The Concept of Complex Numbers What is the concept of complex numbers?

Complex numbers are numbers of the form a b i. where a, b are real numbers, and i is a number that satisfies i 2 1, because the square of any real number is not equal to 1, so i is not a real number, but a new number other than the real number.

In the complex number A bi, a is called the real part of the complex number, b is called the imaginary part of the complex number, and i is called the imaginary unit. When the imaginary part is equal to zero, this complex number is a real number; When the imaginary part is not equal to zero, this complex number is called an imaginary number, and if the real part of the imaginary number is equal to zero, it is called a pure imaginary number. From the above, it can be seen that the complex set contains the set of real numbers, and is therefore an expansion of the set of real numbers.

There are many representations of complex numbers, and the common form z a b i is called algebraic. In addition, there are the following forms.

Geometric forms. The complex number z a b i is represented by the point z ( a ,b ) on the Cartesian coordinate plane. This form makes it possible for the problem of complex numbers to be studied graphically. It is also possible to use the theory of complex numbers to solve some geometric problems.

Vector form. The complex number z a b i is represented by a vector o z starting from the origin o and ending at the point z ( a ,b ). This form allows the addition and subtraction of complex numbers to be properly geometrically explained.

Triangular form. The complex number z a b i is transformed into a triangular form.

z z (where z = in cos isin, is called the modulo (or absolute value) of complex numbers; is the starting edge with the x-axis; The vector o z is the angle of the terminal side, which is called the radial angle of the complex number. This form is convenient for multiplication, division, multiplication, and open square operations of complex numbers.

Exponential form. Replace the trigonometric form z z of the complex number (cos isin in cos isin with e i q, and the complex number is exponential.

z z e i q , the multiplication, division, power, and square of complex numbers can be performed according to the algorithm of power.

Several characteristics of complex number sets that differ from real number sets are: open square operations are always possible; The unary nth complex coefficient equation always has n roots (the double roots are counted as multiple numbers); Complex numbers do not establish a size order.

What is the concept of singular and plural.

Numbers of the form z=a+bi (both a and b are real numbers) are called complex numbers. where A is called the real part, B is called the imaginary part, and I is called the imaginary unit. When the imaginary part of z is b 0, then z is a real number; When the imaginary part of z is b≠0 and the real part is 0, z is often called a pure imaginary number.

A complex number field is an algebraic closure of a real number field, i.e., any complex coefficient polynomial always has roots in the complex number field.

Complex numbers were first introduced in the 16th century by the Italian Milanese scholar Cardan, and through the work of d'Alembert, Dimoff, Euler, Gauss and others, the concept was gradually accepted by mathematicians.

Plural applications. 1. Abnormal points.

At the application level, complex analysis is often used to calculate the anomalous functions of certain real values, which are derived by complex value functions. There are several methods, see Wai Dao Integration Method.

2. Quantum mechanics.

Complex numbers are very important in quantum mechanics because their theory is based on the infinite-dimensional Hilbert space over the complex number field.

3. The theory of relativity.

If you treat time variables as imaginary numbers, you can simplify some of the spatio-temporal metric equations in special and general relativity.

4. Applied Mathematics.

In practical application, to solve a system with a given difference equation model, it is usually necessary to first find all the complex eigenroots r of the eigenequation corresponding to the linear difference equation, and then express the system as a linear combination of the basis functions of the shape f(t) = e.

Complex Numbers of Explanations In some languages, the number of words that belong to two or more is represented by the morphological change of words. For example, in English book (book, singular) means a book, and books (book, plural) refers to two or more books. Numbers of the form a+bi are called complex numbers.

where a, b are real numbers, and i is an imaginary unit. A is called the real part of the complex number, and bi is called the imaginary part of the complex number. For example, 1-3i and 5i are all plural.

Word decomposition Explanation of complex Fu ( Fu Fu ) ù Hui Zen Mill , Hui : Repeatedly. Reciprocate. , Return: Resurrection. Reply.

Revenge. Revert, make the same as before: revert to the old.

Turn. Reinstated. Recovery.

Restoration. Again, do it all over again: review.

Referral. Review. Repetition.

Reconsideration. Many, not singular: the explanation of the number of weights (numbers) ù the quantities that represent, divide or calculate:

Number. Quantity. Numeral.

Number theory (a branch of mathematics that studies the properties of positive integers and the laws related to them). Numerical control. Few, several:

Several people. days. Technical, Academic:

Now the husband is playing the number, and the decimal is also". Destiny, day.

Explanation of complex numbers.

In some languages, the number of oranges, which are indicated by morphological changes of words, etc., belong to two or more oranges. For example, in English book (book, singular) means a book, and books (book, plural) refers to two or more books. Numbers of the form a+bi are called complex numbers.

where a, b are real numbers, and i is an imaginary unit. A is called the real part of the complex number, and bi is called the imaginary part of the complex number. For example, 1-3i and 5i are all plural.

The word breaks down the sedan car.

Explanation of Fu Fu ( Fu Fu ) ù Go back , return : repeatedly. Reciprocate. , Return: Resurrection. Reply.

Revenge. Revert, make the same as before: revert to the old.

Turn. Reinstated. Recovery.

Restoration. Again, do it all over again: review.

Referral. Review. Repetition.

Reconsideration. Many, not singular: the explanation of the number of weights (numbers) ù the quantities that represent, divide or calculate:

Number. Quantity. Numeral.

Number theory (a branch of mathematics that studies the properties of positive integers and the laws related to them). Numerical control. Few, several:

Several people. days. Technical, Academic:

Now the husband is playing the number, and the decimal is also". Destiny, day.

The concept of complex numbers: we call a number of forms such as z=a+bi (a, b are both real numbers) as complex numbers, where a is called the real frontal acacia, b is called the imaginary part, and i is called the imaginary unit.

Since natural numbers are not closed to subtraction (i.e., the result of subtracting a larger natural number from a smaller natural number is not a natural number), we extend the natural numbers to integers in order to close the subtraction operation. Since integers are not closed to division operations (i.e. :

An integer cannot be divisible by another integer, the result is not an integer), and in order to close the division operation, we expand the integer to rational numbers.

Plural

Since rational numbers are not closed to the square operation (i.e., the rational number is open to the power of the integer, and the result can not be a rational number), in order to close the square calculation, we will expand the rational number to a part of the algebraic number. An "algebraic number" is defined as the root of a univariate polynomial equation with integer coefficients (or rational coefficients), which includes a part of a real number and a part of an imaginary number.

On the other hand, rational numbers are not closed to limit operations, and in order to close limit operations, we extend rational numbers to real numbers. As a result, limit, calculus, and infinite series operations can all be operated well. That is to say, the functions defined on the field of real numbers are operated on limits, definite integrals, multiple integrals, infinite series, infinite products, etc., and as long as they do not diverge, the simplified results are within the range of real numbers.

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