Index isexponentiation A parameter in a ⁿ (a ≠ 0), a isbase number, n is the exponent, the exponent is located in the upper right corner of the base, and the power operation means that the base numbers of the exponent are multiplied.When n is apositive integer，aⁿRepresents n a multiplications.When n=0, a ⁿ=1.
The exponent is a parameter in the power operation a ⁿ (a ≠ 0). A is the base number, n is the exponent, and the exponent is located in the upper right corner of the base number.
The exponential function image is shown in Figure 1:
Figure 1
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Someone once askedEinsteinWhat is the most terrible thing in the world?Einstein said:“Compound interestThe scariest. "
Compound interest means that the principal is deposited in the bank at a certain interest rate, and the interest is included in the principal at maturity to continue to be deposited in the bank, with the principal and interest increasing.If the principal is
, the annual interest rate is
，
The money that can be withdrawn from the bank after
。General annual interest rate
It will not exceed 15%, while the index item, namely, the number of years of deposit in the bank
But it grows very fast when
When it is large enough, the sum of principal and interest will reach an extremely large value.Manhattan, New York, was bought by early immigrants from Indians with $200 worth of jewelry. If $200 had been deposited in the bank, the principal and interest would have been higher than the value of all Manhattan properties.If you deposit 1000 yuan in the bank, the annual interest rate is 5%. If compound interest is calculated, you can get it from the bank after 200 years
Yuan, i.e
Yuan.
It is said that in ancient India, a king wanted to reward a prime minister and asked him what he wanted. The prime minister took out a chess board.Smile and say, I just ask you to give me some wheat grains and put one in the first box（
）Put two in the second box（
）Put four in the third box（
）, that is, the first
Put it in a grid
The king thought it was too simple, so he readily agreed.But when he really wanted to carry out this promise, the king had to back down. Why?There are 64 squares on the chess board. According to the requirements of the prime minister, the total number of grains of wheat needed isProportional sequence
The sum of is
Grain.If 50000 grains per kilogram, the total grains needed are
Tons.These grains may not be enough for the whole country. How could the king agree?
Whether it is the fear of compound interest or the cunning of the prime minister, it is because there is a common key factor - the index term
, is an index term
The wonderful role of the "" makes seemingly simple things surprising.[1]
development history
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The formation of the concept of index and power is quite tortuous and slow. There are many kinds of sign of power, and the notation is diversified.
There are at least ten different ways to write the word "power" in ancient China.
Liu HuibyNine Chapters of ArithmeticNote onFang TianIn the rule of finding rectangular area, the chapter wrote: "This product is called Tian power, and the product of multiplying widely is called power (the product of multiplying length and width is called power)." This is the first time that power appears in mathematical literature.
Zhunnanzi Astronomical Instruction talks about music rhythm, and there are some words: "Therefore, the law of Huang Zhong is nine inches, and the tone of the palace; therefore, nine of them, ninety-nine eighty-one, so Huang Zhong has the right to vote... 30% of each of the twelve, so it is set to one and eleven of them, and the score is seventeen thousand seven thousand fourteen seventeen, and the number of Huang Zhong is nine inches.",Its tone is called Gong.Multiply 9 by 81.This number is called the yellow bell number.Each of the 12 laws is based on the principle of sharing profits and losses.Therefore, the product obtained by multiplying 3 by 11 times is equal to 177147 in length. This 177147 is called the big number of Huangzhong, which is different from the 81 number of Huangzhong.Obviously, "set one and eleven three" is the power operation, and 11 is the exponent.The whole sentence contains the formula
, with the preliminary concept of index.
In 1607, Matteo Ricci and Xu Guangqi jointly translated Euclid's《Geometric primitives》In the translation, Xu Guangqi re used the word "power" with a note: "Self multiplying number is power." This is the first time to define the concept of power.
It was not until the 17th century that the index symbol with the meaning of "modern" appeared.Initially, it only represents the number of unknowns, but there is no sign of unknowns.Bilge wrote the Roman numerals on the coefficient numerals to express the times of unknown quantity.Later, Kepler and others adopted this symbol.Romance began to write unknown letters.In 1631, Hariot (1560-1621) improved Veda's notation, which is represented by aa
, represented by aaa
。In 1636, James Hume, a Scot living in Paris, expressed the index by placing a small Roman numeral in the upper right corner of the letter, such as
express
, except that Roman numerals are used, this representation is the same as the exponential representation.Descartes(1596-1650) Indices are represented by small Indian Arabic numerals placed in the upper right corner, which is a common index representation nowadays.[1]
