Open classification: mathematics, mathematicians, real numbers, imaginary numbers.

definition

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Complex number is a general term for real number and imaginary number.

The basic form of complex number is a+bi, where A and B are real numbers, A is called the real part, bi is called the imaginary part, and I is the imaginary part. On the complex plane, a+bi is the point z (a, b). The distance r from z to the origin is called the modulus of z |Z|=√a square +b square.

A+bi: a=0 is a pure imaginary number, b=0 is a real number, and B = 0 is an imaginary number.

The triangular form of a complex number is z = r [cosx+isinx]

Where x and r are real numbers, rcosx is called the real part, irsinx is called the imaginary part, and I is the imaginary unit. The distance between z and the origin r is called the modulus of z, and x is called the radial angle.

origin

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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 called "Cardan Formula" by later generations. He was the first to put negative numbers. When discussing whether 10 can be divided into two parts to make their product equal to 40, he wrote the answer as =40. Although he thinks that the two expressions of sum are meaningless, imaginary and illusory, he still divides 10 into two parts. Let their product be equal to 40. The French mathematician Descartes (1596- 1650) gave the name "imaginary number", and he made "imaginary number" 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 refused to admit imaginary numbers. The German mathematician Leibniz (1646- 17 16) said in 1702: "The imaginary number is a subtle and strange hiding place of God. It "can't have all shapes and shapes, and it's impossible to imagine numbers, because they represent the square root of negative numbers. For such figures, we can only assert that they are neither nothingness, nor higher than nothingness, nor lower than nothingness, and they are purely illusory. "But, really rational things, of course, can stand the test of time and space. The French mathematician D'Alembert (17 17- 1783) pointed out in 1747 that if the imaginary number is operated according to the four algorithms of polynomials, the result will always be in the form of (A and B are real numbers) (note: the mark =-I is not used in the current textbooks). And use = a 1). The French mathematician Dimov (1667- 1754) discovered this formula in 1730, which is the famous Dimov theorem. Euler found the famous relation in 1748, and it was in his differential formula (. The symbol I was first used as the unit of imaginary number. " Imaginary number is not an imaginary number, but it does exist. 1745-181779, the Norwegian surveyor Chensell tried to give an intuitive geometric explanation of this imaginary number, and published his method for the first time, but it was not recognized by the academic circles.

German mathematician Gauss (1777- 1855) published the image 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 a point on a plane. In the rectangular coordinate system, the horizontal axis takes the point A corresponding to the real number A, and the vertical axis takes the point B corresponding to the real number B and passes through it. Their intersection c represents the complex number A+bi. In this way, the plane with all points corresponding to complex numbers is called "complex plane", and later it is also called "Gaussian plane". In 183 1, Gauss represents the complex number A+bi with real arrays (a, b), and establishes some operations of complex numbers. Some operations of complex numbers are also "algebraic" like real numbers. He first put forward the term "complex number" in 1832, which integrated two different representations of the same point on the plane-rectangular coordinate method and polar coordinate method. They are unified into algebraic form and triangular form representing the same complex number, and the points on the number axis correspond to real numbers. Points on a plane correspond to complex numbers. 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 complex numbers by using the one-to-one correspondence between complex numbers and vectors. At this point, the complex number theory has been established relatively completely and systematically.

After many mathematicians' unremitting efforts for a long time, the theory of complex numbers has been deeply explored and developed, and the ghost of imaginary numbers that has been wandering in the field of mathematics for 200 years has unveiled its mysterious veil and revealed its true colors. Original imaginary number is not empty. It has become a member of the number family, and 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.

Specific content and application

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A number in the form of a+bi, where a and b are real numbers and I is a number satisfying I 2 =- 1. Because the square of any real number is not equal to-1, I is not a real number, but a new number other than a real number.

In the complex number A+Bi, A is called the real part of the complex number and B is called the imaginary part of the complex number. The real part and imaginary part of a complex number are represented by Rez and Imz respectively, that is, Rez =a and Imz=b.i are called imaginary parts. When the imaginary part is equal to zero, the complex number is a real number. When the imaginary part is not equal to zero, this complex number is called imaginary number, and if the real part of imaginary number is equal to zero, it is called pure imaginary number.

As can be seen from the above, the complex set contains the real set, so it is the expansion of the real set. Complex numbers come from the need to solve algebraic equations. 16th century, Italian mathematician G. cardano first expressed the root of a cubic equation with a formula, but the formula quoted the form of a negative root, and I = sqrt (- 1) was used as a number to participate in the operation together with other numbers. Because of this, for a long time, the square root of a negative number was not considered as a number, but called an imaginary number. It was not until the19th century that mathematicians scientifically explained that these imaginary numbers participated in the algebraic operation of real numbers and made them widely used in solving equations and other fields that people realized this new number.

The four operations of complex numbers are specified as follows:

(a+bi)+(c+di)=(a+c)+(b+d)i，

(a+bi)-(c+di)=(a-c)+(b-d)i，

(a+bi)？ (c+di)=(ac-bd)+(bc+ad)i，

(c and d are not both zero)

(a+bi)÷(c+di)=(ac+bd/c^2+d^2)+(bc-ad/c^2+d^2)i，

(c+di) is not equal to 0.

There are many expressions of complex numbers, and the common form z = a+bi is called algebraic expression.

In addition, there are the following forms.

(1) geometric form. The complex number z = a+bi is represented by the point Z(a, b) on the rectangular coordinate plane. This form enables complex number problems to be studied by means of graphics, and some geometric problems can be solved by complex number theory 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 addition and subtraction of complex numbers get a proper geometric explanation.

③ Triangle. Complex number z = a+bi is transformed into triangular form.

z = r(cosθ+isθ)

Where r = sqrt (a 2+b 2) is called the modulus (or absolute value) of a complex number; θ is based on the x axis; The vector OZ is the angle of the terminal edge, which is called the radial angle of the complex number. This form is convenient for multiplication, division, power and root operations of complex numbers.

④ exp(iθ) is used to replace cosθ+isθ in the triangular form of complex number z = r(cosθ+isθ), and the complex number is expressed in the exponential form z = rexp (i θ).

Complex triangular operation;

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)], if the triangular form of complex z is r(cosθ+isinθ).

Multiplication, division, power and root operations of complex numbers can be performed according to the operation law of power. Complex number set and real number set have several different characteristics: root operation is always feasible; The unary equation with n complex coefficients always has n roots (multiple roots count as multiples); Complex numbers cannot establish size order.

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Good (2) bad (0) real numbers include rational numbers and irrational numbers. Among them, irrational numbers are infinitely cyclic decimals and mantissas, and rational numbers include integers, fractions and 0.

Mathematically, real numbers are intuitively defined as numbers corresponding to points on the number axis. At first, real numbers were just numbers, and later the concept of imaginary numbers was introduced. The original numbers were called "real numbers"-meaning "real numbers".

Real numbers can be divided into rational numbers and irrational numbers, algebraic numbers and transcendental numbers, or positive numbers, negative numbers and zero. A set of real numbers is usually represented by the letter r or r n, and r n represents an n-dimensional real number space. Real numbers are uncountable. Real number is the core research object of real analysis.

Real numbers can be used to measure continuous quantities. Theoretically, any real number can be expressed by infinite decimal places, and to the right of the decimal point is an infinite series (either cyclic or acyclic). In practice, real numbers tend to approximate a limited number of decimal places (n digits are reserved after decimal point, and n is a positive integer). In the computer field, because computers can only store a limited number of decimal places, real numbers are often represented by floating-point numbers.

(1) inverse number (only two numbers with different signs, we will say that one of them is the inverse number of the other) The inverse number of the real number A is-a.

② Absolute value (the distance between a point corresponding to a number on the number axis and the origin 0) The absolute value of the real number A is: │ A │ = ① When a is a positive number, | A | = A

② When a is 0, |a|=0.

③ When a is negative, | a | =-a.

③ Reciprocal (the product of two real numbers is 1, so these two numbers are reciprocal) The reciprocal of real number A is: 1/a (a≠0).