Imaginary number can refer to the following meanings: (1) [unreliable number]: imaginary number. (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 the imaginary number in this math.
In mathematics, a number with a negative square is defined as a pure imaginary number. 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. But in electronics and other industries, because I is usually used to represent current, the imaginary unit is represented by J.
The practical significance of editing imaginary numbers in this paragraph
We can draw an imaginary system in a 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 are also called real imaginary numbers.
Axis and imaginary axis. Students or scholars who are dissatisfied with the above image explanation can refer to the following topics and explanations: If there is a number, its reciprocal is equal to its reciprocal (or its reciprocal is itself), what is the form of this 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 is x=i (imaginary unit). So if there is an algebraic formula t'=ti, we will understand I as the conversion unit from the unit of t to the unit of t'. Then t'=ti will be understood as -t' = 1/t, that is to say, the expression t' =- 1/t has little significance in geometric space, but if it is understood in time by special relativity, it can be explained that if the relative motion speed can be greater than the speed of light c, then the virtual value generated by the relative time interval is essentially the negative reciprocal of the real value. That is, the so-called time interval value of going back to the past can be calculated from this.
Edit the source of this paragraph
To trace the trajectory of imaginary number, it is necessary to contact the emergence process of real number relative to it. We know that real number corresponds to imaginary number, which includes rational number and irrational number, that is, it is real number. Rational numbers appeared very early, accompanied by people's production practice. The discovery of irrational numbers should be attributed to the Pythagorean school in ancient Greece. The appearance of irrational numbers contradicts democritus's "atomism". According to this theory, the ratio of any two line segments is the number of atoms they contain. However, Pythagorean theorem shows that there are incommensurable line segments. The existence of incommensurable line segments made the mathematicians in ancient Greece feel in a dilemma, because their theory only had the concepts of integer and fraction, and could not fully express the ratio of diagonal to side length of a square, that is to say, in their place, the ratio of diagonal to company commander of a square could not be expressed by any "number". They have discovered the problem of irrational numbers, but let it slip away from them. Even for Diophantine, the greatest algebraic scientist in Greece, the irrational solution of the equation is still called "impossible". The word "imaginary number" was invented by Descartes, a famous mathematician and philosopher in17th century, because the concept at that time thought it was a non-existent real number. 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 of the Renaissance from 65438 to 0545. The formula for solving the general cubic equation is put forward: the cubic equation in the form of x 3+ax+b = 0 is as follows: x = {(-b/2)+[(b 2)/4+(a 3)/27] (1/2)}. When Kadan tried to solve the equation x 3-15x-4 = 0 with this formula, his solution was: x = [2+(-121)] (1/3. 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 said, "All mathematical expressions in the form of √- 1 and √-2 are impossible, imaginary numbers, because they represent the square root of negative numbers. For such figures, we can only assert that they are neither nothing nor more than nothing, nor less than nothing. They are purely illusory. " 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 the properties of this paragraph.
The high power of I will continue to do the following cycle: I1= II 2 =-1I 3 =-II 4 =1I 5 = II 6 =-1... Because of the special operation rules of imaginary numbers, when ω = (-1+,
Edit the operation related to I in this paragraph.
Many real number operations can be extended to I, such as exponent, logarithm and trigonometric function. A number to the power of ni is: x (ni) = cos (ln (x n))+isin (ln (x n)). The Ni power root of a number is: x (1/ni) = cos (ln (x (1/n). The logarithm of I is: log _ I (x) = 2ln (x)/i π. The cosine of I is a real number: cos (I) = cosh (1) = (e+1/e)/2 = (e 2+65438+. The sine of 2e = 1.54308064. I is an imaginary number: sin (I) = sinh (1) I = [(e-1/e)/2] I =1.1752066.
Edit the origin of this paragraph symbol.
1777, the 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 the relevant description of this paragraph.
Original imaginary number: Lawrence Mark Lesser (Armstrong Atlantic State College) Translation: Xu Guoqiang's imaginary number text has been constructed since ancient times, and the word Ai can now be multiplied. 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]. The imaginary numbers invented by Lawrence Mark Lesseland Atlantic State University are multiples of series, and everyone wants to know, "Are they useful 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, all the connections: geometry, trigonometry and call are seen as "I-to-me" [①] See "I-to-me" refers to the application of visible imaginary symbol, and the pun about see eye to eye is a consistent quotation.