Two students, Lin Ming and Zhang Jun, came to the bookstore to buy an interesting math book. They came across the words "folio 787× 1092 1/32" when looking at the publication time of the book, and did not understand its meaning. After returning to school, they went to ask the math teacher. Teacher Liu said with a smile, "This is the mathematics in paper-cutting."
"Paper-cutting is paper-cutting. What does this have to do with mathematics? " These two students are even more confused.
"Don't worry, let me speak slowly!" Teacher Liu patiently went on to say, "787 (mm) and 1092 (mm) represent the width and length of a piece of paper, and the paper that meets this specification is called the whole piece."
Teacher Liu said that he drew a rectangle on the blackboard to represent a whole piece of paper, and more and more people came to listen to the lecture in the classroom.
"Fold this piece of paper in half along the length direction and cut it open, and you will get two pieces of paper of the same size. In terms of length, width and size, we call it two pieces of paper. If you cut two pieces of paper in half along the length, you will get four pieces of paper. According to the above method, if you continue to cut in half, you can get 8 quarts, 16 quarts, 32 quarts, 64 quarts and so on. " Teacher Liu drew a series of large and small rectangles on the blackboard and marked their corresponding numbers, then said:
"The so-called open number is the size of a rectangular piece of paper. How many sheets of paper are opened means that this small rectangular paper is one third of the original whole paper. 1/32 in the book refers to the size of 1/32 of a piece of paper, that is, 32 slits. Books and periodicals have different specifications. Common magazines are mostly in 16 format, and our textbooks are mostly in 32 format. "
Teacher Liu paused and the students were whispering to each other. Lin Ming and Zhang Jun found a new problem and asked the teacher:
"If we know a piece of paper opened on 128, can you tell how many times it was cut from a piece of paper?"
The students' discussion immediately became louder, but Mr. Liu was in no hurry to answer this question. Some students took the initiative to stand up and answer: "128 times!" Caused a burst of laughter. Teacher Liu said in an inspiring way:
"We listed the specifications of paper-cutting, 2, 4, 8, 16, 32, 64, 128 ... and then expressed these values in the form of powers of 2, 22, 23, 24, 25, 26 and 27 ... according to the process and income of paper-cutting. Teacher Liu stopped to let the students think, or did Lin Ming and Zhang Junxian answer:
"The number of paper-cuts is equal to the exponent of the positive integer power of 2. 128 opened because 27= 128, so * * * was cut seven times. " Teacher Liu and his classmates agree with Lin Ming and Zhang Jun's answers.
In this way, the mathematics in paper-cutting is really interesting!
The origin of power and the formation of the concept of power are quite tortuous and slow.
In ancient China, there were at least 10 different ways to write dynamic characters, and the simplest one was "Yi". "Power" covers food with a towel as a noun and covers food with a towel as a verb. Shuo Wen Jie Zi explains: "Yi, Fu also, hangs from one."
Cover something with a square cloth, and the four corners hang down, which becomes the shape of "Xi". Extending this meaning, anything square can also be called a power. Further extension, the rectangular area or the product of two numbers (especially the result of a number squared) is also called power. This promotion started in Liu Hui.
In 263, Liu Hui annotated the Nine Chapters of Arithmetic. Under the rule of finding the rectangular area in the chapter of "Square Field", he wrote: "This is called Tian Mi". He also said that the product of length multiplied by width is called power. This is the first time that power appears in mathematical literature. In the chapter of Pythagorean, Liu Hui expressed Pythagorean theorem as: "Pythagorean power is combined with chord power." The power here refers to the square area or the result of the square area.
More than 300 years later, Li attached great importance to Nine Chapters Arithmetic, but he disagreed with Liu Hui's use of notional words in this way. In the Ming dynasty, some math books didn't use power characters at all.
1607, Matteo Ricci and Xu Guangqi jointly translated Euclid's Elements of Geometry, in which Xu Guangqi re-used the dynamic words. He said: "The number of times you raise yourself is called power." This is the first time to define the concept of power.
On the other hand, the concept of power is also influenced by foreign countries. 159 1 year, the French mathematician Veda once expressed "power" in Latin in his algebraic masterpiece Introduction to Analytical Methods, which was later translated into English. 1935, China published "Mathematical Terminology", which translated "force" into "power" and defined this term from then on.
Zhu Xiao, who just joined the work, had a brainwave when he remembered that the multiplication speed was fast. He told the manager to try it out in the company for a week. The first day's salary is 0.0 1 yuan, and the second day's salary is 0.02 yuan. After that, the amount of money per day is the square of the previous day. No salary is paid within one week, and one-time settlement will be made after one week. Unexpectedly, the manager readily agreed, so they agreed and signed the contract.
Zhu Xiao worked hard for a week. He calculated it carefully: the first day 1, the second day 2, the third day 4, the fourth day 16, the fifth day 256, and the sixth day 65536, which is 655 yuan and 36 cents. After working for six days, he got nearly 700 yuan money.
Six days later, the manager calculated in front of Zhu Xiao and gave Zhu Xiao 3 cents. How did this happen?
It turns out that Zhu Xiao thinks according to the unit, but if the unit is yuan, then the amount of money per day is the square of the previous day, that is to say, the salary on the third day is 0.0004 yuan, and the salary on the fourth day is 0.000000 16 yuan ... So, only 3 cents are actually recorded.
After thinking about it, Zhu Xiao sighed: "Hey! I only thought that the power increased rapidly, but I forgot that if the radix is a decimal between 0 and 1, the power will also drop rapidly! "
The above are some short stories before class that are suitable to be explained in the course "The Power of Power" for your reference.