To graph the complex number [tex]\( -5 - \sqrt{-9} \)[/tex] on the complex plane, let's follow these steps:
### Step 1: Simplify and identify components
First, we recognize that the complex number can be written as:
[tex]\[ -5 - \sqrt{-9} \][/tex]
Recall that [tex]\( \sqrt{-9} \)[/tex] can be expressed as:
[tex]\[ \sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i \][/tex]
Thus, our complex number becomes:
[tex]\[ -5 - 3i \][/tex]
Here, the real part is [tex]\(-5\)[/tex] and the imaginary part is [tex]\(-3\)[/tex].
### Step 2: Plot on the complex plane
The complex plane consists of a horizontal real axis and a vertical imaginary axis.
- The real part ([tex]\(-5\)[/tex]) tells us how far to move along the horizontal axis.
- The imaginary part ([tex]\(-3\)[/tex]) tells us how far to move along the vertical axis.
1. Start at the origin [tex]\((0,0)\)[/tex].
2. Move [tex]\(5\)[/tex] units to the left along the real axis, reaching the point [tex]\((-5,0)\)[/tex].
3. From [tex]\((-5,0)\)[/tex], move [tex]\(3\)[/tex] units down along the imaginary axis to reach the point [tex]\((-5, -3)\)[/tex].
This point [tex]\((-5, -3)\)[/tex] is the graphical representation of the complex number [tex]\(-5 - 3i\)[/tex].
### Step 3: Find the absolute value
The absolute value (or magnitude) of a complex number [tex]\( a + bi \)[/tex] is given by:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
For the complex number [tex]\(-5 - 3i\)[/tex]:
- The real part [tex]\(a = -5\)[/tex]
- The imaginary part [tex]\(b = -3\)[/tex]
So the absolute value is:
[tex]\[ | -5 - 3i | = \sqrt{(-5)^2 + (-3)^2} \][/tex]
[tex]\[ = \sqrt{25 + 9} \][/tex]
[tex]\[ = \sqrt{34} \][/tex]
[tex]\[ \approx 5.830951894845301 \][/tex]
Thus, the absolute value of the complex number [tex]\( -5 - 3i \)[/tex] is approximately [tex]\( 5.831 \)[/tex].
In summary:
1. The complex number [tex]\( -5 - 3i \)[/tex] is plotted at the point [tex]\((-5, -3)\)[/tex] on the complex plane.
2. The absolute value of this complex number is approximately [tex]\( 5.831 \)[/tex].
