yes, that's right. You can accept bets before you know their dates of birth. Many people think that such a coincidence is unlikely. That is why some authors call it the birthday paradox .
D: So that's why I have to read math to earn a couple of dollars?
P: I understand that for you this may be a kind of stimulus, but I hope that mathematics also inspires you and without any monetary perspectives.
R: I wonder what the result will be for other values. n . I'll try to do some more calculations.
P: Good thought. We can even make a graphic image of all your calculations. You can make a graph of the number of people and the probability of coincidence of birthdays, although it can be left at another time.
D: Oh, look, the author did some calculations for me. He says that for n=30 the answer is about 71%; I also received such a figure. And for n=23 It turns out about 50%. It makes sense? I think it has. The more people, the higher the probability of coincidence of birthdays. Hey, I'm ahead of the author. Not bad. Okay, let's continue.
P: Well, now you will tell yourself when to continue.
Ch: It seems that we got to the proof. It should explain why the formula works. What is this Q(n) ? I guess, that P means “probability” (probability), but what does Q ?
P: The author introduces a new value. He uses Q simply because it is the next letter in the alphabet after P but Q(n) - this is also a probability, and having a close relationship to P(n) . It's time to take a moment to think. What Q(n) and why is it equal 1−P(n) ?
H: Q(n) - This is the likelihood that no one in the room does not match birthdays. Why does the author care about this question? Don't we need another chance that the birthdays are the same?
P: Good point. The author does not say this explicitly, but between the lines you can understand that he has no idea how to directly calculate P(n) . Instead, he introduces the probability Q(n) which is supposed to be 1−P(n) . Probably, after that the author should show how to calculate Q(n) . By the way, when you finish an article, you may wonder how to calculate P(n) directly. This is a great continuation for the ideas presented here.
H: Everything has its time.
P: Okay. So now we know Q(n) , what's next?
W: Then we can get P(n) . If a Q(n)=1−P(n) then P(n)=1−q(n) . Ok but why Q(n)=1−P(n) ? The author believes that this is obvious?
P: Yes, he thinks so, and even worse, he doesn’t even tell us that this is obvious. Here is a rule of thumb: when the author clearly says that this is true or obvious , you should spend 15 minutes and convince yourself that this is the way it is. If the author does not even bother to say it, but implies it, then the process will take a little longer.
Q: How do I understand what you need to stop and think?
P: Just be honest with yourself. If in doubt, stop and think. If you are too tired, go watch TV.
D: So why Q(n)=1−P(n) ?
P: Let's imagine a special case. If the probability of coincidence of two or more birthdays is 1/3, then what is the probability of not getting a match?
H: It's 2/3, because the probability of the absence of an event is inverse to the probability of an event occurring.
P: Well, you should be careful when using words like the opposite , but you're right. In fact, you discovered one of the first rules that is being studied in probability theory. Namely, that the probability of the absence of an event is equal to one minus the probability of an event occurring. We now turn to the next paragraph.
H: It seems to explain what equals Q(n) - in a long and complex-looking formula. I will never understand this.
P: Formula for Q(n) hard to understand, and the author relies on your diligence, perseverance and / or knowledge to understand it.
H: It seems to calculate all the probabilities of something and divide them by the total number of probabilities, whatever that means. I have no idea why he does that.
P: Maybe I can help you with some information on this subject. The probability of a certain outcome in mathematics is determined as follows: the total number of possible variants of this outcome is divided by the total number of all outcome variants. For example, the probability of throwing a four at a die is 1/6. Since there are one four and six possible outcomes. What is the probability that you will throw a four or a three?
D: Well, I think 2/6 (or 1/3), because the total number of possible outcomes is still six, but I have two options for a successful outcome.
P: Good. Now a more complex example. What about the probability of throwing four in total when throwing two dice? There are three options to get this amount (1-3, 2-2, 3-1), while the total number of combinations is 36. This is 3/36 or 1/12. Look at the following 6 × 6 table and see for yourself.
What about the probability of throwing seven in the amount?
D: Wait, what does 1-1 mean? Does it not equal 0?
P: Sorry, I'm to blame. I used the minus sign as a dash, just referring to a couple of numbers, so 1-1 means throwing edinichki on both dice.
Q: Could you come up with a better record?
P: Well, maybe I can or should do it, but the commas look worse, and the slash will look like a division, and everything else can also be misleading. In any case, we are not going to publish this transcript.
D: Thank God. Well, now I know what you mean. I can get seven in total in six ways: 1-6, 2-5, 3-4, 4-3, 5-2 or 6-1.The total number of outcomes is still 36, so it turns out 6/36 or 1/6. It is strange, why is the probability of dropping four different from the probability of dropping seven?
P: Because not every amount is equally likely. Here, the situation is different from simply rotating the wheel with numbers from 2 to 12 in equal intervals. In this case, each of the 11 digits has the same probability of dropping 1/11.
W: Well, now I'm an expert. Is the probability calculated simply by counting?
P: Sometimes yes. Although it is difficult to calculate.
Ch: Clear, let's continue. By the way, did the author really expect me to know all this? My friend is taking a course in probability and statistics, but I'm not sure that he knows all these things.
P: A small section of mathematics contains a lot of information. Yes, the author expects the reader to know all of this or that he will find this information and assimilate it, as we did. If I were not here, you would have to ask these questions to yourself and find the answers by thinking, in textbooks and reference books or in consultation with a friend.
H: So the probability of coincidence of birthdays for two people is the number of possible sets ofn birthdays without duplicates, divided by the total number of possible sets ofn birthdays. P: Excellent resume. D: I don't like using
n , so let's use 30. Perhaps then it will be easier to understand the generalization.n .
P: Great thought. It is often useful to analyze a special case in order to understand the general case.
H: So how many sets of 30 birthdays are there at all? I can not count. I think that I will have to further limit the conditions. Let's pretend we only have two people.
P: Good. Now you think like a mathematician. Will considern = 2 .How many possible birthday combinations are there?
D: I count birthdays from 1 to 365, not counting leap years. Then here are all the possible combinations:
P: When you write 1-1, you mean 1-1 = 0, how is subtraction?
Ch: Stop teasing me. You know exactly what I mean.
P: Yes, I know, and I can point out a good choice of recording method. Now how many pairs of birthdays are there?
H: For two people, 365 × 365 variants are obtained.
P: And how many options, if you do not take into account the coinciding birthdays?
H: You can not use 1-1, 2-2, 3-3 ... 365-365, so it goes
The total number of variants is 365 × 364, because each line now has 364 pairs instead of 365.
P: Good. You're in a bit of a hurry, but still 100% right. Now can you summarize for 30 people? What is the total number of possible sets of 30 birthdays? Try to guess. You are good at it.
H: Well, if you try to guess (although this is not really a guess at all, after all, I already know the formula), then I would say that for 30 people you need to multiply 365 × 365 × ... × 365, for the total number of possible sets of DR.
P: Exactly. Mathematicians write 365 30 . And what is the total number of sets of 30 birthdays without repetitions?
H: I know that the answer should be 365 × 364 × 363 × 362 × ... × 336 (that is, start with 365 and multiply each time by a number with subtraction of a unit 30 times), but I'm not sure that I really understand why so Perhaps you should first consider the case with three people and find a way to increase to 30?
P: Brilliant thought. Let's finish for today. The overall picture is clear to you. When you rest and you have more time, you can go back and fill the last gap in understanding.
W: Thank you very much; It was a good experience. See you later.
Reading protocol - a set of strategies that the reader must use to get all the benefits of reading the text. A set of strategies for poetry is different from fiction, and strategies for reading fiction are different from scientific articles. It would be ridiculous to read a fiction book and wonder what sources allowed the author to say that the main character is tanned blond; but it would be wrong to read scientific literature and not ask such a question. This reading protocol is extended to viewing and listening protocols in art and music. In fact, most introductory courses in literature, music, and art are devoted to the study of these protocols.
For mathematics there is a special reading protocol. As we learn to read literature, so must we learn to read math. Schoolchildren should study reading protocol for mathematics in the same way that they learn the rules of reading a novel or a poem, learn to understand music and painting. Edward Rothstein's remarkable Emblems of the Mind reveals the relationship between mathematics and music, implicitly affecting reading protocols for mathematics.
When we read a novel, we are absorbed by the plot and characters. We are trying to follow different storylines and how each of them affects the development of the characters. The characters themselves must become real people for us - both those who delight us and those we despise. We do not stop at every word, but present the words as strokes in a picture. Even if the specific word is unknown, the overall picture is still clear. We rarely interrupt to reflect on a particular phrase or sentence. Instead, we allow the work to captivate us in its stream and quickly get through to the end. This is a useful and relaxing exercise, it gives food for thought.
')
Writers often characterize characters, involving them in carefully selected action episodes, rather than describing carefully selected adjectives. They depict one aspect, then another, then again the first in a new light, and so on. As the overall picture grows, it becomes more and more clear. This is the way to convey complex thoughts that are not precisely defined.
Mathematical ideas are inherently accurate and well defined, so you can give a clear and very short description. Both the mathematical article and the artistic novel tell a certain story and develop complex ideas, but the mathematical article does it with far fewer words and symbols than in the book. The beauty of the book is in the aesthetic way of using language to awaken emotions and to represent things that are not amenable to precise definition. The beauty of a mathematical article is in the elegance and effectiveness of the laconic expression of clear ideas of great complexity.
What are some common mistakes made by people trying to read math? How can these errors be corrected?
Do not lose the big picture
“Reading mathematics is generally not linear ... To understand the text, you need cross-references, fast scanning, pauses and repeated reading” (from the same place)
Do not think that understanding each phrase individually will allow you to understand the whole idea. It's like trying to see a portrait, looking at each square centimeter of the painting. You will see the details, texture and color, but completely miss the portrait. A math article tells a story. Try to understand what this story is before going into details. You can take a closer look later when you build an understanding framework. Do it the same way you could re-read an art book.
Don't be a passive reader
“The three-line proof of the modest theorem is the quintessence of years of work. Reading mathematics ... provides for a return to the thought process that occurred during its recording "(from the same place)
Examine examples for patterns. Try special cases.
A mathematical article usually tells only a small part of a large and extensive history. As a rule, the author wanders for months in the dark, trying to discover something new. In the end, he organizes his discovery in the form of an article where he hides all the mistakes (and their causes) and presents a complete idea in the form of a clean neat stream. To truly understand the idea and recreate what the author has hidden, read between the lines.
Mathematicians are very laconic, but much is hidden behind these few words. The reader must participate. At each stage, he / she has to decide how clear the presented idea is. Ask yourself the following questions:
- Why is this idea right?
- Do I really believe in her?
- Can I convince someone that she is true?
- Why did the author not use a different argument?
- Do I have a better argument or method explaining the idea?
- Why didn't the author explain it in a way that I could understand?
- Is my method incorrect?
- Did I really get the idea?
- Am I missing some nuance?
- The author misses some nuance?
- If I can't understand the meaning, maybe a similar, but simpler idea will be clearer to me?
- Is it really necessary to understand this idea?
- Can I accept this idea without understanding the details, why is it correct?
- Will my understanding of the big picture hurt if I don’t understand the proof of this particular idea?
If you spend too little effort on such participation, then it's like reading a book without concentration. Half an hour later you suddenly realize that you just turned the pages, thinking about yours, and did not remember anything that you read.
Do not read too fast
Reading mathematics too quickly leads to frustration. Half an hour of focusing on the art book will allow the average reader to read 20-60 pages with a full understanding of the text, depending on the book and reading experience. The same half an hour over a mathematical article will give 0-10 lines, depending on the article and the experience of reading mathematics. Work and time can not be replaced. You can speed up mathematical reading with practice, but be careful. As with any skill, trying to do too much too quickly can throw you back and eliminate motivation. Imagine that you exercise vigorously for an hour, if you haven’t been doing two years before. Maybe you will go through the first stage, but it is unlikely to continue further. The disappointment of constant observation of how experienced guys tirelessly perform twice as much as yours, while your whole body aches the next day because of fatigue is too hard to accept.
For example, take this theorem from the treatise “The Case of the Computer” by Levi ben Gershom, written in 1321.
"When you add consecutive numbers, starting with 1, and the number of items is odd, the result is equal to the product of the average and the last." For modern mathematics it will be natural to write the theorem as follows:
sum limits2k+1i=1i=(k+1)(2k+1)
The reader will need about the same time to unravel this small formula as he needs to understand two lines of the text version of the theorem. Example of Levy's theorem:
1+2+3+4+5=3×5.
Make an idea of your own
The best way to understand what you read is to make the idea your own. It means tracing the idea to its roots and rediscovering it yourself. Mathematicians often say that in order to understand something you must first read it, then write it in your own words, and then teach someone else. Each has different tools and a different level of "assimilation" of complex ideas. You need to adapt the idea to your own vision and experience.
“When I take a word, it means what I want.”
(Humpty Dumpty from the book "Alice through the Looking Glass" by Lewis Carroll)
“The meaning will seldom be completely understood, because each symbol or word already constitutes an extraordinary condensation of concept and reference” (from the same place)
A well-written mathematical text will accurately use the word in only one sense, making a distinction, say, between a combination and a permutation (or arrangement ). A clear mathematical definition may imply that a “yellow mad dog” and a “mad yellow dog” are a different arrangement of words, but the same combination. Most English-speaking readers will not agree with this. Exceptional accuracy is completely alien to most fiction and poetry, where the use of various words, synonyms and changeable descriptions is considered good form .
The reader is expected to understand that the absolute value is not some arbitrary value that turns out to be absolute, but the function has nothing to do with anything functional.
A particularly notorious example is the use of the phrase “from this it easily follows that ...” and similar constructions. They mean something like the following:
Now we can verify the validity of the following statement. Verification will require a certain amount of essentially mechanical, although perhaps difficult, work. As an author, I could spend it, but it will take a lot of space and, probably, will not lead to good results, since it is in your best interest to make calculations yourself and clarify what is happening here. I promise that no new ideas are involved. Although, of course, you may need to think a bit to find the right combination of good ideas to apply.
In other words, this phrase, when properly applied, is a signal to the author that there is something tedious and even difficult here, but not suggesting bright insights. Then the reader is free to decide how much he wants to understand all the details himself or if the author’s assurances are enough for him - and then we can say: “Okay, I believe you for the word.”
Now, regardless of your opinion on the appropriateness of the use of such a phrase in a particular situation and on the correctness of its use by the author, you should understand what it really means. The phrase "From this it easily follows that ..." does not mean
If you don't see it right away, you're a fool.
And she does not mean
This should not take more than two minutes.
But a person unfamiliar with a mathematical vocabulary may misunderstand it and will face disappointment. This is in addition to the question that the “tedious task” for one person may be the most difficult problem for another. Therefore, the author must correctly assess their audience.
Know yourself
Texts are written with the expectation of readers of a certain level. Make sure that you are such a reader or you want to do everything to join them.
From Thomas Eliot, "Song of Simeon":
Lord, the winter sun creeps between the snowy peaks,
Roman hyacinths flared in bowls;
Time froze - stubborn dumb lord,
Waiting for the whiff of death with a lightened soul
Like a feather on the senile danii. One.
Dust under the sun and the memory of the back streets
I look for coolness that blows for mortal valleys.
Lord, the Roman hyacinths are blooming in bowls and
The winter sun creeps by the snow hills;
The stubborn season has made stand.
My life is light, waiting for the death wind,
Like a feather on the back of my hand.
Dust in sunlight and memory in corners
Wait for the wind that chills towards the dead land.
For example, Eliot's poem largely implies that readers either know who Simeon is or want to find out. It also assumes that the reader will be either an experienced poetry reader, or one who wants to gain such experience. The author assumes that readers either know or understand the hints here. It goes beyond simple things like who Simeon was. For example, why are Roman hyacinths? Why is it important?
Eliot assumes that the reader will read slowly and pay attention to the pictures: the author compares dust and memory, compares old age with winter, compares the expectation of death with a fluff on the back of the palm, etc. He assumes that the reader will take it as poetry; in a sense, he assumes that the reader is familiar with the whole poetic tradition. He should notice that the odd lines rhyme and the rest do not, etc.
Most importantly, he assumes that the reader will connect to reading not only his mind, but also his emotions and imagination, which will draw the image of an old man, tired of life but forced to cling to it, waiting for some important event.
Most mathematical books are also written with some assumptions about the audience: that they know certain things, that they have reached a certain level of “mathematical maturity”, and so on. Before you begin to read, make sure that the author expects from you.
Math entry example
To experiment with the principles presented here, I included a small mathematical fragment, often called the birthday paradox. The first part is an exact mathematical article describing the problem and its solution. The second part is the reader's fictional attempt to understand the article using the appropriate reading protocol. The subject of the article is probability. It is accessible to the bright and motivated reader, even without a mathematical education.
Birthday paradox
A professor in a class of 30 random students often suggests that at least two people in a class have the same birthday (month and day, not necessarily a year). Will you accept the argument? And if there are fewer students in the class? Would you accept the offer to argue?
Suppose birthdays n people are evenly distributed over 365 days of the year (for simplicity, we do not take into account leap years). Let us prove that the probability of the same birthday (month and day) in at least two of them is:
1− beginpmatrix frac365 times364 times363 times... times(365−n+1)365n endpmatrix
What is the probability that among 30 random people in the room there will be at least two people with the same birthday? For n=30 this probability will be 71%, that is, the professor will win the dispute in 71 cases out of 100 in the long run. As it turns out, with 23 students in a class, the probability is about 50%.
Here is the proof: Let P(n) is the desired probability. Let be Q(n)=1−P(n) it is likely that everyone has different birthdays. Now find Q(n) by calculating the number of sets from n birthdays without duplicates and dividing it by the total number of possible sets of n birthdays Then find P(n) .
Total number of sets from n birthdays without duplicates:
365 times364 times363 times... times(365−n+1)
This is because for the first DR there are probably 365 variants, for the second - 364 and so on for n person. Total amount n birthdays without any restrictions is simply 365n because for each of n Birthdays have 365 options. In this way, Q(n) equals
frac365 times364 times363 times... times(365−n+1)365n
Solution for P(n) gives P(n)=1−q(n) and therefore our result.
Attempts by our reader to understand the paradox of birthdays
In this section, the naive reader tries to understand the meaning of the last few paragraphs. Readers' replicas are a metaphorical expression of his thoughts out loud, and Professional's comments are a research paper that the reader must do. The relevant protocols are in bold and inserted in the appropriate places of the story.
It looks like the Reader seems to grasp things very quickly. But be sure that in reality there is a lot of time between the reader’s comments and that I have left out many of his comments that lead to a dead end. You can understand what he is experiencing only if you read between the lines and imagine his way of thinking. Thinking about it is part of your own efforts.
Know yourself
Reader (W): I don’t know anything about probabilities, can I figure this out?
Professional (P): Let's try. You may have to make big digressions at every step.
D: What does the phrase "30 random (chaotic) students" mean?
"When I take a word, it means what I want, neither more nor
less"
P: Good question. The phrase does not mean that we have 30 people who are crazed or sick in the head. It means that we must assume that the birthdays of these 30 people are independent of each other and that every birthday is equally likely for each person. A little further, the author describes this more formally: “Suppose that birthdays n people are evenly distributed over 365 days of the year. "
Ch: Isn't it obvious? Why specifically to stipulate?
P: Yes, the assumption seems obvious. The author simply sets the basic condition. This sentence ensures that everything is fine, and the solution does not imply any imaginary bizarre science fiction.
Q: What do you mean?
P: For example, the author is not looking for a solution like this: everyone lives in the Land of Independence and was born on the 4th of July, so the chances of coincidence of the birthdays of two or more people are 100%. Mathematicians do not like solutions of this kind. By the way, the assumption also implies that we do not take into account leap years. That is, no one in this task was born February 29th. Read on.
D: I do not understand this long formula, What is n ?
P: The author solves the problem for an arbitrary number of people, not just for 30. From this point on, the author names the number of people. n .
Ch: I still don't understand. What reply?
Do not be a passive reader - try examples
P: Well, if you need an answer only for 30, just install n=30 .
Q: Okay, but it's hard to count. Where is my calculator? Let's see: 365 × 364 × 363 × ... × 336. This is tedious, and the result does not even fit on the screen. It is written here:
2.1710301835085570660575334772481e+$7
Even if knowing the formula, I cannot calculate the result, how can I understand where the formula came from?
P: You are right that this is not an accurate result, but if you continue further and divide, the answer will not be too far from the exact one.
Ch: All this is somehow uncomfortable. I would like to calculate the exact value. Is there another way to make calculations?
P: How many factors do you have in the numerator? How much in the denominator?
D: Do you mean 365 as the first factor, 364 as the second? Then it turns out 30 factors at the top. Also 30 multipliers below (30 copies 365).
P: Now can you calculate the result?
H: Oh, I see. I can pair each numerator from the numerator with each denominator multiplier, so it will turn out 365/365 for the first multiplier, then multiply by 364/365 and so on for all 30 multipliers. So the result will always fit in my calculator. (In a few minutes) ... So, I received .29368 if to round up to five characters.
P: What does this number mean?
Do not lose the big picture
Ch: I forgot what I was doing. So let's see. I was looking for an answer for n=30 . The number 0,29368 is all calculations, except subtraction from one. If you continue, you get 0.70632. Now what does this mean?
P: It is useful to learn more about probabilities here, but simply this means that in a group of 30 people two or more birthdays will coincide in 70,632 out of 100,000 cases, that is, in about 71% of cases.
H: Interesting. I would not have guessed. Would you like to say that in my class of 30 students the probability is quite high that at least two people will have the same birthday?
P: Yes, that's right. You can accept bets before you know their dates of birth. Many people think that such a coincidence is unlikely. That is why some authors call it the birthday paradox .
D: So that's why I have to read math to earn a couple of dollars?
P: I understand that for you this may be a kind of stimulus, but I hope that mathematics also inspires you and without any monetary perspectives.
R: I wonder what the result will be for other values. n . I'll try to do some more calculations.
P: Good thought. We can even make a graphic image of all your calculations. You can make a graph of the number of people and the probability of coincidence of birthdays, although it can be left at another time.
D: Oh, look, the author did some calculations for me. He says that for n=30 the answer is about 71%; I also received such a figure. And for n=23 It turns out about 50%. It makes sense? I think it has. The more people, the higher the probability of coincidence of birthdays. Hey, I'm ahead of the author. Not bad. Okay, let's continue.
P: Well, now you will tell yourself when to continue.
Do not read too fast
Ch: It seems that we got to the proof. It should explain why the formula works. What is this Q(n) ? I guess, that P means “probability” (probability), but what does Q ?
P: The author introduces a new value. He uses Q simply because it is the next letter in the alphabet after P but Q(n) - this is also a probability, and having a close relationship to P(n) . It's time to take a moment to think. What Q(n) and why is it equal 1−P(n) ?
H: Q(n) - This is the likelihood that no one in the room does not match birthdays. Why does the author care about this question? Don't we need another chance that the birthdays are the same?
P: Good point. The author does not say this explicitly, but between the lines you can understand that he has no idea how to directly calculate P(n) . Instead, he introduces the probability Q(n) which is supposed to be 1−P(n) . Probably, after that the author should show how to calculate Q(n) . By the way, when you finish an article, you may wonder how to calculate P(n) directly. This is a great continuation for the ideas presented here.
H: Everything has its time.
P: Okay. So now we know Q(n) , what's next?
W: Then we can get P(n) . If a Q(n)=1−P(n) then P(n)=1−q(n) . Ok but why Q(n)=1−P(n) ? The author believes that this is obvious?
P: Yes, he thinks so, and even worse, he doesn’t even tell us that this is obvious. Here is a rule of thumb: when the author clearly says that this is true or obvious , you should spend 15 minutes and convince yourself that this is the way it is. If the author does not even bother to say it, but implies it, then the process will take a little longer.
Q: How do I understand what you need to stop and think?
P: Just be honest with yourself. If in doubt, stop and think. If you are too tired, go watch TV.
D: So why Q(n)=1−P(n) ?
P: Let's imagine a special case. If the probability of coincidence of two or more birthdays is 1/3, then what is the probability of not getting a match?
H: It's 2/3, because the probability of the absence of an event is inverse to the probability of an event occurring.
Make an idea of your own
P: Well, you should be careful when using words like the opposite , but you're right. In fact, you discovered one of the first rules that is being studied in probability theory. Namely, that the probability of the absence of an event is equal to one minus the probability of an event occurring. We now turn to the next paragraph.
H: It seems to explain what equals Q(n) - in a long and complex-looking formula. I will never understand this.
P: Formula for Q(n) hard to understand, and the author relies on your diligence, perseverance and / or knowledge to understand it.
H: It seems to calculate all the probabilities of something and divide them by the total number of probabilities, whatever that means. I have no idea why he does that.
P: Maybe I can help you with some information on this subject. The probability of a certain outcome in mathematics is determined as follows: the total number of possible variants of this outcome is divided by the total number of all outcome variants. For example, the probability of throwing a four at a die is 1/6. Since there are one four and six possible outcomes. What is the probability that you will throw a four or a three?
D: Well, I think 2/6 (or 1/3), because the total number of possible outcomes is still six, but I have two options for a successful outcome.
P: Good. Now a more complex example. What about the probability of throwing four in total when throwing two dice? There are three options to get this amount (1-3, 2-2, 3-1), while the total number of combinations is 36. This is 3/36 or 1/12. Look at the following 6 × 6 table and see for yourself.
1-1, 1-2, 1-3, 1-4, 1-5, 1-6 2-1, 2-2, 2-3, 2-4, 2-5, 2-6 3-1, 3-2, 3-3, 3-4, 3-5, 3-6 4-1, 4-2, 4-3, 4-4, 4-5, 4-6 5-1, 5-2, 5-3, 5-4, 5-5, 5-6 6-1, 6-2, 6-3, 6-4, 6-5, 6-6
What about the probability of throwing seven in the amount?
D: Wait, what does 1-1 mean? Does it not equal 0?
P: Sorry, I'm to blame. I used the minus sign as a dash, just referring to a couple of numbers, so 1-1 means throwing edinichki on both dice.
Q: Could you come up with a better record?
P: Well, maybe I can or should do it, but the commas look worse, and the slash will look like a division, and everything else can also be misleading. In any case, we are not going to publish this transcript.
D: Thank God. Well, now I know what you mean. I can get seven in total in six ways: 1-6, 2-5, 3-4, 4-3, 5-2 or 6-1.The total number of outcomes is still 36, so it turns out 6/36 or 1/6. It is strange, why is the probability of dropping four different from the probability of dropping seven?
P: Because not every amount is equally likely. Here, the situation is different from simply rotating the wheel with numbers from 2 to 12 in equal intervals. In this case, each of the 11 digits has the same probability of dropping 1/11.
W: Well, now I'm an expert. Is the probability calculated simply by counting?
P: Sometimes yes. Although it is difficult to calculate.
Ch: Clear, let's continue. By the way, did the author really expect me to know all this? My friend is taking a course in probability and statistics, but I'm not sure that he knows all these things.
P: A small section of mathematics contains a lot of information. Yes, the author expects the reader to know all of this or that he will find this information and assimilate it, as we did. If I were not here, you would have to ask these questions to yourself and find the answers by thinking, in textbooks and reference books or in consultation with a friend.
H: So the probability of coincidence of birthdays for two people is the number of possible sets ofn birthdays without duplicates, divided by the total number of possible sets ofn birthdays. P: Excellent resume. D: I don't like using
n , so let's use 30. Perhaps then it will be easier to understand the generalization.n .
P: Great thought. It is often useful to analyze a special case in order to understand the general case.
H: So how many sets of 30 birthdays are there at all? I can not count. I think that I will have to further limit the conditions. Let's pretend we only have two people.
P: Good. Now you think like a mathematician. Will considern = 2 .How many possible birthday combinations are there?
D: I count birthdays from 1 to 365, not counting leap years. Then here are all the possible combinations:
1-1, 1-2, 1-3, ..., 1-365, 2-1, 2-2, 2-3, ..., 2-365, ... 365-1, 365-2, 365-3, ..., 365-365
P: When you write 1-1, you mean 1-1 = 0, how is subtraction?
Ch: Stop teasing me. You know exactly what I mean.
P: Yes, I know, and I can point out a good choice of recording method. Now how many pairs of birthdays are there?
H: For two people, 365 × 365 variants are obtained.
P: And how many options, if you do not take into account the coinciding birthdays?
H: You can not use 1-1, 2-2, 3-3 ... 365-365, so it goes
1-2, 1-3, ..., 1-365, 2-1, 2-3, ..., 2-365, ... 365-1, 365-2, ..., 365-364
The total number of variants is 365 × 364, because each line now has 364 pairs instead of 365.
P: Good. You're in a bit of a hurry, but still 100% right. Now can you summarize for 30 people? What is the total number of possible sets of 30 birthdays? Try to guess. You are good at it.
H: Well, if you try to guess (although this is not really a guess at all, after all, I already know the formula), then I would say that for 30 people you need to multiply 365 × 365 × ... × 365, for the total number of possible sets of DR.
P: Exactly. Mathematicians write 365 30 . And what is the total number of sets of 30 birthdays without repetitions?
H: I know that the answer should be 365 × 364 × 363 × 362 × ... × 336 (that is, start with 365 and multiply each time by a number with subtraction of a unit 30 times), but I'm not sure that I really understand why so Perhaps you should first consider the case with three people and find a way to increase to 30?
P: Brilliant thought. Let's finish for today. The overall picture is clear to you. When you rest and you have more time, you can go back and fill the last gap in understanding.
W: Thank you very much; It was a good experience. See you later.
Source: https://habr.com/ru/post/346228/
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