The Argument of a Complex Number

When you delve into the world of complex numbers, a key concept is the 'argument.' The argument of a complex number is a measure of the angle created between the positive real axis (Re) and the vector that stretches from the origin to the point z=(a;b). This point corresponds with the complex number z=a+bi in the Gaussian plane.

More formally, we denote it as Arg(z), expressed as $$ \theta = \text{Arg(z)} $$ Just as with any angle, it can be measured in either degrees or radians.

Now, let's imagine we have a complex number, which we'll affectionately name z.

This complex number can be presented in an algebraic form, expressed as z = a + bi. Here, "a" represents the real component, and "bi" makes up the imaginary part.

$$ z = a+bi $$

So how does this look on a complex plane? Well, the x-axis measures the coefficient "a" of the real part, and the y-axis takes into account the coefficient "b" of the imaginary part.

To better understand this, think about a vector starting from the origin, the point (0;0), and ending at the point (a,b) which is represented by your complex number, z.

The angle that's formed between the positive real axis and this vector is precisely what we call the argument of the complex number z.

Let's illustrate this with an example: the complex number z=4+3i corresponds to the point (a;b)=(4;3) on the Gauss plane.

You can sketch a vector or segment that connects the origin, O, with the point (4;3). This vector's length represents the modulus.

The angle formed between this vector and the positive x-axis, well, that's the argument of our complex number z=4+3i.

So, how do we calculate the argument of a complex number?

A handy tool for this is the arctangent trigonometric function.

By calculating the arctangent of the ratio of coefficients b/a, you're on your way to finding the argument.

$$ \arg(z) = \arctan( \frac{b}{a} ) $$

Just like that, you've got the argument of your complex number.

Bear in mind, the argument is, after all, an angle. It can be measured in degrees or radians. However, there's a crucial detail to remember: the arctangent function yields values from -π/2 to π/2 (or -90 to 90 degrees). So, if your coefficient "a" is negative, the angle will need a bit of adjustment. Just add π radians (or 180 degrees) to the result.

Using our example, the coefficients of the complex number are a=4 and b=3.

$$ \arg(z) = \arctan( \frac{3}{4} ) = 36.87° $$

And there you have it. The argument of the complex number z=4+3i is 36,87° gradi.

One final point before wrapping up - the argument of a complex number is not unique. If you add or subtract multiples of 2π (or 360 degrees), you'll stumble upon other valid arguments for the same complex number. For instance, our complex number z=4+3i has an argument of θ=36.87°. However, it's just as valid to indicate any other angle with a value of $$ \theta = 36,87° + k \cdot 360° $$ where k can be any integer.