91Ó°ÊÓ

The magnitude of a complex number gives an intuitive sense of how far the point representing the complex number is from the origin on the complex plane. Imagine a coordinate system where every point is a complex number, marked by its real part on the x-axis and the imaginary part on the y-axis.
To find the magnitude of a complex number, we use the formula:

• \( a \) is the real part, and
• \( b \) is the imaginary part.

The formula is:

• \[ r = \sqrt{a^2 + b^2} \]

If the complex number is \(-2 - 7i\), plug these values into the formula:

• \( a = -2 \)
• \( b = -7 \)

This results in:

• \[ r = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \]

The magnitude, \( \sqrt{53} \), tells us the distance from the origin to the point \(-2 - 7i\) on the complex plane.

The argument of a complex number is like a compass that points you to the right direction on the complex plane. It is the angle formed with the positive real axis. To find the argument of a complex number, we use the formula:

• \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \)

Where \( a \) and \( b \) are the real and imaginary parts, respectively. For the complex number \(-2 - 7i\):

• Real part \( a = -2 \)
• Imaginary part \( b = -7 \)
• The formula becomes \( \theta = \tan^{-1} \left( \frac{-7}{-2} \right) \)

However, the initial calculation gives a value that must be adjusted because both parts are negative. This places the complex number in the third quadrant of the complex plane.To correct the angle:

• \( \theta = \pi + \tan^{-1} \left( \frac{7}{2} \right) \)

This adjustment ensures we have the proper direction on the plane.

In the third quadrant of the complex plane, both the real part and the imaginary part of a complex number are negative. This part of the plane is like the bottom left of a graph where both axes are below zero. Understanding which quadrant a complex number lies in helps in accurately determining the argument, \( \theta \), of the complex number. When you're looking at a number like \(-2 - 7i\):

• The real part \( -2 \) is on the negative side of the x-axis.
• The imaginary part \( -7 \) means it is also on the negative side of the y-axis.

Thus, this complex number resides in the third quadrant. Knowing this, we adjust our initial angle calculation from step two to ensure that the angle correctly accounts for its position:

• Adding \( \pi \) to compensate for the third quadrant position
• Ensures the angle correctly reflects its true position

This ensures that when using \( \theta = \pi + \tan^{-1} \left( \frac{7}{2} \right) \), the angle measures correctly counterclockwise from the positive x-axis.