complexThe modulus and radiance of are two basic elements of the complex triangular form. The length of the vector corresponding to the complex number is called the amplitude of the complex number. The angle between the vector and the positive direction of the real axis is the radiance of the complex number.The magnitude of the spoke angle is infinite, but the principal value of the spoke angle is uniquely determined.By using the modulus and argument of complex numbers, complex numbers can be expressed as triangular expressions and exponential expressions, and can be transformed with algebraic expressions to facilitate the discussion of different problems.
Chinese name
Radial angle
Foreign name
argument
Directions
Counterclockwise is positive, clockwise is negative
The points in are one-to-one, so you can use the coordinates as
Point of
To represent the complex number, when
The points on the axis correspond to real numbers, called
The axis is real,
The point on the axis (except the origin) corresponds to a pure imaginary number, called
The axis is imaginary, and the plane representing complex numbers like this is calledComplex plane。
complex
You can also use vectors
To indicate that,
And
Are vectors
stay
Shaft and
Projection on the axis.Thus, plural
Just like the vector on the plane
A one-to-one correspondence has been established.
vector
The length of is called complexmodelOr absolute value, recorded as
, so
Current point
Not the origin, that is, the complex number
When, vector
And
The included angle in the positive direction of the axis is called complex number
OfRadial angle, recorded as
。The symbol of the spoke angle is specified as: turning counterclockwise from the positive real axis
Positive, turn clockwise
Is negative.[1]
Principal value of spoke angle
Announce
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Obviously a non-zero complex number
There are infinitely many values for the radiance of
Integer multiple of, but
Only one value in
Meet conditions
, said
Plural
The main spoke angle of, recorded as
, so
When
When,
The spoke angle of is meaningless.
complex
Primary and secondary angles of
Principal value of arc tangent
There are the following relationships:[1]
Radial angle
Triangular and exponential expressions
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It can be seen from the relationship between rectangular coordinates and polar coordinates that non-zero finite complex numbers
Can use its mold
And spoke angle
To represent, that is
Using Euler formula
have to
The first and third formulas are called non-zero complex numbers respectively
Triangular expression and exponential expression of, which can be transformed with algebraic expression to facilitate the discussion of different problems.[1]
