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The basic concept of plural numbers

1.Plural form

Remember“.a+bi( a,b The number of the real numbers is called the plural.


2.Imaginary unit

 A similar unit can be pushed to PI, in which we use pi to represent PI, where Pi is 3.1415926…

Alike,a+bi The I in it represents a unit of imaginary numbers. “.


3.The concept of the real part and the imaginary part

staya+bi In this, we call ita It is the real part of the plural.

              b It is the imaginary part of the plural.


4.The distinction between pure imaginary numbers, plural numbers, and real numbers

We take shape as a formbi The number of numbers becamePure imaginary number. Such as: 5I, 0.8i, 0.3i.

Then the plural is mentioned above, and then the Wayne diagram on the three is like this:

The basic concept of plural numbers is like the previous book.



A reference to a complex number in geometry

1.Complex plane

Two coordinate axes are introduced in the plane.

The abscissa unit is a real number, representing the real part of the complex number.

Then the unit of the longitudinal axis is a plural uniti .



2.The relationship between the complex and the vector

In the coordinate system of the complex plane, the complex numbera+bi Can be expressed as a vector(a,b) 

For example, a vector in a graph coordinate axis is a complex numbera+bi.


3.Conjugate complex number

Conjugate plural: for the complex number z=a+bi,For another pluralz′=a-bi,We call itzz′ Forzz The conjugate complex numberz 。

It is easy to find that a plural is equivalent to the real part of the conjugate complex, and the imaginary part is opposite.


After introducing the relationship between complex numbers and vectors, we can better understand the concept of conjugate complex numbers and explain them by vectors.The conjugate complex is two vectors of the same size and about X axis symmetry.


4.The amplitude of the complex number


We can take the plural.z Writez=r×(cosθ+isinθ),Among themr Pluralz Model lengthθ Pluralz The angle of the radiant.


The amplitude of complex numbersθ In ordinary plane rectangular coordinates, the inclination angles of straight lines are quite similar.

A complex number has multiple radiant angles, and these values differ.2π.

Among them, we will

θ[π,π)θ∈[−π,π) Itθθ Called the radiant angleMain value.



Basic operations of complex numbers

For the complex number (a, b) and (C, d).Satisfy the addition rule (a, b) + (C, d)=(a±c,c±d).

Satisfy the rule of multiplication (a, b) x (C, d)=(ac−bd,bc+ad) .

For division, the numerator denominator must be multiplied by the conjugate complex number of denominator, and the denominator will be real.

  The geometric meaning of complex multiplication is the multiplication of modular length and the addition of angles.

                                                ——From NaVi_Awson


Unit root of complex number

On the unit root of the complex number

The complex root of the equation Z isnSubunit root.
The distribution of these roots in the complex plane is like this.
When n=1, it has a unique root (1, 0), because the vector itself is 1..
When n=2, it has two roots, namely (-1, 0) and (1,0).
When n=3, it has three roots.
Then generalization, when there are n times, there are n unit roots.
Then, how do we express the unit roots in the complex plane?
As in figure n=5, the 5 red dots are all the units of the unit.
From this we can introduce the distribution law of unit roots in complex plane.
1.byn Sub squareThe unit circle will be divided inton Equal.
2.n The point is connected to a positive n edge.
3.n The starting point of the point is: (1,0)
4.Divide n according to 360 degrees, then move from the start point.
And then there is another concept:
Root source
In a nutshell, the root of the source is the equation about Zz^n=1 A root Z1, we think it is a root source.
When and only as follows:
     z1 onlyn Sub squareThe result of the operation is 1.
Such as:
  For the case of n = 2, only (-1, 0) is its original root.
  Because in n&lt, = 2, (1, 0) the 1 power and the 2 power are also 1.
  So (-1, 0) is the root of the n=2.
Other situations can be made by analogy.

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