Modulus and argument of a complex number

Modulus of a Complex Number

The modulus of a complex number is a measure of its size or magnitude.
It’s calculated as ** a + bi = √(a² + b²), where **a is the real part and b is the imaginary part.
Conceptually, this is the distance of the point (a, b) from the origin, when plotted on the Argand diagram.
For example, the modulus of 3 + 4i is √(3² + 4²) = 5.
The modulus of a complex number is always non-negative, with 0 occurring only if the complex number is 0.

The argument of a complex number, also known as the phase, gives the angle that the complex number makes with the real axis on the Argand diagram.
It is usually denoted by arg(a + bi), and is calculated using trigonometry arg(a + bi) = tan⁻¹(b / a).
This calculation may need to be adjusted by 180° or π radians depending on the quadrant of the complex number.
There are two conventions when dealing with arguments:
- Principal argument is typically between -180° and 180° (-π and π in radians).
- While argument can take on any real value as it allows for complete circles around the origin.
For example, the argument of 1 + √3i is 60° or π/3 radians.
Multiply the argument by -1 (or add π radians) to find the angle for the conjugate complex number.

The Polar Form of a complex number combines modulus and argument and is often useful in calculations.
It is denoted as z = r(cos θ + isin θ), where r is the modulus of z, θ is the argument of z, i is the imaginary unit and cos θ + isin θ is also called cis θ.
To convert from rectangular (Cartesian) to polar form, calculate the modulus and argument of the complex number.
Converting back into Cartesian form involves using the trigonometric identities: cos θ = a/r and sin θ = b/r.