An imaginary number is a number whose square is negative. The term imaginary numbers was coined by Descartes, a famous mathematician in the 17th century, because the concept at that time was that they were real numbers that did not exist. Later it was discovered that imaginary numbers can correspond to the vertical axis on the plane, which is equally true as the real numbers corresponding to the horizontal axis on the plane.
Table of contents
Brief introduction
Formula trigonometric functions
Four arithmetic operations
***Yoke complex numbers
p>
Power
The actual meaning of imaginary numbers in mathematics
Origin
Properties of i
Related operations
Origin of symbols
Related description
Brief introduction
Formula trigonometric functions
Four arithmetic operations
***Yoke complex numbers
Power
The actual meaning of imaginary numbers in mathematics
Origin
Properties of i
< p>Related operationsOrigin of symbols
Related description
Expand and edit this paragraph for a brief introduction
Real axis and imaginary axis
Imaginary numbers can refer to the following meanings: (1)[unreliable figure]: false and unreal numbers. (2)[imaginary part]: In the complex number a+bi, b is called the imaginary part, and a is called the real part. (3) [imaginary number]: A word in Chinese that does not indicate a specific number. If the square of a number is negative, that number is an imaginary number; all imaginary numbers are complex numbers. The term "imaginary numbers" was coined by Descartes, a famous mathematician in the 17th century, because the concept at that time was that they were real numbers that did not exist. Later it was discovered that imaginary numbers can correspond to the vertical axis on the plane, which is equally real as the real numbers corresponding to the horizontal axis on the plane. The plane formed by the imaginary number axis and the real number axis is called the complex number plane, and each point on the complex plane corresponds to a complex number.
Edit this formula
Trigonometric functions
sin(a+bi)=sinacosbi+sinbicosa =sinachb+ishbcosa cos(a-bi)=coscosbi+ sinbisina =cosachb+ishbsina 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)/(c^2+d^2)+(bc-ad)/(c^2+d^2)i 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=r^n (isinna+cosna)
***Yoke plural
_(a+bi)=a-bi _(z1+z2)=_z1+_z2 _(z1-z2) =_z1-_z2 _(z1z2)=_z1_z2 _(z^n)=(_z)^n _z1/z2=_z1/_z2 _z*z=|z|^2∈R
Power< /p>
z^mz^n=z^(m+n) z^m/z^n=z^(m-n) (z^m)^n=z^mn z1^mz2^m=( z1z2)^m (z^m)^1/n=z^m/n z*z*z*…*z (n pieces)=z^n z1^n=z2-->z2=z1^1/ n logai(x)=1/ iπ/2 ln(x)+logx(e) a^(ai+b)=a^ai*a^b = a^b[cosln(x^n) + i sinln( x^n). ]
Edit this paragraph Imaginary numbers in mathematics
In mathematics, numbers whose squares are negative are defined as pure imaginary numbers. Definition. is i^2=-1. However, there is no arithmetic root for imaginary numbers, so ±√(-1)=±i. For z=a+bi, it can also be expressed in the form of e raised to the power of iA, where e is a constant, i is the unit of imaginary numbers, and A is the argument of the imaginary number, which can be expressed as z=cosA+isinA. A pair of real numbers and imaginary numbers is regarded as a number in the range of complex numbers and is named a complex number. Imaginary numbers are neither positive nor negative. Complex numbers that are not real numbers, even purely imaginary numbers, cannot be compared. This kind of number has a special symbol "i" (imaginary), which is called the imaginary unit. However, in industries such as electronics, because i is usually used to represent current, the imaginary unit is represented by j.
Practical significance
We can draw the imaginary number system in a plane rectangular coordinate system. If the horizontal axis represents all real numbers, then the vertical axis can represent imaginary numbers. Each point on the entire plane corresponds to a complex number, which is called the complex plane. The horizontal and vertical axes are also called the real-imaginary axis and the imaginary axis.
Students or scholars who are not satisfied with the above image explanation can refer to the following questions and explanations: If there is a number whose reciprocal is equal to its opposite (or the opposite of 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). Therefore, if there is an algebraic formula t'=ti, we understand i as From the unit of t to the unit of t', t'=ti will be understood as -t' = 1/t, that is, t' = - 1/t. This expression has no meaning in geometric space. Large, but if combined with the special theory of relativity and understood in terms of time, it can be explained that if the relative motion speed can be greater than the speed of light c, the imaginary value generated by the relative time interval is essentially the negative reciprocal of the real value. In other words, the so-called time interval value for returning to the past can be calculated from this. Imaginary numbers have become a core tool in the design of microchips and digital compression algorithms. Imaginary numbers are the theoretical basis of quantum mechanics that triggered the electronics revolution.
Origin
To trace the trajectory of the emergence of imaginary numbers, we must contact the emergence process of real numbers relative to them. We know that real numbers correspond to imaginary numbers, which include rational numbers and irrational numbers, which means that they are real numbers. Rational numbers appeared very early, and they came into being along with people's production practice. The discovery of irrational numbers should be attributed to the Pythagoreans of ancient Greece. The emergence of irrational numbers conflicts with Democritus's "atomic theory". According to this theory, the ratio of any two line segments is simply the number of atoms they contain. The Pythagorean theorem shows that there are incommensurable line segments. The existence of incommensurable line segments put ancient Greek mathematicians in a dilemma, because their theories only had the concepts of integers and fractions, and they could not fully express the ratio of the diagonal to the side length of a square. That is to say, in their , the ratio of the diagonal of a square to its continuous length cannot be represented by any "number". They had already discovered the problem of irrational numbers in West Asia, but let it slip away quietly from their side. Even to Diophantus, the greatest algebraist in Greece, the irrational number solution to the equation was still called "impossible" of". The term "imaginary numbers" was coined by Descartes, a famous mathematician and philosopher in the 17th century, because the concept at that time believed that these were real numbers that did not exist. Later it was discovered that imaginary numbers can correspond to the vertical axis on the plane, which is equally real as the real numbers corresponding to the horizontal axis on the plane. People found that even if all rational 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 range of real numbers. The great Indian mathematician Vaskara in the 12th century believed that this equation had no solution. He believed 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 twofold; a positive number and a negative number. Negative numbers have no square roots, so negative numbers are not square numbers. This amounts to denying the existence of negative square roots of equations. In the 16th century, Italian mathematician Cardano recorded 1545R15-15m as the earliest imaginary number symbol in his book "Dashu" ("Mathematical Canon"). But he thinks this is just a formal expression. In 1637, the French mathematician Descartes gave the name "imaginary numbers" for the first time in his "Geometry" and corresponded to "real numbers". In 1545, Cardano of Milan, Italy, published one of the most important algebraic works of the Renaissance, proposing a formula for solving general cubic equations: Solution to a cubic equation of the form: x^3+ax+b=0 As follows: x={(-b/2)+[(b^2)/4+(a^3)/27]^(1/2)}^(1/3)+{(-b/2) -[(b^2)/4+(a^3)/27]^(1/2)}^(1/3)When Cardin tried to use this formula to solve the equation x^3-15x-4=0 His solution is: x=[2+(-121)^(1/2)]^(1/3)+[2-(-121)^(1/2)]^(1/3) In that Negative numbers themselves are questionable, and the square roots of negative numbers are even more absurd. Therefore 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 Cardin was not enthusiastic about explaining the occurrence of (-121)^(1/2). Considered "elusive and useless."
It was not until the beginning of the 19th century that Gauss systematically used the symbol i and advocated using even numbers (a, b) to represent a + bi, called complex numbers, and imaginary numbers gradually became popular. Because when imaginary numbers broke into the field of numbers, people knew nothing about their actual use. In real life, there seemed to be no quantity that could be expressed by complex numbers. Therefore, for a long time, people had various doubts and misunderstandings about it. . Descartes' original intention of calling "imaginary numbers" was to mean that they were false; Leibniz believed: "Imaginary numbers are wonderful and strange shelters of the gods. They are almost amphibious creatures that both exist and do not exist." Although Euler was Imaginary numbers are used in many places, but he also said: "All mathematical formulas in the shape of √-1, √-2 are impossible, imaginary numbers, because they represent the square root of a negative number. For this type of We can only conclude that they are neither nothing nor more than nothing, nor less than nothing. They are purely illusory."