91Ó°ÊÓ

Complex numbers are numbers that have both a real and an imaginary part. They are expressed in the form \( z = x + yi \), where \( x \) represents the real part and \( yi \) represents the imaginary part. Imaginary numbers are multiples of \( i \), the imaginary unit, which is defined by \( i^2 = -1 \).

Complex numbers can be visualized as points or vectors on the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part. By allowing numbers to be expressed in this way, many more mathematical problems can be solved.

In our exercise, the complex number is given as \( -3a - 4ai \). Here, \( -3a \) is the real part and \( -4a \) is the imaginary part. This means the number is represented as a point in the third quadrant of the complex plane.

The magnitude of a complex number, also known as the modulus, is a measure of the distance of the complex number from the origin on the complex plane. It is denoted as \(|z|\) and is calculated using the formula \( |z| = \sqrt{x^2 + y^2} \), where \( x \) is the real part and \( y \) is the imaginary part.

For our complex number \( -3a - 4ai \), the magnitude is found by substituting \( x = -3a \) and \( y = -4a \) into the formula:

\[ |z| = \sqrt{(-3a)^2 + (-4a)^2} = \sqrt{9a^2 + 16a^2} = \sqrt{25a^2} = 5a \]

This magnitude tells us how far away the point \( -3a - 4ai \) is from the origin, and is an important component in expressing the complex number in its polar form.

The argument of a complex number is the angle formed with the positive real axis, and it helps to describe the direction of the vector in the complex plane. It is denoted by \( \theta \) and can be calculated using the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

For our example \( z = -3a - 4ai \), we substitute \( x = -3a \) and \( y = -4a \) to find:

\[ \theta = \tan^{-1}\left(\frac{-4a}{-3a}\right) = \tan^{-1}\left(\frac{4}{3}\right) \]

However, since this complex number lies in the third quadrant of the complex plane, we must adjust the angle by adding \( \pi \) to account for the correct direction:

\[ \theta = \pi + \tan^{-1}\left(\frac{4}{3}\right) \]

This adjusted angle will help us fully express our complex number in polar form.

The complex plane is divided into four quadrants, much like a typical Cartesian coordinate system. The third quadrant is defined by negative real parts and negative imaginary parts. So, all complex numbers located in the third quadrant have both their real part \( x \) and their imaginary part \( y \) less than zero.

In the case of \( -3a - 4ai \), both \( -3a \) (real part) and \( -4a \) (imaginary part) are negative, clearly placing the number in the third quadrant. When dealing with angles, this placement is crucial because it affects the calculation of the argument. In particular, angles in the third quadrant increase from \( \pi \) to \( 1.5\pi \), requiring an adjustment when converting into polar form.

Understanding which quadrant a complex number is in allows for accurate calculations of the argument, giving us the correct angle needed for the polar representation.