The number set is extended to the real number range, but some operations are still impossible. For example, the quadratic equation with discriminant less than 0 still has no solution. Therefore, the data set is expanded again to reach a complex range.
We define a number in the form of z=a+bi as a complex number, where I is defined as 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 imaginary number Z and write it as REZ = A..
The real number b is called the imaginary part of the imaginary number z, and is recorded as imz = B.
Yi Zhi: When b=0 and z=a+ib=a+0, then the complex number becomes a real number;
When a=0 and z=a+bi=0+bi, we call it pure imaginary number.
Let z=a+bi be a complex number, then let the complex number z' = a-bi be a * * yoke complex number of z.
Definition: the modulus (absolute value) of a complex number = √ (A2+B2) (see below for the definition).
The set of complex numbers is represented by C. Obviously, R∩C=R (that is, R is the proper subset of C).
Four operations of complex numbers (algebraic expressions);
(a+bi)+(c+di)=(a+c)+(b+d)i，
(a+bi)-(c+di)=(a-c)+(b-d)i，
(a+bi)？ 6? 1(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.
Other expressions of plural numbers
There are many representations of complex numbers, and the common form z=a+bi is called algebraic form.
Here are some other expressions of the plural.
(1) geometric form.
In the rectangular coordinate system, the coordinate system formed by taking X as the real axis, Y as the imaginary axis and O as the origin is called the complex plane (see the attached figure of this entry).
In this way, all complex numbers can be uniquely determined by the point representation on the complex plane.
The complex number z=a+bi is represented 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 addition and subtraction of complex numbers get a proper geometric explanation.
③ Triangle. The complex number z = a+bi is transformed into a triangular form.
z=r(cosθ+sinθi)
Where r = sqrt (A 2+B 2) is called the module (i.e. 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 complex multiplication, division, multiplication and root calculation.
④ Exponential form. Replace cos θ+isinθ in the triangular form z = r (cos θ+isinθ) with exp(iθ), and the complex number will be expressed as 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, multiplication and root of complex numbers can be carried out according to the arithmetic of power. Complex set and real 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.
An Important Theorem in Complex Numbers: Demory Theorem
If there is a complex number z=cosθ+isinθ, then z n = cos (nθ)+isin (nθ).
If z n = a, then z = n √ a [cos (2kπ/n)+isin (2kπ/n)], n ∈ n, n = 1, 2,3 ... (n-1).