About the origin
16th century Italian Milan scholar Jerome Cardan (1501-1576) published the general solution of cubic equation in his book "Important Art" in 1545, which was later called "Cardan Formula". He was the first mathematician to write the square root of a negative number as a formula. When discussing whether it is possible to divide 10 into two parts to make their product equal to 40, he wrote the answer as =40. Although he thought the expressions sum were meaningless, fictitious and illusory, he divided 10 into two parts and made their product equal to 40. The French mathematician Descartes (1596-1650) gave the name "imaginary number", and he made it correspond to "real number" in geometry (published in 1637). Since then, imaginary numbers have spread.
A new star, imaginary number, was found in the number system, which caused a chaos in mathematics. Many great mathematicians do not admit imaginary numbers. German mathematician Leibniz (1646-16) said in 1702: "imaginary number is a subtle and strange hiding place for gods, and it is probably an amphibian in the field of existence and falsehood." Swiss mathematician Euler (1707- 1783) said; "In all forms, it is impossible to learn mathematics. Imagine numbers, because they represent the square root of a negative number. For such figures, we can only assert that they are neither nothing nor more than nothing, nor less than nothing. They are purely illusory. " However, truth can stand the test of time and space and finally occupies its own place. The French mathematician D'Alembert (1717-1783) pointed out in 1747 that if the imaginary number is operated according to the four algorithms of polynomials, its result will always be in the form of (A and B are both real numbers) (Note: the symbol =-is not used in the current textbooks. The French mathematician Demofer (1667- 1754) discovered this formula in 1730, which is the famous Demofer theorem. Euler found the famous relation in 1748. In his article Differential Formula (1777), he first expressed the square root of 1 with I, and he pioneered the use of the symbol I as the unit of imaginary number. "Imaginary number" is not an imaginary number, but it does exist. 1745- 18 18, a Norwegian surveyor tried to give an intuitive geometric explanation of the imaginary number 1779, and published his practice for the first time, but it did not get the attention of academic circles.
The German mathematician Forrest Gump (1777- 1855) published the graphic representation of imaginary numbers in 1806, that is, all real numbers can be represented by a number axis, and similarly, imaginary numbers can also be represented by points on a plane. In the rectangular coordinate system, take the point A corresponding to the real number A on the horizontal axis and the point B corresponding to the real number B on the vertical axis, and draw a straight line parallel to the coordinate axis through these two points, and their intersection point C represents the complex number A+Bi. In this way, the plane whose points correspond to complex numbers is called "complex plane", and later it is also called "Forrest plane". 183 1 year, Gauss expressed the complex number A+Bi with real array (a, b), and established some operations of complex numbers, making some operations of complex numbers "algebraic" like real numbers. He first put forward the term "complex number" in 1832, and also integrated two different representations of the same point on the plane-rectangular coordinate method and polar coordinate method. Unifying the algebraic form and triangular form representing the same complex number, the points on the number axis correspond to the real number-1, and the points extended to the plane correspond to the complex number-1. Gauss regarded the complex number not only as a point on the plane, but also as a vector, and expounded the geometric addition and multiplication of the complex number by using the corresponding relationship between the complex number and the vector. At this point, the complex number theory has been established completely and systematically.
After many mathematicians' unremitting efforts for a long time, the complex number theory has been deeply discussed and developed, which makes the ghost of imaginary number, which has been wandering in the field of mathematics for 200 years, unveil its mysterious veil and reveal its true colors. Original imaginary number is not empty. The imaginary number has become a member of the family of number systems, so the real number set has been extended to the complex number set.
With the progress of science and technology, complex number theory becomes more and more important. It is not only of great significance to the development of mathematics itself, but also plays an important role in proving the basic theorem of wing lift, showing its power in solving the seepage problem of dams and providing an important theoretical basis for the construction of giant hydropower stations.
Related definition
Complex number concept
The number set is extended to the real number range, but some operations are still impossible. For example, the univariate quadratic equation with discriminant less than 0 still has no solution, so the number set is expanded again to reach the complex range.
A number in the form of z=a+bi is called a complex number, where I is an imaginary unit and I 2 = I× I =-1(A and B are arbitrary real numbers).
We call the real number A in the complex number z=a+bi the real part of the complex number Z, and write it as REZ = A..
The real number b is called the imaginary part of the complex number z, and is recorded as imz = B.
It is known that when b=0 and z=a, the complex number becomes a real number; If and only if a=b=0, it is a real number 0;
When a=0 and b≠0 and z=bi, we call it pure imaginary number.
Modulus of complex number
The value of the positive square root of the sum of squares of the real part and imaginary part of a complex number is called the module of a complex number, and is expressed as ∣z∣.
That is to say, for the complex number z=a+bi, its modulus
∣z∣=√(a^2+b^2)
The set of complex numbers is represented by C, and the set of real numbers is represented by R. Obviously, R is the proper subset of C.
The complex set is out of order, and the size order cannot be established.
Complex number of * * * yoke
For the complex number z=a+bi, the complex number z'=a-bi is called the * * * yoke complex number of z, that is, the two real parts are equal and the imaginary part (imaginary part is not equal to 0) is a conjugate complex number. The * * * yoke of the complex number z is denoted as zˊ. The representation method is to add a horizontal line above the letter Z, that is, the * * * yoke symbol.
According to the definition, if z=a+bi(a, b∈R), then z = a-bi (a, b ∈ r). * * * The point corresponding to the complex number of the yoke is symmetrical about the real axis. Two complex numbers: x+yi and x-yi are called * * * yoke complex numbers. Their real parts are equal, but their imaginary parts are opposite. On the complex plane. The point representing the complex number of two yokes is symmetrical about X, which is the origin of the word "* * * yoke". Two cows pull a plow in parallel with a beam on their shoulders. This beam is called a "yoke". If X+Yi is represented by z, then adding a "1" to z is X-Yi, and vice versa.
* * * yoke complex numbers have some interesting properties:
︱x+yi︱=︱x-yi︱
(x+yi)*(x-yi)=x^2+y^2
Independent variable of complex number
In a complex function, the independent variable z can be written as z= r*(cosθ+i sinθ). R is the modulus of z, that is, r = | z |θ is the radial angle of z, and the radian angle between 0 and 2π is called the principal value of radian angle, and it is recorded as arg(z).
About operation
Addition rule
The addition rule of complex numbers: let z 1 = a+bi and z2 = c+di be any two complex numbers. The real part of sum is the sum of the original two complex real parts, and its imaginary part is the sum of the original two imaginary parts. The sum of two complex numbers or a complex number.
That is, (a+bi) (c+di) = (a c)+(b d) i.
Multiplication rule
Complex multiplication rule: two complex numbers are multiplied, similar to two polynomials. In the result, I 2 =- 1, and the real part and imaginary part are merged respectively. The product of two complex numbers is still a complex number.
That is, (a+bi) (c+di) = (AC-BD)+(BC+ad) I.
Division rule
Definition of complex division: The complex number x+yi(x, y∈R) satisfying (c+di)(x+yi)=(a+bi) is the quotient of the complex number A+bi divided by the complex number C+di.
Operation method: the numerator and denominator are multiplied by the * * * yoke complex number of the denominator at the same time, and then operated according to the multiplication rule.
That is, (a+bi)/(c+di)
=[(a+bi)(c-di)]/[(c+di)(c-di)]
=[(ac+bd)+(bc-ad)i]/(c^2+d^2).
Prescription rule
If z n = r(cosθ+isθ), then
z = n√r[cos(2kπ+θ)/n+isin(2kπ+θ)/n](k = 0， 1，2，3……n- 1)
Operating rule
Additive commutative law: z 1+z2=z2+z 1.
Multiplicative commutative law: z 1*z2=z2*z 1.
Additive associative law: (z1+z2)+z3 = z1+(z2+z3)
Multiplicative association law: (z1* z2) * z3 = z1* (z2 * z3)
Distribution law: z1* (z2+z3) = z1* z2+z1* z3.
Multiplication of I
I (4n+ 1) = I, I (4n+2) =- 1, I (4n+3) =-I, I 4n = 1 (where n∈Z).
Dimofor theorem
For the complex number z = r(cosθ+isθ), there exists the n power of z.
Z n = (r n) * [cos (nθ)+isin (nθ)] (where n is a positive integer)
Complex triangle
Let the triangular forms of complex numbers z 1 and z2 be r1(cos θ1+is θ1) and R2(cosθ2+isθ2), then z1z 2 = r1r 2 [cos
Z1÷ z2 = (r1÷ R2) [cos (θ1-θ 2)+isin (θ1-θ 2)] (in the complex plane, it is modular division and angular reduction. )
Complex sets are different from real sets in several characteristics: root operation is always feasible (excluding pure imaginary sets)
The unary equation with n complex coefficients always has n roots (multiple roots count as multiples); Complex numbers cannot establish size order.
Complex number and geometry
Introduction to complex plane
The points on the horizontal axis of the complex plane correspond to all real numbers, so it is called the real axis, and the points on the vertical axis (except the origin) correspond to all pure imaginary numbers, so it is called the imaginary axis. On the complex plane, the complex number also corresponds to the plane vector from the origin to the point z=x+iy, so the complex number z can also be represented by the vector z (as shown in the right picture). The length of the vector is called the modulus or absolute value of z, and it is denoted as | z | = r = √ (x 2+y 2).
In addition to the works of Wiesel (1745- 18 17) and Argonne (1768- 1822), there are also Coates (1707-/kloc). Vandermonde (1735- 1796) also realized that points on the plane can correspond to complex numbers one by one, which was confirmed by the fact that they regarded the roots of binomial equations as vertices of regular polygons. However, Gauss's contribution is very important in this respect. His famous basic theorem of algebra is derived on the premise that points on the coordinate plane can correspond to complex numbers one by one. 183 1 year, Gauss explained in detail in Journal of Gottingen that the complex number a+bi is expressed as a point (A, B) on a plane, thus clarifying the concept of the complex plane, and he integrated the rectangular coordinates representing the plane points with the polar coordinates. Unified in two representations of the same complex number-algebraic form and triangular form. Gauss also gave the name "complex number". Because of Gauss's outstanding contribution, later generations often call it the complex plane. Complex plane characteristics: the plane that establishes rectangular coordinate system to represent complex numbers is called complex plane, the X axis is called real axis, the Y axis is called imaginary axis except the origin, the origin represents real number 0, and the origin is not on the imaginary axis. Every point on the complex plane has a unique complex number corresponding to it, and conversely, every complex number has a unique point corresponding to it, so the complex set C is one-to-one corresponding to the set of all points on the complex plane.
Geometric representation
① Geometric form
The complex number z=a+bi is uniquely determined by the point z(a, b) on the complex plane. This form enables the complex number problem to be studied with the help of graphics. We can also use complex number theory to solve some geometric problems in turn.
② Vector form. The complex number z=a+bi is represented by the vector OZ, starting from the origin o and ending at the point Z(a, b). This form makes the four operations of complex numbers get proper geometric explanation.
③ Triangle. The complex number z=a+bi is transformed into a triangular form.
z = r(cosθ+isθ)
Where r = √ (A 2+B 2) is the modulus (i.e. absolute value) of a complex number.
θ is the angle with X axis as the starting edge and ray OZ as the ending edge, which is called complex angle, and the principal value of the angle is recorded as arg(z).
This form is convenient for complex multiplication, division, multiplication and root calculation.
④ Exponential form. Replace cosθ+isθ in the triangular form z = r(cosθ+isθ) of a complex number with exp(iθ), and the complex number will be expressed in the exponential form z=rexp(iθ).
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