The full name of e is the base of natural logarithm, not natural logarithm, and natural logarithm is ln. Generally speaking, the base e of natural logarithm was discovered by Euler (1707- 1783, Switzerland) while studying calculus. E = lim (1+ 1/x) x, the extreme value when x approaches positive infinity. In the calculation, e =1+1(1! )+ 1/(2! )+ 1/(3! ) ..., the more entries, the more accurate. Compared with pi mentioned last time, e is not as important to human beings as π. But e is everywhere. -Ancient people's understanding of E Around 65438 BC+0700 BC, the ancient Babylonians once asked a question: If you lend money to others at an annual interest rate of 20%, how much money do you have one year later? This problem is nothing more than a simple formula:1x (1+0.2)1=1.2 If the interest is compounded every six months, the sum of the principal and interest in the first year is1x (1+0.2/2). That is1x (1+0.2/4) 4 =1.21550625. If you compound interest once a month, it is1.2193910849. Compound interest once a day. The sum of principal and interest in the first year is 1.22 1399696,1.221402717,1.22/kloc-respectively. As can be seen from the above calculation, the annual interest rate remains unchanged, compound interest by stages, the number of periods increases, and the principal and interest and growth are slow; However, no matter how the number of installments increases, the sum of principal and interest will not increase indefinitely, but there is an "upper limit" that can never be exceeded. This cap is the sum of the principal and interest of the first year when compound interest has been compounded. In mathematical language, it is the limit of the sum of principal and interest in the first year when the number of periods tends to infinity. With a little knowledge of calculus, we can work out that this limit is equal to E0.2 =1.221402 7581. The Babylonians didn't know the problem of continuous compound interest. Obviously, it was very painful to discuss such a large decimal in ancient times. -The Bernoulli family's contribution to E was studied by the famous Swiss mathematician Jacob Bernoulli (1683). But he only put forward a formula that the number should be between 2 and 3, and did not get complete data. Because there was no concept of limit at that time. By the way, the Bernoulli family has trained eight talented scientists for three generations. This Jacob Bernoulli was fascinated by the number of winners and losers in gambling games and wrote his masterpiece "Guessing". He also solved catenary problem (1690), curvature radius formula (1694), Bernoulli lemniscate problem (1694), Bernoulli differential equation (1695) and isoperimetric problem (65433). In addition, he loves logarithmic spiral very much. One of the most talked-about anecdotes is that Jacob is obsessed with studying logarithmic spiral, which started with 169 1. He found that the logarithmic spiral was still a logarithmic spiral after various transformations. For example, its inflection point line and extension line are logarithmic spirals, and the trajectory from the pole to the tangent is vertical. The reflection line obtained by reflecting the logarithmic spiral with the pole as the luminous point, and the curves tangent to all these reflection lines (back rays) are logarithmic spirals. He was amazed at the magic of this curve, and even asked future generations to carve the logarithmic spiral on their tombstones in his will, accompanied by a eulogy "Even if it changes, it will still be me" to symbolize immortality after death. There is also a johann bernoulli, who not only solved the catenary problem (169 1), but also put forward the Robida rule (1694), the steepest descent line (1696) and the geodesic problem (1697). In addition to the publication of the Course of Integral calculus (1742), there is also a greatest contribution to the field of human mathematics, which is to cultivate a good student-Euler. Students studying physics have also heard of another Bernoulli: daniel bernoulli, who is the son of John above. This man has made great contributions to fluid mechanics. The transverse vibration of elastic string (174 1 ~ 1743) is studied, and the propagation law of sound in air (1762) is put forward. His works also cover astronomy (1734), gravitation (1728), Shengxi Lake (1740), magnetism (1743, 1746) and vibration theory (65438). Far away, let's go back to natural logarithm. -The Birth of the Genius Euler Now, it's Euler's turn to play. Before, let's introduce this Mr. Euler in a little space. Euler's life is legendary. He began to teach himself algebra before he was ten years old. You know, many European knights were illiterate at that time. He was promoted by johann bernoulli in college, and then daniel bernoulli recommended him to the Academy of Sciences in Petersburg. It can be said that the Bernoulli family is a noble person of Euler. Euler can calculate the orbit of a comet in three days. 177 1 year, a fire broke out in Petersburg and Euler's study was destroyed. However, he was blind and spent a year rewriting most of his papers from memory. Euler wrote 886 books and papers. After his death, the Academy of Sciences in Petersburg spent 47 years sorting them out. Euler can recite the first 10 power of the first 100 prime numbers. Euler created many new symbols, such as π( 1736), i( 1777), e( 1748), sin and cos( 1748), tg( 1753). F(x)( 1734) has Euler's name in almost every mathematical field, from Euler line of elementary geometry, euler theorem of polyhedron, Euler transformation formula of solid analytic geometry, Euler solution of quartic equation to Euler function in number theory, Euler equation of differential equation, Euler constant of series theory, Euler equation of variational method, Euler formula of complex variable function and so on. His contribution to mathematical analysis is more original, and the book Introduction to Infinitesimal Analysis is his epoch-making masterpiece. Goldbach conjecture was also put forward in his correspondence with Goldbach. Euler also completed the precise theory of the moon's motion around the earth for the first time, founded the mechanics disciplines such as analytical mechanics and rigid body mechanics, and deepened the design and calculation theory of telescopes and microscopes. Euler first defined logarithm as the inverse operation of power, and discovered for the first time that logarithm has infinite values. He proved that any nonzero real number r has infinite logarithm. Euler made trigonometry a systematic science. He first gave the definition of trigonometric function by ratio, but he always used the length of line segment as the definition before. Euler's definition makes trigonometry jump out of the circle of studying triangular tables only. Euler analyzed and studied the whole trigonometry. Before that, every formula was only derived from the chart, and most of them were expressed by narration. Euler analytically deduced all the triangular formulas from the first few formulas, and got many new formulas. Euler used A, B and C to represent the three sides of a triangle, and A, B and C to represent the angle opposite to the first side, thus greatly simplifying the narrative. Euler's famous formula relates trigonometric function to exponential function. If you don't want to read the above paragraph, you don't have to read it. You all studied math in high school. Under the guidance of the teacher, Euler quickly put forward a formula to express the base of natural logarithm by the sum of the reciprocal of infinite factorial. With the formula, it is much simpler. It is said that he calculated to 23 decimal places by hand. Considering that this awesome person has an excellent memory, such a thing seems normal. The appearance of natural logarithm not only solved the catenary equation, but also had great significance to the popular astronomy at that time-western astrology. Logarithm enables complex multiplication to be transformed into simple addition, just look up the logarithm table. At the same time, logarithmic ruler came into being. Of course, with the popularity of calculators today, this kind of thing is rarely used. -parting line, version c # includes