In high school mathematics, everyone has learned the concept of logarithm and used logarithmic tables. The logarithm table in the textbook is based on 10, which is called common logarithm. As mentioned in the textbook, there is a logarithm whose base is irrational number E = 2.7 1828 ... called natural logarithm, and this E is the protagonist of our story. I don't know if it makes you more confused. Is it more natural to use such a strange number as the base in decimal system than to use 10 as the base? What is even more curious is, what is the story of such a strange number?
This will start from ancient times. This number was mentioned at least half a century before the invention of calculus, so although it often appears in calculus, it was not born with calculus. So under what circumstances did it appear? One possible explanation is that this figure is related to the calculation of interest.
We all know what compound interest is, that is, interest can be regenerated with the principal. But the sum of principal and interest depends on the interest period. In a year, interest can be calculated once a year, once every six months, once a season, once a month or even once a day. Of course, the shorter the interest period, the higher the sum of principal and interest. Some people are therefore curious, what will happen if the interest period is shortened indefinitely, such as once every minute, even once every second, or all the time (theoretically)? Will the principal and interest rate increase indefinitely? The answer is no, its value will stabilize and approach a limit value, and the number e appears in the limit value (of course, this number was not called e at that time). So in today's mathematical language, e can be defined as a limit value, but there was no concept of limit at that time, so the value of e should be observed, not obtained through strict proof.
All-encompassing e
I'm afraid readers have been thinking that just calculating interest should not be enough to tell a whole book. Of course not. Interest is only a small part. Surprisingly, this figure, which is closely related to the calculation of compound interest, is actually related to many problems in different branches of mathematics. When discussing the origin of e, there are actually many other possibilities besides compound interest calculation. Although the questions are different, the answers are all from different routes to the number e. For example, one of the famous problems is to find the area under the hyperbola y =1/x. What is the relationship between hyperbola and compound interest calculation? No matter how you look sideways, how you look vertically and how you sit and think, you can't figure out why, can you? But this area is calculated, but it is closely related to E. I just gave an example, and there are more in this book.
It would be embarrassing if the whole book only talked about mathematics and told stories. In fact, while discussing mathematics, the author interspersed many interesting related stories. For example, do you know who invented the first logarithmic table? It's John Napier. Never heard of it? This is normal, and I only met him after reading this book. It's important to ask the next question. Do you know how long it took Napier to build the whole logarithmic table? Please note that this happened at the end of16th century17th century, not to mention computers and computers. There is no computing tool at all. All calculations can only be done slowly with paper and pencil, not logarithm, so as to simplify the calculation. So it took Napier twenty years to build his logarithmic table, which is incredible! Try to imagine doing the same boring calculation every day for 20 years. This kind of boring day is by no means unbearable for ordinary people. But Napier survived, and his efforts paid off-Logarithm was eagerly welcomed, adopted by many scientists in Europe and even China, and even Napier was praised from all over the world. Among the earliest users of logarithms were the famous astronomer Kepler, who used logarithms to simplify the complicated calculation of planetary orbits.
There are many interesting facts in Mao Qi's Theory that we can't read in ordinary math textbooks. For example, who wrote the first calculus textbook? (If you have suffered from the calculus course, you also want to know who is the "initiator", right? ) It's Mr. Robida. Yes, it's the policy of Robida Hospital. But it was johann bernoulli who first discovered Robida's law. But this has nothing to do with plagiarism. There is an agreement between them.
Speaking of Bernoulli, there is a story. This family is really amazing. Other families can laugh at a genius, and their genius is described as "mass production". Bernoulli has been active in the field of mathematics for one hundred years, and many of their achievements (not only in the field of mathematics) are as thick as a book even if they are listed casually. However, there is another thing that this family is good at, and that is unpleasant, and that is quarreling. It's not enough for family to quarrel, but also quarrel with people outside (it can be said that it is "what it looks like"). Even father and son won the grand prize, and father was very dissatisfied. He felt that it should be his own, and he was so angry that he drove his son out of the house. Compared with many modern "dutiful sons", this father should feel ashamed.
The "influence" of e is not limited to the field of mathematics. In nature, the seed arrangement of sunflower and the pattern on nautilus shell are all spiral shapes, and the equation of spiral is defined by E, which is also used to construct the scale. If both ends of a chain are fixed and loosely suspended, it needs to be used if its shape is expressed by mathematical formula. Isn't it wonderful that all these problems that have nothing to do with calculating interest rates or hyperbolic areas are related to E?
Mathematics is actually not that difficult!
Each of us has read a lot of mathematics when growing up, but in many people's minds, mathematics seems to be a boring or even terrible subject. Especially when it comes to college calculus, definitions, theorems and formulas are everywhere and daunting. One of the reasons why we are afraid of a subject is that we have a sense of distance. Those things in calculus seem to come out of thin air, and I have no feeling for them. The feeling has nothing to do with me. If we know how calculus evolved, who invented it and what happened when it was invented (who invented calculus has been debated for many years, which has a great influence on the development of mathematics),