What are base, exponent, power and power?

Exponent is a parameter in the power operation an(a≠0), where a is the base, n is the index, and the index is located in the upper right corner of the base. Power operation means the multiplication of the index base. When n is a positive integer, an represents a multiplication of n. When n=0, an= 1.

Chinese name

index

Foreign name

Index [2]

Belonging discipline

mathematics

Basic explanation

Number of power operations

write

The index is located in the upper right corner of the base.

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Defining the story development process of power operation logarithmic operation exponential function TA said

definition

Exponent is a parameter in the power operation an(a≠0), where a is the base, n is the exponent, and the exponent is located in the upper right corner of the base.

When indexing

When,

When indexing

And n is an integer,

When indexing

When,

When indexing

When, call Fang.

When indexing

When, called a cube.

Power operation

Power operation (exponential operation) is a mathematical operation about power. Same base power, constant base, exponential addition; Same base powers divides, the base remains the same, and the exponent is subtracted. Power of power, constant radix, exponential multiplication. Below a≠0.

1)

2)

3)

4)

5)

Logarithmic operation

if

, namely

about

Power equals

(

and

), and then calculate

It's called.

For the bottom,

Logarithm of, recorded as

Among them,

It is called the base of logarithm,

It's called a real number,

It's called "for the sake of

For the bottom,

Logarithm of ". Therefore, in some cases (base >；; 0, instead of 1), the exponent in exponential operation can be obtained by logarithmic operation.

exponential function

Generally speaking, it is shaped like

(

and

)(

) function is called exponential function, that is, a function with exponent as independent variable and constant base greater than 0 and not equal to 1 is called exponential function and is one of elementary functions.

The exponential function image is shown in figure 1:

Figure 1

story

Someone once asked Einstein, what is the most terrible thing in the world? Einstein said, "Compound interest is the most terrible thing."

Compound interest means that the principal is deposited in the bank at a certain interest rate, and the interest at maturity is included in the principal and continues to be deposited in the bank, and the principal and interest are increasing. If the principal is

, the annual interest rate is

,

The money that can be withdrawn from the bank after the year is

. General annual interest rate

It will not exceed 15%, and the indicator item is the number of years of deposit in the bank.

But it grew rapidly 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 settlers from Indians with $200 worth of jewelry. If you deposit $200 in the bank, the principal and interest will be higher than all the properties in Manhattan. If you deposit 1000 yuan in the bank with an annual interest rate of 5%, you can get it from the bank after 200 years with compound interest.

Yuan, that's

Yuan.

Legend has it that in ancient India, a king wanted to reward a prime minister and ask him what he wanted. The Prime Minister took out a chessboard. Laugh and say, I just ask you to give me some wheat and put one in the first box (

), put two (

), put four (

), which is the first one.

Put it in the grid

Until the wheat grains in each grid are put away. The king thought it was too simple, so he readily agreed. But when the promise really came true, the king had to break his word. Why? The chessboard has 64 squares, and the total number of grains required by the prime minister is geometric series.

The sum of is

Grain. If there are 50,000 grains in 1 kg, the total grain required is

Tons. These grains may not be enough to bring all the wheat in the country. How can the king promise?

Whether it is the horror of compound interest or the cunning of the prime minister, it is because it contains the key factor of * * *-index term.

, is an exponential term.

The wonderful function of makes seemingly simple things amazing. [ 1]

Development history

The formation of the concepts of exponent and power is quite tortuous and slow. There are many symbols of power, and the symbols are diversified.

There are at least ten different ways to write the word "right" in ancient China.

Liu Hui made a note for "Nine Chapters Arithmetic", in which he wrote: "This product is called the field power, and everything is multiplied by it (the product of the multiplication of length and width is called the power)." This is the first time that power appears in mathematical literature.

"Zhu Nanzi's Astronomical Training" talks about the law of music, which has the following words: "Therefore, the law of Huang Zhong is nine inches, which is a palace tune; So nine is nine and nine is eighty-one, so Huang Zhongzhi has the right to vote ... Twelve is thirty percent each, so one is eleven plus thirteen, which is 177,147. " The pipe that emits the rhythm of the yellow bell is 9 inches long, and its tone is called gong. Multiply by 9 to get 8 1. 8 1 This number is called Huangling. Every law of 12 is based on the principle of three-point profit and loss. Therefore, multiply 3 by 1 1 times and divide the product by 177 147. This 177 147 is called the Huangling, which is different from the Huangling of 8 1. Obviously, "putting one and eleven and one third" is a power operation, and 1 1 is an exponent. The whole sentence contains an expression.

With the original concept of index.

1607, Matteo Ricci and Xu Guangqi jointly translated Euclid's Elements of Geometry. In the translation, Xu Guangqi re-used the word power and added a note: "Multiplying a number by itself is called power." This is the first time to define the concept of power.

It was not until the seventeenth century that the index symbols with "modern" significance appeared. At first, it only indicates the number of unknowns, and there is no sign of unknowns. Bilgi wrote Roman numerals on the coefficient figures to indicate the number of unknowns. Later, Kepler and others also adopted this symbol. The Romans began to write an unknown number of letters. 163 1 year, harriot (1560- 162 1) improved David's notation and expressed it as aa.

, expressed by aaa.

. 1636, James Hume, a scot living in Paris, expressed the index in the form of small Roman numerals in the upper right corner of letters, such as

express

This notation is the same as the exponential notation except that Roman numerals are used. Descartes (1596- 1650) put the smaller Indian Arabic numerals in the upper right corner to represent the index, which is the commonly used index representation at present. [ 1]

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reference data

[ 1]? She huaiqing. A brief history of exponential and logarithmic development [J]. Journal of Chizhou University, 2006,20 (5): 9-10.

[2]? Comparison of mathematical terms between Chinese and English. Baidu library? [reference date 2021-07-11]

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