(2) [imaginary part]: When a+bi and b in the complex number are not equal to zero, bi is called imaginary.

(3) [imaginary number]: Chinese words that do not represent specific numbers. [Edit this paragraph] Imaginary numbers in mathematics In mathematics, numbers with negative squares are defined as pure imaginary numbers. All imaginary numbers are complex numbers. Defined as I 2 =-1. But the imaginary number has no arithmetic root, 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 imaginary unit, and A is imaginary amplitude, which can be expressed as z=cosA+isinA. A pair of numbers consisting of real numbers and imaginary numbers is regarded as a number within the range of complex numbers, so it is called a complex number. Imaginary number is neither positive nor negative. Complex numbers that are not real numbers, even pure imaginary numbers, cannot compare sizes.

This number has a special symbol "I" (imaginary number), which is called imaginary unit. However, in electronics and other industries, because I is usually used to represent the current, the imaginary unit is represented by J. We can draw the imaginary number system in the plane rectangular coordinate system. If the horizontal axis represents all real numbers, then the vertical axis can represent imaginary numbers. Every point on the whole plane corresponds to a complex number, which is called a complex plane. The horizontal and vertical axes have also been renamed as real and imaginary axes. [Edit this paragraph] The origin of the word "imaginary number" was invented by Descartes, a famous mathematician and philosopher in the17th century, because the concept at that time thought it was a real number that did not exist. Later, it was found that the imaginary number can correspond to the vertical axis on the plane, which is as real as the real number corresponding to the horizontal axis on the plane.

It is found that even if all rational numbers and irrational numbers are used, the problem of solving algebraic equations cannot be solved in length. The simplest quadratic equation like x 2+ 1=0 has no solution in the real number range. 12 century Indian mathematician Bashgaro thinks this equation has no solution. He thinks that the square of a positive number is a positive number and the square of a negative number is also a positive number. Therefore, the square root of a positive number is double; A positive number and a negative number, negative numbers have no square root, so negative numbers are not squares. This is tantamount to denying the existence of negative roots of the equation.

In16th century, the Italian mathematician Cardin recorded it as1545r15-15m in his book Dafa (Da Yan Shu), which is the earliest imaginative symbol. But he thinks this is just a formal expression. 1637, the French mathematician Descartes gave the name of "imaginary number" for the first time in Geometry, corresponding to "real number".

Cardin of Milan, Italy, published the most important algebraic works in the Renaissance in 1545, and put forward the formula for solving the general cubic equation:

The cubic equation in the form of x 3+ax+b = 0 can be solved as follows: x = {(-b/2)+[(b 2)/4+(a 3)/27] (1/2)} (1/3).

When Kadan tried to solve the equation x 3- 15x-4 = 0 with this formula, his solution was: x = [2+(-121)] (1/3)+.

At that time, the negative number itself was questionable, and the square root of the negative number was even more absurd. So Cardin's formula gives x=(2+j)+(2-j)=4. It is easy to prove that x=4 is indeed the root of the original equation, but Kadan did not enthusiastically explain the appearance of (-12 1) (1/2). Think of it as "unpredictable and useless."

It was not until the beginning of19th century that Gauss systematically used this symbol, and advocated using a number pair (a, b) to represent a+bi, which was called a complex number, and the imaginary number gradually became popular.

Because imaginary number has entered the field of numbers, people know nothing about its practical use, and there seems to be no quantity expressed by complex numbers in real life, so people have all kinds of doubts and misunderstandings about it for a long time. Descartes called it "imaginary number" because it is false; Leibniz thinks: "imaginary number is a wonderful and strange hiding place for gods." It is almost an amphibian that exists and does not exist. " Although Euler used imaginary numbers in many places, he also said that everything is similar.

After Euler, the Norwegian surveyor Wiesel proposed that the complex number (a+bi) should be represented by points on the plane. Later, Gauss put forward the concept of complex plane, which finally made complex numbers have a foothold and opened the way for the application of complex numbers. At present, vector (vector) is generally represented by complex numbers, which are widely used in water conservancy, cartography, aviation and other fields, and imaginary numbers are increasingly showing their rich contents. [Edit this paragraph] My attribute, my high power, will continue to do the following cycle:

I1= me

i^2 = - 1

I^3 =-me

i^4 = 1

I^5 = me

i^6 = - 1 ...

Because of the special operation rules of imaginary numbers, the symbol I appeared.

When ω=(- 1+√3i)/2 or ω=(- 1-√3i)/2:

ω^2 + ω + 1 = 0

ω^3 = 1

Many real number operations can be extended to I, such as exponent, logarithm and trigonometric function.

The ni power of a number is:

X (ni) = cos (ln (x n)+my 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)/ i*pi。

The cosine of I is a real number:

cos(I)= cosh( 1)=(e+ 1/e)/2 =(e^2+ 1)/2e = 1.54308064。

The sine of I is an imaginary number:

sin(I)= sinh( 1)* I =(e- 1/e)/2 } * I = 1. 17520 1 19i。

The wonderful relationship between I, E, π, 0 and 1;

e^(i*π)+ 1=0

Ii I = e (-π÷ 2) [Edit this paragraph] Symbol origin 1777 Swiss mathematician Euler (or translated as Euler) began to use the symbol I to represent imaginary units. Then people organically combine imaginary numbers with real numbers and write them in the form of a+bi (A and B are both real numbers, when A equals 0, they are pure imaginary numbers, when ab is not equal to 0, they are complex numbers, and when B equals 0, they are real numbers).

Usually, we use the symbol C to represent the complex set and the symbol R to represent the real set. [Edit this paragraph] Original description of imaginary number: Lawrence Mark Lesser (Armstrong Atlantic State College)

Translation: Xu Guoqiang

Since ancient times, the word ai can be used. Everyone was surprised when asked. Where can there be real energy in life? Oh, I tried to adjust it. I was shocked and patted the night light. With or without transistors, AC circuits are willing to be salty. If you ask ridiculous questions, negative values will increase your doubts. Emotions are used to listening at first, which is related to negative numbers. It's a bit complicated to integrate into the academic field. But looking at the geometric triangle, the lush wormwood also means this [1].

Lawrence Marc Lesseronstrom Atlantic State University

Imaginary numbers, multiples of series, everyone wants to know, "Are they used in real life?" Ok, try the amplifier I am using now-AC! You say this is ridiculous, the root of this-1 But the same thing has been heard about the number-1! Imaginary number is a bit complicated, but in real mathematics, everything is linked together: geometry, triangle and call are all seen in "i to i"

[①] See "I to me." refers to the application of visible imaginary symbols, and homonym pun see eye to eye means consistency [1]

References:

Network Journal of Humanities Mathematics, 22 issues, 48 pages.

Open classification: words, mathematics, vocabulary, numbers, plural.