Argument Of Complex Number
Argument Of Complex Number is the measure of the angle made by the line representing the complex number, with the positive xaxis of the argand plane. The argument of the complex number Z = a + ib is the angle θ which is the inverse of the tan function of the imaginary part divided by the real part of the complex number.
Argument of Complex Number = θ = Tan^{1}(b/a)
The argument of a complex number gives the relationship between the real part and the imaginary part of the complex number. Let us learn more about the principle and general argument of the complex number, applications of the argument of the complex number, with the help of examples, FAQs.
What Is Argument Of Complex Number?
Argument of complex number is the angle made by the line representation of the complex number, with the positive xaxis of the argand plane. Any complex number can be represented in the argand plane with the real part marked along the xaxis and the imaginary part marked along the yaxis. And the complex number Z = a + ib can be represented as a point A(a, b) in the argand plane, and the angle made by the line OA with the positive xais is the argument of the complex number.
For a complex number Z = a + ib, the argument of the complex number is the angle measure, which is equal to the inverse of the trigonometric tan function of the imaginary part, divided by the real part of the complex number.
Argument of Complex Number = θ = Tan^{1}(b/a)
Principle Vs General Argument Of Complex Number
The argument of the complex number is measured as an angle made by the line representation of this complex number with the positive xaxis. This angle based on its values has principle value and the general value, which is resulting in the principle argument and general argument of the complex number. The trigonometric value of Tan is used to find the argument of the complex number, and hence it is based on the general solution of the trigonometric tangent function.
Principle Argument Of Complex Number = π < θ < π
The principle argument of complex numbers has values from π < θ <π. Further, It is 0 < θ < π, if taken in the first two quadrants where the angle is measured with respect to the positive xaxis in the anticlockwise direction. And it is π < θ < 0 in the third and fourth quadrant, with respect to the positive xaxis, where the angle is measured along the clockwise direction. Further, the general argument of the complex number is 2nπ + θ.
General Argument Of Complex Number = 2nπ + θ
Thus the argument of the complex number is based on the trigonometric function, and hence it has the principle and general argument.
Modulus And Argument of Complex Number
The argument of the complex number and the modulus of the complex number are the two important characteristic features that completely define the complex number in the argand plane. The modulus of the complex number is the distance of the complex number from the origin, and the argument of the complex number is the angle made by the complex number with the positive axis of the argand plane.
Modulus Of Complex Number: The modulus of a complex number is the distance of the complex number from the origin in the argand plane. For a complex number Z = a + ib the modulus of the complex number is represented as Z, and we have Z = \(\sqrt {a^2 + b^2}\). It is the distance between the origin (0, 0), and the point (a, b) in the complex plane. Further, we can also define the modulus of a complex number as the square root of the sum of the squares of the real part and the imaginary part of the complex number.
Argument Of Complex Number: The argument of the complex number Z = a + ib is represented as arg Z. The complex number Z = a + ib is represented as a point A(a, b) in the argand plane with the origin O(a, 0). And the angle made by the line OA with the positive xaxis in the anticlockwise direction is called the argument of the complex number. The argument of the complex number is θ = Tan^{1}(b/a).
Applications Of Argument Of Complex Number
The argument of a complex number has numerous applications in transforming the complex number to polar form, and also in finding the relationship between the real part and the imaginary part of the complex number. The argument value of the complex number is the angle θ which helps in identifying in the real part is greater or the imaginary part is greater. For θ = 45º the real part is equal to the imaginary part, for 0º < θ < 45º the real part is greater than the imaginary part, and for 45º < θ < 90º the imaginary part is greater than the real part.
Polar Form of Complex Number: The polar form of the complex number is P = r(Cosθ + iSinθ). Here θ is the argument of the complex number, and r is the argument of the complex number. Polar form is another important form of representing the complex number in the argand plane. The polar form of the complex number represented in cartesian form is (rCosθ, rSinθ).
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Examples on Argument Of Complex Number

Example 1: Find the argument of the complex number Z = 5 + 4i
Solution:
The given complex number is Z = 5 + 4i
This can be compared with the complex number Z = a + ib, and we have the argument of the complex number as θ = Tan^{1}(b/a).
θ = Tan^{1}(4/5)
Therefore the argument of the complex number is Tan^{1}(4/5).

Example 2: Find the argument of the complex number having an imaginary part of 2 units and the real part of the complex number as \(2\sqrt 3\).
Solution:
For the given complex number Z = a + ib, the real part is a = \(2\sqrt 3\), and the imaginary part is b = 2.
The argument of the complex number is θ = Tan^{1}(b/a)
θ = Tan^{1}(\(\dfrac{2}{2\sqrt 3}\))
θ = Tan^{1}(\(\dfrac{1}{\sqrt 3}\))
θ = 30º
Therefore the argument of this complex number is 30º.
FAQs on Argument Of Complex Number
What Is An Argument Of Complex Number?
Argument of complex number is the angle made by the line representation of the complex number, with the positive xaxis of the argand plane. Any complex number can be represented in the argand plane with the real part marked along the xaxis and the imaginary part marked along the yaxis. For the complex number Z = a + ib the argument of the complex number is θ = Tan^{1}(b/a)
How Do You Find The Argument Of Complex Number?
The argument of the complex number is the angle made by the complex number representation with the xaxis of the argand plane. The argument θ of the complex number Z = a + ib is equal to the inverse tan of the imaginary part (b) divided by the real part(a) of the complex number. The argument of a complex number is θ = Tan^{1}(b/a).
What Is The Use Of Argument Of Complex Number?
The argument of a complex number is useful to find the proportional relationship between the real part and the imaginary part of the complex number. The argument of the complex number also is helpful in writing the complex number in polar form. The complex number Z = a + ib is written in polar form as Z = r(Cosθ + iSinθ), and here r is the magnitude of the complex number, θ is the argument of the complex number.
What Is The Difference Between General And Principle Argument Of Complex Number?
The principle argument of the complex number is the first value of the complex number which is between 0 and 2π. ( 0 < θ < 2π). And the higher θ values of the complex number are called the general argument of the complex number and θ = 2nπ + θ
What Are The Properties Of Argument Of A Complex Number?
The two important properties of the argument of the complex number are as follows.
 arg(Z_{1}.Z_{2}) = arg(Z_{1}) + arg(Z_{2})
 arg(Z_{1}/Z_{2}) = arg(Z_{1})  arg(Z_{2})
How Is Argumet Of Complex Number Related To Modulus Of Complex Number?
The argument of complex number and the modulus of complex number are two distinct characteristics of the complex number. The modulus of the complex number gives the distance of the complex number representation from the origin, and the argument of complex number gives the inclination of the the complex number in the argand plane. For a complex number Z = a + ib, the modulus of the complex number is represented as Z = \(\sqrt {a^2 + b^2}\), and the argument of the complex number is θ = Tan^{1}(b/a).
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