The essence of mathematics seems to me to be a steep mountain rising above the clouds, towering, unattainable, yet fascinating. I've tried to climb to the top, but the sheer slope and intricacies always make me halfway through. I know that this is not an easy task. Behind the simple word of mathematics lies a deep and mysterious knowledge, which is like an endless labyrinth that tempts us to explore the mystery at its core. Some people try to avoid math when choosing a college major because they are afraid of it. They would rather choose any other major than face this diabolical subject. I know their fears and frustrations, as I felt when I was a student. However, for most of us, learning math isn't just about solving complex math problems or becoming mathematicians. It is to develop our mathematical thinking. Mathematical thinking, which is a unique way of thinking. It gives us the ability to see the world in numbers and logic, enabling us to better understand and analyze problems. This way of thinking is ubiquitous in our daily lives, whether it's making decisions, solving problems, or trending in the future, it's an indispensable tool for us. So, how do you sharpen your mathematical thinking? First, be brave enough to face a problem head-on and not be intimidated by its superficial complexity. You know, it is these problems that constantly inspire us to learn more and make continuous progress. Second, we need to focus on understanding the essence of the problem, rather than just rote memorization. Only by truly understanding the core of the problem can we be able to reach out and respond flexibly to various situations. Finally, be good at induction. Through continuous reflection and refinement, we can grasp the essence of mathematical thinking and apply it more effectively to solve practical problems. This article may be a little lengthy and informative, but I will use all my knowledge to present it in a more interesting way. Trust me, read patiently, and you will surely gain some unexpected revelations. 
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1. Mathematics, the product of logic.
Oh, why do you still look resistant to this reaction? Mathematics is an indispensable tool in our daily lives. You've been studying mathematics for more than ten years, and now you don't know what it's for, isn't it a bit of a waste? You see, although the Chinese language is difficult to learn, at least after learning it well, you can write a touching love letter that will make your heart go further. It's the same with English, and although it's difficult, at least you don't have to ask for a translator when you travel abroad, so you can easily handle it yourself. Mathematics, which may have always been simple in your eyes, is nothing more than addition, subtraction, multiplication and division. However, when trigonometry appears in your life, you will find that it is also full of fun and charm. So, stop resisting math. It is not just knowledge in books, it is an indispensable part of our lives. Try to appreciate it, to understand it, and you will find that mathematics is actually a very interesting subject. Why is math daunting? Think about it, mathematics is like a majestic castle, so how did this castle accumulate bit by bit? In this temple of mathematics, there are many axioms, like the cornerstone of a castle. For example, a straight line between two points is the shortest; The line segment can be extended into a straight line indefinitely; With one end of the line segment as the center and the line segment as the radius, a perfect circle can be drawn; All right angles are congruent; If two straight lines intersect a third straight line, and the sum of the inner angles of the same side is less than the sum of the two right angles, then the two straight lines must intersect. These axioms can be dizzying, but as you may have noticed, these are actually the five basic axioms of Euclidean geometry. From these five axioms, we can derive countless theorems through pure logical reasoning. For example, the angle of each straight line is 180 degrees; The sum of the inner angles of the triangle is 180 degrees; Passing a point outside the line, only one line is parallel to the known line. These theorems make up the magnificent castle of Euclidean geometry. Have you noticed that mathematics is actually a product of logic? However, if I were to take a random piece of masonry from this castle and give it to you, could you tell me what problem it was used for? Do you know from which axiom it is derived? Do you understand how it's built? It doesn't matter if you don't know, we just have to remember it. However, who can remember the entire castle? When you are confused when you study, you will naturally feel like the world is spinning when you take the exam. Therefore, mathematics is daunting because it requires us to understand and memorize a large number of axioms and theorems, and the process of deriving these axioms and theorems is extremely complex. Only when we truly understand and master the logical system of mathematics can we truly appreciate the charm of mathematics and no longer be afraid of this majestic castle. But do you believe it? All math is about solving interesting problems. Clause.
Second, the base system is the wisdom of counting.
First, let's take a look at why computers use binary. In short, this is due to the fact that the various gate circuits of the computer have only two states, "on" and "off", which correspond to the binary "0" and "1", which is both concise and clear. However, why did most countries end up opting for the decimal system? This is because to solve the counting problem, our ancestors made use of ten fingers for counting, thus forming the decimal system. So, why are there still decimal and sexagesimal systems? In fact, the birth of these decimal numbers is also based on human wisdom and practical needs. For example, the decimal and sexagesimal systems are used for the timing of constellations, zodiac signs, and alarm clocks, respectively. The emergence of these decimal numbers is not because a certain race is born with twelve or sixty fingers, but because human beings have found the most suitable way to count themselves through continuous practice and innovation over a long period of time. In addition, we cannot ignore the importance of the decimal system. Humans have a total of twenty fingers with hands and feet, which also laid the foundation for the emergence of the 20decimal notation. The Mayans used this method of base notation. However, why isn't the decimal system as widely used as other decimal systems? This is probably because counting with feet is not very civilized. To sum up, binary, decimal, twelve, sexagesimal and twenty are all the crystallization of wisdom invented by human beings in order to solve practical problems. The advent of these decimal numbers not only enriches the way we count, but also makes our lives more convenient. Imagine if a math teacher in elementary school could use a lesson to guide us to understand the principles of these decimal numbers by breaking our fingers, would we remember this knowledge more deeply? I firmly believe that the answer is yes.
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3. Oral arithmetic is the rhythm of Chinese.
Perhaps, you feel like your math grades are a disaster. But you know what? Compared with your classmates in other countries around the world, you have a unique advantage. Why? Because of Chinese, it is helping you learn mathematics. Seeing this, you may be a little confused. What does learning mathematics have to do with Chinese? Let me tell you that Chinese is our mother tongue, and the rich connotation and unique charm it contains are a great help for us in learning mathematics. Imagine that when you calculate that 13 times 9 equals 117, have you ever thought that this calculation process is actually hidden in our common Chinese expressions? For example, what we often say is "three nine twenty-seven" and "one nine gets nine", which actually contains the rules of multiplication. This unique way of oral arithmetic, which is based on the rhythm and expression of Chinese, makes us more handy in the study of mathematics. But you know what? Not all languages can carry the logic and operations of mathematics so easily. In many other languages around the world, they don't have a ninety-nine multiplication table like Chinese, how do they do the calculations? Take, for example, the ancient Egyptians and Russians, who solved the problem of multiplication in their own way. The ancient Egyptians calculated the area by stacking stones, while the Russians wrote down a string of numbers on a piece of paper and got the answer by adding and subtracting. While these methods are unique and effective, they are not comparable to our ninety-nine multiplication tables in terms of efficiency and accuracy. The reason behind this is actually the difference in language. Numbers in Chinese are monosyllabic, which makes it easier for us to remember when reciting the ninety-nine multiplication table. In other languages, numbers may be two-syllable or even multi-syllable, which undoubtedly increases the difficulty of memorization. Therefore, when you think that mathematics is a disaster, you might as well think about it from another perspective: your mother tongue, Chinese, is actually a great assistant for you to learn mathematics. Not only does it make it easier to learn math, but it also gives us the opportunity to get a glimpse into a deeper layer of math. So, the next time you are struggling with math, you might as well thank the existence of the Chinese language, because it allows you to go further and more steadily on the road of mathematics. Clause.
Fourth, conditional probability is the best trick.
I still vividly recall my first math class at Nanjing University, which was a course called "Probability". When you hear "probability", you may think that it is just something you learned in high school. However, it wasn't until we delved deeper into Conditional Probability that we realized the mystery of this. "Conditional probability" is a seemingly simple concept, but it hides deep thinking. Imagine a couple of parents who have two small children. Now that one of the children is known to be a girl, what is the probability that the other child is also a girl? At first glance, you may think it's 50%, but it's not that simple. From the most basic point of view, we know that the sex of having a child is random, so there are four possibilities: male, male, female, and female, each of which is 25%. However, there is a key clue in the title: it is already known that one of the children is a girl. This means that we can't take into account the situation of men and men, so there are only three cases left: men and women, and women and men. Then, the probability that the other child is also a girl is one in three. That's the beauty of conditional probability, which allows us to see the likelihood of an event occurring under certain conditions. Further, the application of conditional probability goes far beyond that. For example, in a survey, we may conduct a survey under different conditions, which can affect our results. For example, in our work, we may select potential target customers based on certain conditions, which involves the application of conditional probability. However, the most interesting thing is that conditional probabilities can even be used to increase their success rate. They may use some means to screen out their target customers who are easily deceived, so as to increase the success rate of the scam. This seemingly clever trick actually makes use of the principle of conditional probability. So, you see, conditional probability is not a sophisticated mathematical concept, it is actually hidden in our daily lives. As long as we look closely, we can see its existence and impact. 
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5. Mathematical expectations are a gambler's labyrinth.
When it comes to probability, we naturally think of a classic experiment: a numbered cube repeatability probabilistic experiment, also known as a "bet size". This experiment is like a scene from a TV show, where the dealer shakes the dice cup and then asks you to place a bet. You may be wondering, should you buy big or buy three sixes? This actually involves a mathematical concept – mathematical expectation. Let's do the math. If you bet big, the probability of winning is 486%, the income is 1 yuan; If you don't win, you will lose 1 yuan. It seems to be balanced, but after calculation, it is found that the mathematical expectation value of buying a large is -00278 yuan, which means that in the long run, every time you gamble, you will lose 00278 yuan. Let's take a look at buying three sixes. If you win, you can earn 149 yuan; If you don't win, you will lose 1 yuan. At first glance, it may seem like a good opportunity, but in reality, the probability of winning three sixes is only 046%。After calculation, it is found that the mathematical expectation value of buying three sixes is -0$31, which means that for every bet, you will lose 031 yuan. You will find that whether you choose to buy big or buy three sixes, you will lose money in the long run. Therefore, in the face of this gambler's labyrinth, the wise choice is, of course, to "gamble for a long time and lose" and withdraw in time. Clause.
Sixth, the law of large numbers is the good luck of the system.
After going deeper**, the probabilities and mathematical expectations in math books are not just for doing bad things. In fact, they are the basis for business operations. Venture capital, for example, cleverly uses these concepts to generate returns. Imagine if a startup had a 5% chance of success and could bring 20 times the profit after success, would it be worth the investment? From the point of view of mathematical expectations, the answer is yes. However, the probability of success in self-employment is only 5%, which is negligible and full of uncertainty. So, how do you turn this uncertainty into stable gains? The answer lies in the law of large numbers. Imagine that when you toss an even coin, there is a 50% probability that heads or tails will be face up. But if you toss multiple times in a row, the number of heads and tails will tend to be closer to the desired value as the number of times you toss them. Venture capitalists use this principle to increase the probability of investment success. If the probability of success of a single entrepreneur is only 5%, then investing in 20 such entrepreneurs increases the probability of at least 1 person succeeding to 642%。In this way, by diversifying their investments, they can greatly increase their probability of success. In the business world, every entrepreneur has a different probability of success. But overall, starting a business is a life-or-death adventure. The task of investors is to rely on their own experience and vision to screen out those entrepreneurs with higher success rates and greater return multiples for investment. And to ensure a high probability of success, they tend to diversify their risk by investing in many candidates at once. In this way, they are able to turn individual uncertainty into group certainty, reaping rich rewards through the law of large numbers. 
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7. Calculus is a dynamic vision.
When you were an innocent student, when you said to your math teacher in confusion, "Teacher, I always think math is useless." Most math teachers will give you a heartfelt answer: "Learning math is to develop your mathematical thinking." So, what exactly is mathematical thinking? It is like a blueprint for the construction of an edifice, clearly depicting the growth of the enterprise. Another example is the probability system, which teaches us that the right thing is worth repeating. Calculus, algebra, and game theory, which seem to be esoteric fields, are also valuable tools for exercising mathematical thinking. Calculus, a theory invented by Newton, has an extremely concise and beautiful way of thinking. Calculus uses the concept of infinitesimal to reveal the laws of moments, while integration reflects the cumulative effect of these transient variables. Imagine that a stationary object accelerates instantaneously when pushed, but acceleration does not produce velocity instantaneously and takes a period of accumulation. That's the beauty of calculus, which allows us to understand how the world works from a microscopic perspective. What's more, understanding calculus allows us to see things in a dynamic light. Just like your hard work at night, one night's hard work will not immediately translate into ability, it will take a period of accumulation. In the same way, having the ability does not mean that the results will be achieved immediately, and it will take time to precipitate. This process from effort to ability and then to achievement is a point effect. It is not a day's work, this is the wisdom of life. However, there are always people in life who are in a hurry and ask suspiciously: "I work so hard, why don't the leaders appreciate me?" "They ignore the accumulation and precipitation of the process. Similarly, some people may have been doing well at work, but one day they start to slack off, only to find themselves less capable after a few months. They forget that all successes and failures do not happen instantaneously, but are the result of accumulation over a long period of time. Therefore, learning mathematics is not just about preparing for exams, but also about developing our mathematical thinking so that we can understand the world from a deeper and more comprehensive perspective. Revolutionize the way we understand limits, derivatives, differentiation, and integration.
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8. Algebra is the value of direction.
After the exploration of calculus, let's dig deeper into the treasure of algebra. Natural numbers, like clear streams, trickles, nourish our mathematical world. They are the cornerstone of mathematics. However, natural numbers are only the tip of the iceberg, and the ocean of mathematics contains even richer treasures. Integer, they are like a magnificent sea, deep and vast. The addition of negative numbers causes the integer to extend on the number line. They are like a tapestry of mathematics, delicate and colorful. Rational numbers, they are the soul of mathematics. Scores, these seemingly mundane numbers, but they sow the seeds of continuity in our hearts. They make our digital world change from discrete to continuous, so that every point on the number axis is filled with the rhythm of life. However, rational numbers are not the end of mathematics. Irrational numbers, these infinite non-cyclic decimals, like mysterious stars dot the night sky of mathematics. They challenge our perceptions and break our preconceived notions of numbers. and root numbers, representatives of these irrational numbers, they are complex and mysterious, symbolizing the infinite possibilities and complexities of this world. In addition to size, number has another important property: direction. Vectors, these directional numbers, are the compass of mathematics. They not only describe the size, but also reveal the direction. In this complex world, the existence of direction makes mathematics have richer expression and deeper connotation. When we are faced with a complex situation, how do we choose? How can you tell? Perhaps, we can learn from the laws of survival in nature. When multiple forces act on the same object, the result is not always a simple battle of victory and defeat. They hold each other back, influence each other, and the end result is often the one that is relatively right. This may be the wisdom of mathematics, which teaches us how to find law from complexity, and how to find order from chaos. Let's delve into this complex and mysterious world of mathematics, feel its rhythm, and understand its language. In this journey full of challenges and opportunities, let's keep exploring, learning and growing. In the field I know, mathematics is an indispensable benefactor in my life. It's not only something I love to study, it's also a tool I use to make a living. It shapes the way I think and becomes the underlying logic for me to deal with problems. Just as a poet loves poetry and a family is obsessed with musical notes, my love for mathematics grows day by day, and I can't let it go. (end)