Einstein was not lively when he was a child, and he couldn't speak when he was over three years old. His parents were worried that he was dumb and took him to see a doctor. Fortunately, Einstein was not dumb, but he didn't speak fluently until he was nine years old. Everything he says must be carefully thought over.
At the age of four or five, Einstein was once ill in bed, and his father gave him a compass. When he found that the compass always pointed in a fixed direction, he was very surprised and felt that there must be something hidden behind this phenomenon. He happily played with the compass for several days, pestering his father and uncle Jacob to ask a series of questions. Although he can't even say the word "magnetic" well, he stubbornly wants to know why the compass can guide the direction. This deep and lasting impression was vividly recalled by Einstein until he was 67 years old.
When Einstein was in primary school and middle school, his homework was normal. Because he is slow and doesn't like people, his teachers and classmates don't like him. The teachers who taught him Greek and Latin hated him even more. He once publicly scolded him: "Einstein, you will never be a successful person when you grow up." And I want to kick him out of school for fear that his class will affect other students.
Einstein's uncle Jacob is in charge of technical affairs in the electric appliance factory, while Einstein's father is in charge of business contact. Jacob is an engineer. He loves mathematics very much. When Einstein came to ask him questions, he always introduced his mathematics knowledge to him in very simple and popular language. Under the influence of his uncle, Einstein was enlightened by science and philosophy earlier.
My father's business is not good, but he is optimistic and kind. Inviting poor students from Munich to dinner at home every night is tantamount to helping them. One of them is a pair of Jewish brothers Max and Bernard from Lithuania. They are all medical students. They like reading books and have a wide range of interests. They were invited to eat at Einstein's house and became good friends with shy little Einstein with black hair and brown eyes.
Max can be said to be Einstein's "first teacher". He lent him some popular natural science books. When Einstein was twelve years old, Max gave him a plane geometry textbook from speke. Einstein recalled this sacred little book in his later years and said, "There are many conclusions in this book, such as that the three heights of a triangle intersect at one point. Although they are not obvious in themselves, they can be reliably proved, so any doubt seems impossible. This clarity and reliability left an indescribable impression on me. "
Einstein was also very lucky to know the main achievements and methods in the field of natural science from an excellent popular reading. Popular reading not only increased Einstein's knowledge, but also touched the curious heartstrings of young people and caused him to think deeply about the problem.
At the age of sixteen, Einstein applied for the engineering department of the Federal Institute of Technology in Zurich, Switzerland, but failed in the entrance examination. He accepted the suggestion of Professor Weber, the president of the Federal Institute of Technology and a famous physicist in the school, and completed the middle school course in Arau, Switzerland, with a view to obtaining a middle school degree.
1896 5438+00 In June, Einstein stepped into the school gate of Zurich University of Technology and studied mathematics and physics in the normal department. He is very disgusted with the indoctrination education in school, which makes people have no time and interest to think about other problems. Fortunately, compulsory education that stifles real scientific motivation is much less in Zurich Federal Institute of Technology than in other universities. Einstein made full use of the free air in the school and devoted himself to the subject he loved. At school, he widely read the works of masters of physics such as Helmholtz and Hertz, and he was most fascinated by Maxwell's electromagnetic theory. He has the ability of self-study, the habit of analyzing problems and the ability of independent thinking.
Early work
1900, Einstein graduated from Zurich University of Technology. He was refused to stay in school because he was not enthusiastic about some courses and indifferent to the teachers. He couldn't find a job and made a living as a tutor and substitute teacher. After a year and a half of unemployment, Marcel Grossman, a classmate who cares about and understands his talent, extended a helping hand to him. Grossman managed to persuade his father to introduce Einstein to the Swiss Patent Office as a technician.
Einstein thanked Grossman for his lifelong help. In his mourning letter to Grossman, he said that when he graduated from college, he was "suddenly abandoned by everyone and faced with life at a loss." He helped me. Through him and his father, I later went to Haller (then director of the Swiss Patent Office) and entered the patent office. It's kind of like saving my life. Without him, I probably won't starve to death, but my spirit will be depressed. "
1902 February 2 1 day, Einstein obtained Swiss nationality and moved to Bern, waiting for the recruitment of the patent office. 1902 On June 23rd, Einstein was officially employed as a third-class technician by the Patent Office, whose job was to examine all kinds of technological inventions applying for patent rights. 1903, he married his college classmate mileva Marie.
From 1900 to 1904, Einstein wrote a paper every year and published it in the German journal of physics. The first two articles are about the thermodynamics of liquid level and electrolysis, trying to give chemistry a mechanical basis. Later, it was found that this road was not feasible, and the mechanical basis of thermodynamics was studied instead. 190 1 puts forward some basic theories of statistical mechanics, and the three papers in 1902- 1904 all belong to this field.
1904 seriously discusses the fluctuation phenomenon predicted by statistical mechanics and finds that energy fluctuation depends on Boltzmann constant. It not only applies this result to mechanical systems and thermal phenomena, but also boldly applies it to radiation phenomena, and obtains the fluctuation formula of radiation energy, thus deducing Wien's displacement law. The study of fluctuation phenomenon made him make a major breakthrough in both radiation theory and molecular motion theory in 1905.
1905 miracle
From 65438 to 0905, Einstein created an unprecedented miracle in the history of science. This year, he wrote six papers. During the half year from March to September, he made four epoch-making contributions in three fields in his spare time besides working eight hours a day in the patent office. He published four important papers on quantum theory of light, molecular size measurement, Brownian motion theory and special relativity.
1905 In March, Einstein sent the paper he thought was correct to the editorial department of the German Journal of Physics. He said shyly to the editor, "I will be very happy if you can find room for me to publish this paper in your annual report." This "embarrassing" paper is called "a speculative view on the generation and transformation of light".
In this paper, the quantum concept put forward by Planck in 1900 is extended to the propagation of light in space, and the optical quantum hypothesis is put forward. It is considered that: for time average, light behaves as fluctuation; For instantaneous values, light appears as particles. This is the first time in history to reveal the unity of fluctuation of microscopic objects and particles, that is, wave-particle duality.
At the end of this article, he explained the photoelectric effect with the concept of optical quantum, and deduced the relationship between the maximum energy of photoelectrons and the frequency of incident light. This relationship was not confirmed by Millikan's experiment until 10 years later. 192 1 year, Einstein won the nobel prize in physics for his achievement of "the discovery of the law of photoelectric effect".
But this is really just the beginning. Albert Einstein went hand in hand in the fields of light, heat and electrophysics, and it was out of control. 1905 In April, Einstein completed a new method to determine the molecular size, and in May, he completed the movement of suspended particles in still liquid required by the theory of thermomolecular motion. These are two papers about Brownian motion. Einstein's purpose at that time was to determine the actual size of molecules by observing the irregular motion of suspended particles caused by the fluctuation of molecular motion, thus solving the problem of whether atoms exist in scientific and philosophical circles for more than half a century.
Three years later, French physicist Perrin confirmed Einstein's theoretical prediction with accurate experiments. This justifies the objective existence of atoms and molecules, which also makes ostwald, the German chemist who is most firmly opposed to the founder of atomism and energy theory, take the initiative to announce in 1908 that "the atomic hypothesis has become a scientific theory with a solid foundation".
1In June, 905, Einstein finished his long paper "On Electrodynamics of Moving Objects", which initiated a new era of physics and put forward the special theory of relativity completely. This is the result of Einstein's brewing and exploration in 10, which largely solved the crisis of classical physics at the end of 19, changed the space-time view of Newtonian mechanics, exposed the equivalence of matter and energy, and created a brand-new world of physics, which is the greatest revolution in the field of modern physics.
Special relativity can not only explain all phenomena that classical physics can explain, but also explain some physical phenomena that classical physics can't, and predict many new effects. The most important conclusion of the special theory of relativity is that the principle of conservation of mass has lost its independence and merged with the law of conservation of energy, so that mass and energy can be transformed into each other. Others include the slow scale of the clock, the constant speed of light, the zero rest mass of photons and so on. Classical mechanics has become the limit case of relativistic mechanics at low speed. So mechanics and electromagnetism are unified on the basis of kinematics.
1905 In September, Einstein wrote a short article, is the inertia of an object related to the energy it contains? ",as an inference of relativity. Mass-energy equivalence is the theoretical basis of nuclear physics and particle physics, and it also paved the way for the release and utilization of nuclear energy in the 1940s.
In this short period of six months, Einstein's breakthrough achievements in science can be said to be "unprecedented". Even if he gave up physics research, even if he only accomplished any of the above three achievements, Einstein would leave an extremely important mark on the history of physics development. Einstein dispelled the "dark clouds in the clear sky of physics" and ushered in a more brilliant new era of physics.
Exploration of general relativity
After the establishment of the special theory of relativity, Einstein was not satisfied and tried to extend the application of the principle of relativity to non-inertial systems. He found a breakthrough from Galileo's discovery that the acceleration of objects in the gravitational field is the same, and put forward the equivalence principle in 1907. This year, Minkowski, his university teacher and famous geometer, put forward the four-dimensional spatial representation of special relativity, which provided a useful mathematical tool for the further development of relativity. Unfortunately, Einstein didn't realize its value at that time.
The discovery of the principle of equivalence was regarded by Einstein as the happiest thought in his life, but his later work was very hard and he took a big detour. 19 1 1 year, he analyzed the rigid rotating disk and realized that Euclidean geometry in the gravitational field was not strictly effective. At the same time, it is found that Lorentz variation is not universal, and the equivalence principle is only effective in an infinitesimal region. At this time, Einstein already had the idea of general relativity, but he still lacked the necessary mathematical foundation to establish it.
19 12, Einstein returned to his alma mater in Zurich. With the help of his classmate Grossman, a professor of mathematics at his alma mater, he found a mathematical tool to establish general relativity in Riemannian geometry and tensor analysis. After a year of hard cooperation, they published an important paper "Outline of General Relativity and Gravity Theory" in 19 13, and put forward the gauge field gravity theory. This is the first time to combine gravity with scale, which makes Riemann geometry have real physical significance.
However, the gravitational field equation they got at that time was covariant only for linear transformation, and it was not covariant under any coordinate transformation required by the principle of general relativity. This is because Einstein was not familiar with tensor operation at that time, and mistakenly thought that as long as the conservation law was observed, the choice of coordinate system would be limited, and in order to maintain causality, the requirement of universal covariation would be abandoned.
A long and difficult exploration
After the completion of the general theory of relativity, Einstein still felt dissatisfied, so he wanted to extend the general theory of relativity to include not only the gravitational field, but also the electromagnetic field. He believes that this is the third stage of the development of relativity, namely the unified field theory.
After 1925, Einstein went all out to explore the unified field theory In the first few years, he was optimistic that victory was in sight; It turned out to be a lot more difficult. He thinks that the existing mathematical tools are not enough. Exploration of 1928 to pure mathematics. He tried various methods, but he didn't get any results with real physical significance.
During the 30 years from 1925 to 1955, except for the completeness of quantum mechanics, gravitational waves and general relativity, Einstein devoted almost all his scientific creative energy to the exploration of unified field theory.
1937, with the cooperation of two assistants, he deduced the equation of motion from the gravitational field equation of general relativity, further revealing the unity of time and space, matter and motion, which was a major development of general relativity and the last major achievement Einstein made in scientific creation activities.
He never succeeded in the unified field theory. He never gets discouraged and always starts from scratch with confidence. Because he was far away from the mainstream of physics research at that time, he went alone to overcome the problems that could not be solved at that time. So contrary to the situation in the 1920s, he was very isolated in physics in his later years. However, he is still fearless and unswervingly follows his own path. Until the day before his death, he was still in his hospital bed preparing to continue his mathematical calculation of the unified field theory.
The style of the greatest scientist
Einstein won many awards and honorary doctorates for his achievements in science. If the average person would hang these things high. But Einstein put everything on it, including the Nobel Prize certificate, in a messy box without looking. Neifield said that he sometimes felt that Einstein might not even know what the Nobel Prize meant. It is said that on the day of receiving the award, his expression was as calm as usual, and he did not show any special happiness or excitement.
Einstein lived in Switzerland as a teenager and lived as a poor student. He doesn't ask much about material life. He was satisfied with a plate of spaghetti with a little sauce. After becoming famous, he became a professor and later immigrated to the United States to escape Nazi persecution. He has the conditions to live a good material life, but he still lives a simple and simple life like a poor student. When Einstein came to work at the Institute of Advanced Science in Princeton, the authorities offered him a fairly high salary-annual salary 16000 US dollars, but he said, "Can you give me less money?" Give me three thousand dollars is enough. "
Einstein didn't pay attention to his clothes either. He wore a black leather coat, and for many years he didn't wear socks or a tie. His trousers sometimes wore neither belts nor suspenders. When he was discussing with others in front of the blackboard, he wrote on the blackboard and pulled his pants with his hands as if they were going to slide down. This situation is a bit funny, but his hair is long and unadorned. This is amazing for students of such an aristocratic institution as Princeton University. No wonder they want God to tell him to cut his hair. Einstein was a thrifty man. He writes on both sides of the calculation paper, and he cuts open many envelopes sent to him as calculation draft paper to prevent them from losing their reusable value before entering the wastebasket. Einstein often sits second and third class when he goes out, and usually only eats some simple food.
1In July, 909, Einstein was invited to attend the 350th anniversary celebration of the University of Geneva and the celebration in memory of Calvin, the founder of the University, and accepted the honorary doctorate awarded by the University of Geneva. In the celebration parade, school dignitaries and government officials all wore tuxedos, top hats, or medieval rusty golden robes and flat silk hats, while Einstein wore a suit he usually wore in the street and a straw hat. Einstein was very dissatisfied with the grand banquet held in this celebration. He said to the person sitting next to him, "If Calvin were alive, he would make a lot of firewood, because such a luxurious feast would burn us all to death." Einstein himself once said, "Comfort and happiness have never been an end for me. I call these ethical foundations the ideal of swineherd ... ". He even refused to be placed in the upper class, but lived in a different position and was angry at the special care given to him by society.
Einstein is a person who cherishes time very much. He doesn't like social activities and parties. He once sarcastically said, "This is feeding time to the zoo." He is absorbed in his studies and doesn't want to waste precious time on meaningless social conversations. He doesn't want to hear flattery and praise. He believes: "A person who benefits the whole world with great creative ideas does not need to be praised by future generations. His achievements themselves have given him higher returns. " 1929 In March, in order to avoid celebrating his 50th birthday, a few days before his birthday, he secretly went to a gardener's farmhouse in the suburbs of Berlin to live in seclusion.
As a great master of science in the physics revolution, Einstein never considered himself a superman. He realized that the road he took was an extension of the road taken by his predecessors, and the new era of science was a reasonable development based on the work of his predecessors, so he always appreciated the contributions of his predecessors with gratitude and admiration. When talking about the establishment of the theory of relativity, he said: "Relativity can really be said to be the last stroke of Maxwell and Lorenz's great thoughts, because it tries to extend field physics to all phenomena including gravity." Einstein wrote to friends who praised his achievements several times in his letter: "I am fully aware that I have no special talents: interest, single-mindedness, tenacious work and self-criticism have enabled me to reach my ideal state."
Concerned about the fate of all mankind.
Einstein loved science as well as human beings. He didn't isolate himself from society because he was immersed in scientific research. He always cared about human civilization and progress and fought bravely for it. He said, "One can only find the meaning of a short and risky life if he devotes himself to society", and he did it himself.
1914 In April, Einstein accepted the invitation of the German scientific community and moved to Berlin. In August, World War I broke out. Although he lived in the birthplace of war and was surrounded by war advocates, he firmly showed his anti-war attitude. In September, Einstein participated in launching the anti-war group "New Motherland Alliance". Although the organization was declared illegal and a large number of its members were arrested and persecuted and went underground, Einstein resolutely participated in the secret activities of the organization.
On June+10, 5438, under the manipulation and instigation of militarists, the German scientific and cultural circles published the Declaration of Civilized World, defending the aggressive war launched by Germany, arguing that Germany is above all else and the whole world should accept the "true German spirit". Ninety-three people signed the declaration, all of whom were prestigious scientists, artists and priests in Germany at that time. Even Nernst, Roentgen, ostwald and Planck signed it. When asked for Einstein's autograph, he flatly refused, but at the same time resolutely signed his name on the anti-war letter to Europeans. This move shocked the whole world.
19 17, after the victory of the Soviet socialist revolution led by Lenin, Einstein enthusiastically supported this great revolution and praised it as a great social experiment of decisive significance to the whole world, saying: "I respect Lenin because he is a man with complete self-sacrifice spirit and devoted himself wholeheartedly to realizing social justice. I don't think his method is practical, but one thing is certain: people like him are defenders and reconstructors of human conscience. "
19181kloc-0/October, German workers and soldiers, influenced and inspired by the victory of the October Revolution in Russia, launched an uprising and overthrew the German emperor William II. On the third day after stepping down, Einstein wrote two postcards to his mother, cheering "a great event happened ... what an honor to experience it personally!" "
From the 1920s to the early 1930s, Einstein was basically an absolute pacifist. However, the fact that wars of aggression and plunder are constantly happening has broken his beautiful dream. Especially after Hitler came to power in 1933, Germany became increasingly fascist, which made Einstein realize that a new barbaric war was inevitable and prompted him to change his point of view. He made it clear: "When the law and human dignity must be defended, we must fight. Since the danger of fascism came, I no longer believe that absolute negative pacifism is effective. As long as fascism rules Europe, there will be no peace. "
Because of Einstein's progressive activities and because he was a Jew, he was listed as an important persecution object by the German Nazis. Fortunately, he left Germany to give lectures in the United States at the end of 1932 and was not killed. His house in Berlin was raided and destroyed, his property was confiscated and his works were burned. The Nazis offered a reward of 20,000 marks to kill him. Faced with the danger of Nazi assassination, Einstein did not have the slightest fear, but fought more resolutely. When his close friend Laue wrote to persuade him to take a wise attitude on political issues, he ignored his personal safety and shouted, pointing out that fascism means war, and peace must be defended by armed forces, calling on the American people to rise up against fascism.
In the process of striving for the cause of human progress, Einstein always cared about the oppressed and enslaved countries and nations. He opposed the atrocities of fascist extermination of Jews and spoke loudly for the right of Jews to exist. But he also opposed narrow Jewish nationalism, hoping to see Jews "reach a fair and reasonable agreement with Arabs on the basis of peace, instead of creating a Jewish state." He opposed the racial discrimination policy of the United States, supported the liberation movement of blacks, and called for "the firm efforts made by blacks in this direction should be praised and supported by everyone."
In the 1950s, McCarthyists in the United States made waves. Senator McCarthy called himself "the first enemy of the United States". Some fanatics even rumoured that he was a member of the production of * * *, and said that his former assistant infield knew the materials of the atomic bomb from him and was going to provide them to the Soviet Union. In fact, except that he suggested to President Roosevelt in a letter written on August 2 1939 that this issue should be studied, he had no idea that the American government was secretly engaged in the manufacture of atomic bombs, and some friends of Einstein who were engaged in this work kept it a secret from him. But when he learned that Germany did not build an atomic bomb, but the United States did, he felt heavy and uneasy. He said that if he knew that Germany would not make an atomic bomb, he would not do anything to "open this Pandora's Box".
Einstein was very sad when he later learned on the radio that the United States dropped atomic bombs on Hiroshima and Nagasaki, causing many civilian casualties. Later, he wrote a letter to American citizens, in which he said: "We scientists who liberated this great power should give priority to taking responsibility for everything, and must limit that atomic energy should never be used to kill all mankind, but should be used to enhance human happiness." From 65438 to 0955, Einstein and Russell jointly published the Russell-Einstein Declaration, which opposed nuclear war and called for world peace.
1949, Einstein wrote an article "Why is socialism? Newspapers. Here, he put forward a view that seems to be correct now! "Planned economy is socialism. The planned economy itself may be accompanied by complete slavery of individuals. The establishment of socialism needs to solve some extremely difficult social and political problems. In view of the high concentration of political power and economic power, how can we prevent administrative personnel from becoming unlimited and arrogant? How to ensure a democratic balance for administrative power while protecting individual rights? "
Superstar fall
1955 On April 8th, Albert Einstein, the greatest scientist in human history, died in Princeton, USA due to ruptured aortic aneurysm.
A few days before Einstein died, he recorded a radio broadcast to Israel. He said: "The biggest problem of our time is that human beings are divided into two camps that are hostile to each other: the production world and the so-called free world. Because the meanings of "freedom" and "property" are difficult for me to understand, I prefer to use the power conflict between "East" and "West". But the earth is round, so the true and accurate meanings of "East" and "West" are not clear. "
Einstein should not be vain before his death, let alone mourn after his death. He left a will asking for no obituary and no funeral. He gave his brain to medical research, cremated his body, and secretly scattered his ashes in a river that no one knew. He doesn't want a grave or a monument. When his body was sent to the crematorium for cremation, only his closest 12 people accompanied him, and no one else knew the time and place of cremation. Before Einstein died, he left his house on Merseyside Street 1 12 in Princeton to Miss Ducas, his secretary who had worked with him for decades, and stressed: "Don't turn this house into a museum." He doesn't want to turn Merseyside Street into a pilgrimage site. He has never worshipped idols in his life, and he doesn't want future people to regard him as an idol.
Einstein once said, "I am only a small part of nature". He dedicated everything to the journey of human freedom from nature, and finally even his ashes returned to the embrace of nature. But just as Infield felt when he first came into contact with him: "True greatness and true nobility always walk side by side", Einstein's great achievements and spirit have been left to mankind.
anecdote
Einstein's truancy
1895 spring, Einstein 16 years old. According to the laws of Germany at that time, boys could leave Germany before 17 years old and did not have to come back for military service. Because he hated militarism and couldn't bear to stay in the barracks alone-just like Louis Polder Middle School, Einstein decided to leave Germany and go to Italy to reunite with his parents without consulting his parents.
However, what if I drop out of school and can't get my diploma later? Einstein, who has always been honest and simple, came up with a self-righteous idea in a hurry. He asked the math teacher to give him a certificate to prove that his math scores were excellent and he reached the university level early. I got a sick note from a familiar doctor, saying that it was neurasthenia and I needed to go home and rest. Einstein thought that with these two proofs, he could escape from this disgusting place.
Who knows, before he applied, the dean called him and ordered him to drop out of school on the grounds that he corrupted the class spirit and disobeyed the school discipline.
Einstein blushed. For whatever reason, he was willing to leave this middle school and didn't care about anything. He just suddenly felt guilty that he had come up with a cunning idea but it didn't come true. Einstein felt guilty every time he mentioned it later. Perhaps this kind of thing is far from his frank and sincere personality.
Mr Weber's eyes
At the age of sixteen, Einstein applied for the engineering department of the Federal Institute of Technology in Zurich, Switzerland, but failed in the entrance examination. Mr. Weber, a physicist, read his papers on mathematics and physics. He was very discerning and praised him: "You are a very clever boy, Einstein, a very clever boy, but you have a big shortcoming: you don't want to show yourself."
Mr Webber is right. Einstein is a "genius" in mathematics. He taught himself analytic geometry and calculus at the age of 12 to 16. He is also "unrepentant" for not wanting to show his "shortcomings". In his letter to friends in his later years, he said: "When I was young, my needs and expectations for life were that I could do research quietly in a corner, and the public would not pay full attention to me, but now I can't."
Charlie Chaplin, a famous film actor, invited Einstein and his wife to the movies on the day of the premiere of his film City Lights in Hollywood. When Einstein and Chaplin got off the bus, many people found Einstein coming to the theater. Everyone gathered around and cheered, focusing on him instead of Chaplin. Einstein didn't like this scene and asked Chaplin, "What does this mean?" Chaplin immediately comforted him: "Nothing."
This is how the theory of relativity was discovered.
Mrs Einstein once told Charlie Chaplin about Einstein's work when he discovered the theory of relativity. Later, Chaplin recorded this in his autobiography. This story shows how Einstein worked at this historic moment of discovery:
"As usual, the doctor went downstairs to have breakfast in his robe, but nothing moved that day. I think something must have happened. I asked him what made him's spaced out.
He replied, "dear! I have a clever idea. "
After drinking coffee, he went to the piano and began to play. Stop several times to record something on paper, and then repeatedly say, "I have a clever idea, a wonderful idea."
He said, "It's hard. I still need to work."
He continued to play the piano and wrote for half an hour. Then he went upstairs to his research room and told me not to disturb him. He stayed in the house for two weeks. Every day I go upstairs to feed him. In the evening, he took a walk as exercise, and then came back to continue his work.
At last he came downstairs, pale. "This is my discovery!" He put two pieces of paper on the table, which is his theory of relativity. "
Secret of success
Once, an American journalist asked Einstein what was the secret of his success. He replied: "As early as 190 1, when I was a 22-year-old young man, I had discovered the formula of success. I can tell you the secret of this formula, that is, A=X+Y+Z! A is success, X is hard work, Y is knowing how to rest, and Z is talking less nonsense! This formula is useful to me, and I think it is also useful to many people. "