Modulus and Argument

• The concept of “Modulus and Argument” originates from complex numbers.
• A complex number can be represented in the form z = x + yi, where x and y are real numbers, and i is the imaginary unit. x is the real part, and y is the imaginary part.
• In this context, “modulus” refers to the magnitude or absolute value of a complex number.
• The “argument” refers to the angle that the line connecting the origin and the point representing the complex number makes with the positive x-axis. This angle is usually denoted by ‘arg(z)’ or sometimes by ‘θ’.

• The modulus can be found by using the formula r =

  z

  = √(x^2 + y^2).

• To calculate the argument, use the formula θ = tan^(-1)(y/x). This will give the principal argument, which lies between -π and π.
• “Modulus-Argument” form (also known as “polar form”) of a complex number is given by r(cos θ + i sin θ) or re^(iθ) where r > 0 is the modulus and θ is the argument of a given complex number.
• Note that when the argument is positive, the rotation is counter-clockwise. When the argument is negative, the rotation is clockwise.
• While adding or subtracting complex numbers, it’s more efficient to use the Cartesian form (x + yi). For multiplication or division, using the Modulus-Argument form simplifies the process.

• Also note that if z1 and z2 are two complex numbers, then

  z1z2

  =

  z1

  z2

  and arg(z1z2) = arg(z1) + arg(z2).

• Be aware also of the De Moivre’s Theorem, which states that (cos θ + i sin θ)^n = cos nθ + i sin nθ. This theorem is extremely useful in simplifying the powers of complex numbers.
• Also, consider extending your understanding by exploring the geometric interpretations of the arithmetic operations on complex numbers represented in the Modulus-Argument form.
• Practice numerous questions on this topic to gain proficiency in quickly identifying the modulus and argument of complex numbers.