imaginary number

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In mathematics, imaginary numbers are numbers in the form of a+b × i, where a and b are real number , and b ≠ 0, i ²=- 1. The term imaginary number is a famous mathematician in the 17th century Descartes It was founded because the concept at that time believed that it was a number that did not exist. Later, it was found that the real part a of the imaginary number a+b × i could correspond to the Horizontal axis , imaginary part b can correspond to Longitudinal axis , so the imaginary number a+b × i can correspond to the point (a, b) in the plane.

The imaginary number bi can be added to the real number a to form the complex number in the form a+b × i, where the real numbers a and b are respectively referred to as the real part and imaginary part 。 Some authors use the term pure imaginary number to refer to so-called imaginary numbers, which means any number with a non-zero imaginary part complex 。^[1]

Chinese name: imaginary number
Foreign name: imaginary number
Definition: A number whose square is negative or whose root is negative
inventor: Rene Descartes

Unit: i
Mathematical Application: All imaginary numbers are complex numbers, which broadens the field of mathematics
Example: Virtual time
Discipline: mathematics 、 Physics Broad Philosophy
Composition: Real part, imaginary part

catalog

one formula
two definition
three origin
four Symbol

nine expression

formula

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trigonometric function

sin(a+bi)=sin(a)cos(bi)+sin(bi)cos(a)

=sin(a)cosh(b)+isinh(b)cos(a)

cos(a-bi)=cos(a)cos(bi)+sin(bi)sin(a)

=cos(a)cosh(b)+isinh(b)sin(a)

tan(a+bi)=sin(a+bi)/cos(a+bi)

cot(a+bi)=cos(a+bi)/sin(a+bi)

sec(a+bi)=1/cos(a+bi)

csc(a+bi)=1/sin(a+bi)

Four arithmetic operations

(a+bi)±(c+di)=(a±c)+(b±d)i

(a+bi)(c+di）=(ac-bd)+(ad+bc)i

(a+bi)/(c+di)=(ac+bd)/(c²+d²)+(bc-ad)i/(c²+d²)

r1(isina+cosa)r2(isinb+cosb)=r1r2[cos(a+b)+isin(a+b)]

r1(isina+cosa）/r2(isinb+cosb)=r1/r2[cos(a-b)+isin(a-b)]

r(isina+cosa) n =

(isinna+cosna)

Conjugate complex

z one =a+bi, z two =a-bi

-(z one +z two )=-z one +-z two

-(z one -z two )=-z one -(-z two )

-(z one z two )=-z one -z two

-(z n )=(-z) n

-z one /z two =-z one /-z two

-z two =|z|²∈ R^[5]

Power

z m · z n =z m+n

z m /z n =z m-n

(z m ) n =z mn

z one m · z two m =(z one z two ) m

(z m ) 1/n =z m/n

z · z · z… · Z (n)=z n

z one n =z two -->z1=z two 1/n

ln(a+bi)=ln(a^2+b^2)/2+i Arctan(b/a)

logai(x)=ln(x)/[ iπ/2+ lna]

x ai+b =x ai · x b =e ialn(x) · x b =x b [cos(alnx) + i sin(alnx). ]

definition

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In mathematics, the power of even exponent is negative The number of is defined as Pure imaginary number 。 All imaginary numbers are complex 。 It is defined as i ²=- 1. But there is no arithmetic root for imaginary numbers, so ± √ (- 1)=± i. For z=a+bi, it can also be expressed in the form of iA power of e, where e is constant , i is the imaginary unit, A is the argument of the imaginary number, which can be expressed as z=cosA+isinA. A logarithm consisting of real and imaginary numbers is regarded as a number within the range of complex numbers, and is called complex number. There are no positive or negative imaginary numbers. Complex numbers that are not real numbers, even pure imaginary numbers, cannot be compared in size.^[2]

origin

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To trace the track of the appearance of imaginary numbers, it is necessary to contact the emergence process of its relative real numbers. Real numbers correspond to imaginary numbers, including rational numbers and Irrational number That is to say, it is a real number.

Rational number It comes into being with people's production practice.

Real and imaginary axes

The discovery of irrational numbers should be attributed to ancient Greece Pythagorean school 。 The emergence of irrational numbers and Democritus'“ Atomism ”Conflicts occur. According to this theory, any two line segment The ratio of is just the longitude of the number of atoms they contain. and Pythagorean theorem It shows that there are incommensurable line segments.

The existence of incommensurable line segments left ancient Greek mathematicians in a dilemma, because they only had the concepts of integer and fraction in their theory, and they could not fully express the ratio of square diagonal to side length, that is, in their view, the ratio of square diagonal to side length could not be expressed by any "number". West Asia has discovered the problem of irrational numbers, but they let it slip away from them, even to the greatest algebraist in Greece Difantu There, the irrational solution of the equation is still called "impossible".

The term "imaginary number" is a famous mathematician and philosopher in the 17th century Descartes It was created because the concept at that time believed that it was a real number that did not exist. Later, it was found that the imaginary number can correspond to the vertical axis of the plane, which is as true as the real number on the horizontal axis of the corresponding plane.

It is found that even if all rational numbers and Irrational number And cannot solve the problem of solving algebraic equations. The simplest one like x ²+1=0 Quadratic equation , there is no solution in the real number range. Boshgalo, an Indian mathematician in the 12th century, believed that equation There is no solution. He thinks Positive number The square of is a positive number, negative The square of is also a positive number, so the square of a positive number square root It is dual; A positive number and a negative number, negative numbers have no square root, so negative numbers are not Square number 。 This is tantamount to denying the existence of the negative square root of the equation.

In the 16th century, Italian mathematicians Caldano In his book "Dashu" ("Mathematical Canon"), it was recorded as 1545R15-15m, which is the earliest imaginary number mark. But he thought it was just a formal expression. French mathematician in 1637 Descartes In his "Geometry", he gave the name of "imaginary number" for the first time and corresponded to "real number".

1545, Milan, Italy Caldano Published Renaissance The most important algebra book proposes a solution formula for solving the general cubic equation:

Shape: x three +The solution of the cubic equation ax+b=0 is as follows:

x={(-b/2)+[(b two )/4+(a three )/27] 1/2 } 1/3 +{(-b/2)-[(b two )/4+(a three )/27] 1/2 } 1/3

When Cardin Try to use this formula to solve equation x three -When 15x-4=0, his solution is: x=[2+(- 121) ^ (1/2)] ^ (1/3)+[2 - (- 121) ^ (1/2)] ^ (1/3)

In those days, negative numbers themselves were doubtful, and the square root of negative numbers was even more ridiculous. therefore Cardin The formula of gives x=(2+j)+(2-j)=4. It is easy to prove that x=4 is indeed the root of the original equation, but Cardin never enthusiastically explained (- 121) 1/2 The occurrence of. Think of it as "unpredictable and useless".

It was not until the beginning of the 19th century that Gauss systematically used the symbol "i" and advocated using number pairs (a, b) to represent a+bi, called complex numbers, that imaginary numbers gradually became popular.

When the imaginary number intruded into the field of number, people knew nothing about its practical use, and there seemed to be no quantity expressed in the plural in real life, so for a long time, people had all kinds of doubts and misunderstandings about it. Descartes's original meaning of "imaginary number" is that it is false; Leibniz believed that: "The imaginary number is a wonderful and strange sanctuary of the gods, which is almost an amphibian that exists and does not exist." Although Euler used the imaginary number in many places, he said: "All mathematical expressions such as √ - 1, √ - 2 are impossible. Imaginary numbers, because what they represent is negative The square root of. For this Number of classes We can only assert that they are neither nothing, nor more than nothing, nor less than nothing. They are pure illusions. "

After Euler, the Norwegian surveyor Wiesel proposed that the complex number (a+bi) be represented by points on the plane. Later, Gauss proposed Complex plane The concept of "0" has finally established a foothold for complex numbers and opened the way for the application of complex numbers. Now, complex numbers are generally used to indicate vector (directional quantity), which is widely used in water conservancy, cartography and aeronautics, and imaginary numbers increasingly show their rich content.

In 1843, William Rowan Hamilton（ William Rowan Hamilton ）The concept of imaginary number axis in the plane is extended to the four-dimensional space imagined by quaternion, in which three dimensions are similar to the imaginary number in the complex number field.

With the development of quotient ring of polynomial ring, the concept of imaginary number becomes more obvious, but other imaginary numbers can also be found, such as j of tessarines with the square of+1. This idea first appeared in James Cockle's article from 1848.

Symbol

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1777, Swiss mathematician Euler Euler began to use symbols i A unit that represents an imaginary number. Later generations organically combined imaginary numbers and real numbers, and wrote a+bi Form（ a、b Is a real number, a When it is equal to 0, it is called a pure imaginary number; when a and b are not equal to 0, it is called a complex number, b When it is equal to 0, it is a real number).

In engineering operations, in order not to be confused with other symbols (such as current symbols), sometimes letters such as j or k are used to represent the unit of imaginary numbers.

Generally, the complex number set is represented by the symbol C, and the complex number set is represented by the symbol R Set of real numbers 。

Practical significance

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imaginary number

stay Rectangular coordinate system Draw the imaginary number system in. If using Horizontal axis express All real numbers , then the vertical axis can represent an imaginary number. Each point on the whole plane corresponds to a complex number, called Complex plane 。 Horizontal axis and Longitudinal axis Also called Real axis and Imaginary axis 。 At this time, the P coordinate of a point is P (a, bi), and multiplying the coordinate by i means that the point rotates 90 degrees counterclockwise around the center of the circle.

Students or scholars who are not satisfied with the above image interpretation can refer to the following topics and instructions:

If there is a number whose reciprocal is equal to its opposite number (or the opposite of its reciprocal is itself), what form is the number?

According to this requirement, the following equation can be given:

-x = (1/x)

It is not difficult to know that the solution of this equation x=± i (imaginary unit)

Thus, if t '=t × i, i is understood as the conversion unit from the unit of t to the unit of t', then t '=t × i will be understood as

-t' = 1/t

I.e

t' = - 1/t

This expression has little significance in geometric space, but if it is understood in time with special relativity, it can explain that if the relative speed of motion can be greater than the speed of light c, the virtual value generated by the relative time interval is actually the negative reciprocal of the real value. That is, the so-called time interval back to the past can be calculated from this.

The imaginary number has become the core tool in the design of microchips and digital compression algorithms. The imaginary number is the theoretical basis of quantum mechanics that triggered the electronics revolution.

The imaginary number is an abstract concept used to express the factors that cannot form abstract concepts in things.^[3]

The value of all things can be expressed as: a+bi, not just real numbers.^[4]

Nature of i

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The higher power of i will continue to make the following cycle:

i one = i

i two = - 1

i three = - i

i four = 1

i five = i

i six = - 1

...

i n It has periodicity, and the minimum positive period is 4.

∴ i 4n =1

i 4n+1 =i

i 4n+2 =-1

i 4n+3 =-i

Because of the special operation rules of imaginary numbers, the symbol i appears

When ω=- 1/2+(√ 3)/2i or ω=- 1/2 - (√ 3)/2i:

ω two + ω + 1 = 0

ω three = 1

Related operations

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Many real number operations can be generalized to i, such as index 、 logarithm and trigonometric function 。

The ni power of a number is:

x ni = cos(ln(x n )) + i sin(ln(x n )).

The ni power root of a number is:

x 1/ni = cos(ln(x 1/n )) - i sin(ln((x 1/n )).

The logarithm based on i is:

log_i(x) = 2 ln(x)/ iπ.

The cosine of i is a real number:

cos(i) = cosh(1) = (e + 1/e)/2 = (e² + 1) /2e = 1.54308064.

I's sine Is an imaginary number:

sin(i) = sinh(1) i =[(e - 1/e)/ 2]i = 1.17520119 i.

i,e, π. The wonderful relationship between 0 and 1:

e iπ +1=0

i i =e -π/2

Related description

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Imaginary original: Lawrence Mark Leyser (Armstrong Atlantic State College)

Translator: Xu Guoqiang

The empty text has been empty since ancient times, and the word Ai can be multiplied now. Everyone asked is surprised. Where is the real power in life? Alas, it's a small test to tune and play, and I'm surprised that it's a night light. Whether the triode is used or not, the AC circuit will be stable. Asking absurdity and righteousness by the king, and finding the root of negative value increases doubts. Sentiments are used to hearing at the beginning, which is related to negative numbers. It is a bit complicated and integrated with the academic field, and hundreds of ways to join together are pleasing to have friends. But looking at the geometric triangle, vigorous wormwood means the same inheritance [①].

IMAGINARY by Lawrence Mark LesserArmstrong Atlantic State University

Imaginary numbers, multiples of iEverybody wonders, "are they used in real life?"Well, try the amplifier I'm using right now -- A.C.!You say it's absurd,this root of minus one.but the same things once were heardAbout the number negative one!Imaginary numbers are a bit complex,But in real mathematics, everything connects:Geometry, trig and call all see "i to i."

[①] see "i to i." means visible Imaginary number sign The application of homophonic pun see eye to eye has caused controversy

expression

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a=a+i

It means nothing contact Of concept No matter how a changes, i will not change.

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