【英語要約】とてつもない数学

Review

Suddenly, I would like you to consider the next problem.

"Is there more than one person with exactly the same number of hairs in Yokohama? However, the population of Yokohama is about 3.5 million, and the maximum number of people's hair is 150,000."

When we consider the number of hairs that can be cut off in a day and the small number of small hairs, it is humanity to think, "I don't know." The correct answer is "I". It seems like I'm saying "Isn't it appropriate?", but it can be proved by using the mathematical "Pigeon nest principle".

Suppose there are 3 nests for 4 pigeons. If all four birds enter the nest, there will be a "shared room" that can accommodate two or more pigeons. This is the "pigeon nest principle". Applying the problem at the beginning, the population of Yokohama (pigs) is larger than the number of hairs (nests) per person. Therefore, multiple people enter the same nest (the number of hairs). Therefore, it can be said mathematically that there is 100%.

Mathematics solves such complex-looking problems in an instant. It is a study that combines ultimate rationality and beauty. Many people are "numerically allergic" because of their weakness in mathematics. However, the logical thinking, statistics, and design thinking that are indispensable to the modern business person can all be traced back to mathematics. Mathematics is an essential education for us who live in the present.

This book is a guide like a guide who invites you to a profound "mathematical swamp" from a light edge. Whether you like math or hate it, it's worth reading. I would like you to experience the enormous fun of mathematics by hand.

Main points of this book

Point 1

By converting huge numbers into "numbers per unit amount", it is possible to make them understand with specificity.

Point 2

The classic theory of mathematics, "Original", is a must-read book for Western elites, as it contains the basics of logical thinking. “Logical power” is indispensable as a quality of a leader.

Point 3

With Benford's Law, you'll be able to see through fraudulent books and votes.

Point 4

Just because there is a correlation does not mean that there is a causal relationship. Literacy is required to read statistics.

Ridiculously interesting mathematics

How big is one trillion?

Can you imagine the size of "1 trillion"? There is rarely an opportunity to see "◯ trillion", and it is difficult to realize the size.

Let's think about how long it will take to count from 1 to 1 trillion. Since 1 hour is 3600 seconds and 24 hours a day, it is about 90,000 seconds a day. If you can count up to about 100,000 per day, it will be 36.5 million in one year and about 100 million in three years. It takes about 30,000 years to count up to 1 trillion. 30,000 years ago, the Neanderthals were extinct.

By the way, 1 trillion meters is about 25,000 orbits of the earth, which is about 6.7 times the distance from the earth to the sun. With such meanings, it will be possible for the first time to realize that "1 trillion" is a tremendously large number.

Meaning making use of the size per unit amount is indispensable for persuading others. In 2008, Steve Jobs introduced at an Apple event that the first iPhone "sold 4 million units within 200 days of its release." At that time, he added, "Equivalent to selling 20,000 units every day." Jobs succeeded in letting the audience understand the meaning of the size of the numbers in an instant by making good use of the size per unit quantity of "20,000 units per day".

The technique of reducing the whole number into smaller numbers to make it easier to visualize large numbers is used in various situations.

Explosive power of exponentiation

Multiplying the same number repeatedly, such as "2 x 2 x 2," is called exponentiation. When the number of multiplications increases, it will change explosively from the middle.

For example, the thickness when you fold a newspaper. Assuming that the thickness of newspaper is 0.1 mm, the thickness when bent n times is 0.1×2n (mm). It is about 164 cm in 14 times, which exceeds the average height of an adult woman. But after this, it's sharp. Approximately 170 kilometers (distance between Tokyo and Atami) in 30 times, and distance to the moon (approximately 380,000 kilometers) in 42 times. You can imagine that exponentiation will explode from the middle.

Social phenomena can also be described using exponential functions that extend powers. In the "Population theory," British economist Malthus, who was active in the 18th and 19th centuries, predicted that "the population will increase geometrically in the future." "Increase in geometric progression" is to repeatedly multiply the first number by "1, 3, 9, 27, …" and the same number. The way to increase is nothing but the way to increase power.

In fact, since the end of the 19th century, the world population has been increasing at a rate that can be called a “population explosion”. The world population, which was about 1 billion in 1800, increased to 6.1 billion in 2000, 200 years later. It is said that it will reach 10 billion by 2056.

With a simple elementary function called an exponential function, it is possible to express the activities of social activities by human free will. I feel the tremendous potential of mathematics there.

Wonder of integer

The whole number obtained by increasing (1, 2, 3, …) or decreasing (-1, -2, -3 …) by 1 from 0 and 0 is called an integer. Integers are familiar to us, but their nature is a mystery.

Integers have various characters and have various names such as natural numbers, prime numbers, even numbers, odd numbers, triangular numbers, square numbers, fraternity numbers, and Pythagoras numbers. Among them is the “perfect number”.

A perfect number is a range of positive integers, and for an integer, the sum of all divisors not including itself refers to the number when it matches the original number. The minimum perfect number is 6. Except for 6, the divisor of 6 is 1, 2, and 3, and when these are added together, it becomes 6.

There are only four complete numbers less than 10,000: 6, 28, 496, 8128. Only 51 complete numbers have been found in studies going on from BC to the present, with the 51st perfect number having more than 49 million digits. It has not been proved, but it is expected to be infinite.

By the way, the fact that the first perfect number is 6 is said to be related to the fact that God created the world in 6 days. The multiple of 6 is a convenient number that can be divided by various numbers, and in fact, many numbers around us (12 months, 24 hours, 60 minutes, 360 degrees, etc.) are multiples of 6.

The next perfect number, 28, is the total number of protons and neutrons at which the nucleus is particularly stable. It matches the number of bones excluding the hyoid bones and the number of teeth excluding wisdom teeth that make up the adult skull. In addition, since 28 years have passed and 7 leap years are straddled, the relationship between the month, day, and the day of the week goes round. In other words, the calendar of 28 years ago can be used as it is.

[Must read point!] The origin of logical thinking

Essential reading of the Western elite "Original theory"

The original theory is said to have been written by Euclid of ancient Greece around the 3rd century BC. The oldest mathematical text in the world, it was used as a textbook in high schools around the world at least 100 years ago. Aside from the Bible, it is the best worldwide bestseller ever.

Why did "Original" spread so widely? This is because not only mathematics but also the method of logical thinking that is applicable to all fields is written. In Western countries that inherited ancient Greek culture, logical thinking has been respected since ancient times. Rather than sense and flattery, it is believed that the “logical power” that persuades the surrounding people and understands the claims of the opponent is the necessary quality for the leader. "Original theory" is the best textbook for learning logical power, and it is still an essential education for modern Western elites.

The logical thinking method described in "Principle" is a method of "starting from definition and axioms and accumulating correct propositions". The "definition" is the "meaning of words" and is intended to avoid ambiguity and misunderstanding of words used in discussions. "Axiom" is "a promise as a premise", a common recognition that serves as a starting point. And "proposition" refers to "a thing that can be objectively judged to be true or false". Only "correct propositions" can be discussed and accumulated. The conclusions obtained by accumulating false propositions cannot be said to be logically correct.

Definition → Axiom → Proposition → Conclusion

Let's think about logical thinking in concrete situations. Let's discuss the pros and cons of making cell phone calls on the train. Here, the "state in which there is a person around" is defined as "the state in which the voice of another person can be heard". The "axiom" that presupposes the discussion is that "don't bother others." The "proposition" that can objectively judge whether the defined state causes trouble to others can be examined as follows. First of all, if a person continues to be able to hear only one side of the conversation like a telephone call, he or she unconsciously wants to know the content of the conversation, and is distracted. This is called "cognitive takeover" in psychology. Human beings are irritated by this "takeover of cognitive function". Therefore, the conclusion to be drawn is that "you should not call when there are people around you."

It is said that the philosopher Plato first declared the above logical thinking method of "definition → axiom → (correct) proposition → conclusion". The Principles did not write new facts discovered by Euclid himself. It is a clear description of geometry developed from Pythagoras, systematically according to Plato's law of logical thinking.

With AI and machine learning dominating, logical thinking is becoming increasingly important. However, “logical thinking” has not yet penetrated in Japan compared to the West. It may be related to the fact that we Japanese have not read the original theory.

Incredibly useful math

What is the most frequent number?

There are many numbers around us. Business results, population, address, stock prices, etc. are all numbers. Needless to say, they are made up of combinations of 0-9. So what is the highest number among all numbers? In fact, it has a remarkable regularity.

The highest number at the beginning is 1, which accounts for about 30% of the total. The ratio decreases as the number increases, such as 2 for 17.6% and 3 for 12.5%, and the ratio of numbers starting with 9 is only about 5%.

This is called "Benford's Law," advocated by American physicist Frank Benford in 1938. Benford has reached this law by collecting more than 20,000 samples of molecular weight, river basin area, newspaper articles, pressure, and design.

In the natural world, an exponential increase that doubles at regular time intervals is often seen, like the growth of bacteria. In this way of increasing, the period in which the first number is "1" is particularly longer than the other numbers.

In addition, when numbers are given in order like member numbers, when it is expected that the numbers are evenly distributed in a certain range like the length of a river, and in randomly collected data, Benford's law is used. Holds.

How to spot the financial results

Benford's law can also be useful in our social life. Haru Valian, who is known as "the economist who made Google the number one in the world", said that this law "can spot out the powdered accounts."

When trying to disguise the amount of money in a company's books, etc., if you do not know Benford's law, you may write the leading numbers in a distribution that is too even or too biased. You can discover that the data is fake because it deviates greatly from the law.

In fact, there are cases where this led to the discovery of fraudulent books. In the early 1990s, an accounting school lecturer, Marc Niglini, asked the question, "Make sure that the highest digit of each company's balance shows a distribution according to Benford's law." Then a student discovered that the numbers in the books of a hardware store run by a relative were completely different from the law.

Benford's law is now used not only for auditing audits, but also for verifying election fraud.

Statistics lie

It is said that there is a correlation that if one increases, the other also increases. Sometimes we look for correlations to find surprising combinations in large amounts of data. However, there are two points to note in that case.

First, the correlation obtained is only the result of the survey. For example, suppose a student at a cram school has a correlation that "the higher the English score, the higher the math score." However, I have not examined the whole country, so I do not know if it applies to all Japanese students.

Second, even if a correlation is found between two quantities, they are not necessarily causal. It is assumed that there is a positive correlation between "newspaper purchasing" and "annual income". You may look at this and expect that if you read a newspaper, your annual income will increase! However, rising social status and higher annual income may have only increased the need to read newspapers as a social topic. It's very difficult to see if a causal relationship really holds.

"Statistics" is overwhelmingly persuasive because it is transmitted via numbers and graphs. I also feel the air that does not allow refutation. But statistics are not always correct. Data may be biased, appropriate processing may not have been performed, or the data itself may have been tampered with.

From now on, there will be a world where the correct statistics and the incorrect statistics are confused. We must have true statistical literacy and be able to derive the truly necessary and correct information from the treasure trove of data.

Recommendation of reading

Many interesting math quizzes such as those introduced in this review are scattered in this book, and it is fun to solve them with chopstick rest. There are lots of stories that you'll want to tell people, "I see!" such as "How to solve Tsurugame calculation efficiently" and "Multiply with your fingers." It is easy to understand with illustrations and illustrations, and I flip through and search for interesting problems. It is certain that even people who are not good at mathematics will be addicted to it. There are also articles using Fermi estimation such as "How many manholes are there in Tokyo?"

It's a great book that, while having fun, you can learn the knowledge of mathematics, history, trivia, and the applied power of numbers.

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