The base of the natural logarithm, e = 2.718281828..., is an important constant in mathematics. Why was this constant introduced? What are its important applications? Let's take a closer look.
First, consider an ideal compound interest problem: Imagine a bank with an annual interest rate of 1%. If a customer deposits 1 yuan at the beginning of the year, the total principal and interest at the end of the year would be 1 + 1 = 2 yuan. If the bank applies the same interest rate to shorter-term deposits, calculating the interest based on the proportion of the deposit term in a year, for example, a half-year interest rate of 1/2, a three-month interest rate of 1/4, and a one-month interest rate of 1/12, etc. Now, customers have a better deposit strategy: If they deposit 1 yuan for half a year and immediately renew the principal and interest upon maturity, the total principal and interest at the end of the year would be If the deposit term is three months, the total principal and interest at the end of the year would be If the deposit term is one month, the total principal and interest at the end of the year should be It can be seen that the shorter the deposit term, the higher the total principal and interest at the end of the year according to the above compound interest calculation. Suppose the bank allows any short deposit term, such as one day, one hour, or one minute, or even shorter, then what would be the total principal and interest after one year? Would depositors become extremely wealthy as a result?
In 1697, Swiss mathematician Jacob Bernoulli considered this problem. By dividing the year into n equal parts and calculating the compound interest based on the n-year term, the total principal and interest after one year would be Jacob Bernoulli proved that when n becomes increasingly large and tends to infinity, the above total principal and interest would increase but not infinitely, but would increasingly approach a fixed value between 2 and 3. Now we know that this value is equal to 2.718281828... In mathematical terms, when n tends to infinity, the limit of the sequence exists. This tells us that no matter how short the deposit term is, the total principal and interest at the end of the year will not exceed the limit value of the sequence 2.718281828... If compound interest is calculated every moment, the total principal and interest at the end of the year will be this limit value.
Later, Swiss mathematician Euler encountered this limit value again when studying the exponential function and logarithmic function, and recorded it as e. He also proved that e is an irrational number. More importantly, Euler proved that for any real number x, when n tends to infinity, the limit of the sequence is e^x, that is, the exponential function with base e. It just generalizes the previous compound interest calculation example: the principal is still 1, but the annual interest rate is changed to x. Under this condition, if compound interest is calculated every moment, the total principal and interest at the end of the year will be e^x. However, here the "interest rate" x can be positive or negative, and is any non-zero real number.
The above bank is indeed a fictional mathematical model, but there are indeed many actual natural phenomena that conform to this model. As long as a quantity increases (or decreases) by a fixed rate every moment (or decreases), the change of this quantity will conform to the above exponential growth (or decrease) law. For example, the mass of radioactive substances decays with time, the intensity of sound waves or light waves attenuates with the propagation distance, the relationship between human responses to various stimuli and the intensity of stimuli, etc., all conform to the above exponential rate. Malthus' conclusion about the exponential growth of population also relies on the same mathematical model.
Euler also proved that the instantaneous rate of change (that is, the derivative) of the exponential function e^x is itself. For example, if the velocity of a moving object at time t is e^t, then the acceleration at time t is also e^t. This unique property of the function e^x makes e^x play a special role in calculus, solving differential equations, and other applications, thus in various theoretical derivations and engineering calculations.During the calculation, the number e and e^x constantly appear.
With the advancement of technology and the development of mathematics, the use of the number e has become increasingly widespread. The number e can be found everywhere in various fields of mathematics. One of the important reasons for this is the unique properties of the exponential function with e as the base.