Modulus and Argument of a Complex Number

The number, which represents the distance to the origin, and the angle the line forms with the positive axis, measured clockwise, make up the modulus-argument form of a complex number. The modulus of the complex number is the length of the line segment. The argument of the complex number, z, is the angle between the positive axis and the line segment. Using trigonometry, the modulus and argument are extremely straightforward to determine.

The square root of the sum of the squares of the real and imaginary parts of a complex number is the modulus of the complex number. 

The angle between the positive real axis and the line connecting the origin and z, depicted as a point in the complex plane, is the argument of a complex number z, called arg in mathematics.

The concept of modulus and argument of complex numbers will be explained in this article.

Modulus of a complex number

The square root of the sum of the squares of the real and imaginary parts of a complex number is the modulus of the complex number. If z is a complex number, its modulus is defined as [Re(z)]² + [Im(z)]² and symbolised by |z|. The distance between the origin (0, 0) and the point (a, b) in the complex plane is the modulus of a complex number z = a + ib. Because the distance is the modulus of a complex number, its value is always non-negative.

Complex Number Modulus Formula

The formula |z| = √(x²+y²) gives the modulus of a complex number z = x + iy, denoted by |z|, where x is the real component and y is the imaginary part of the complex number z. The conjugate of z can also be used to derive the modulus of a complex number z.

Graphing the Modulus of a Complex Number

The distance between the complex number’s coordinates and the origin on a complex plane is called the modulus of the complex number when it is shown on a graph. The modulus of a complex number is the distance of a complex number represented as a point on the argand plane (a, b). This distance is defined as r = √(a²+b²).and is measured from the origin (0, 0) to the point (a, b).

Argument of a complex number

The angle formed by the line representing the complex number and the positive x-axis of the argand plane is known as the argument of complex number. The angle, which is the inverse of the tan function of the imaginary part divided by the real part of the complex number, is the argument of the complex number Z = a + ib.

Principal argument of a complex number

The angle  θ of a complex number’s polar representation is its argument, z = a+ib. This is a multi-valued angle. If  is the complex number z argument, then  θ+2nπ,  n is an integer, and will also be an argument of that complex number.

The principal argument of a complex number, on the other hand, is the unique value of such that –π<θ ≤π.

As a result, a complex number’s principal argument is always a single data point, whereas the argument of a complex number has numerous data points due to its integral multiple of  2π.

Complex Number argument Formula

P is a point in the Argand plane that represents a non-zero complex number z = a+ib. The polar form of the complex number is z = r(cos  + i sin ) when OP makes an angle with the positive direction of the x-axis.

A complex number’s general argument is represented as θ+2nπ, where n is an integer.

arg (z) = θ =arg (a+ib)  

Conclusion

In this article we learned that, The modulus of the complex number and the argument of the complex number are two important characteristics that characterise the complex number in the argand plane. The argument of a complex number is the angle formed by the complex number with the positive axis of the argand plane, while the modulus of a complex number is its distance from the origin.