To determine the quadrant in which the complex number [tex]\( 6 - 8i \)[/tex] is located, we need to analyze its real and imaginary parts.
A complex number is given in the form [tex]\( a + bi \)[/tex], where:
- [tex]\( a \)[/tex] is the real part.
- [tex]\( b \)[/tex] is the imaginary part.
For the number [tex]\( 6 - 8i \)[/tex]:
- The real part ([tex]\( a \)[/tex]) is [tex]\( 6 \)[/tex].
- The imaginary part ([tex]\( b \)[/tex]) is [tex]\( -8 \)[/tex].
Now, we need to determine the quadrant based on these parts:
1. Quadrant I: In this quadrant, both the real part and the imaginary part are positive ([tex]\( a > 0 \)[/tex] and [tex]\( b > 0 \)[/tex]).
2. Quadrant II: In this quadrant, the real part is negative and the imaginary part is positive ([tex]\( a < 0 \)[/tex] and [tex]\( b > 0 \)[/tex]).
3. Quadrant III: In this quadrant, both the real part and the imaginary part are negative ([tex]\( a < 0 \)[/tex] and [tex]\( b < 0 \)[/tex]).
4. Quadrant IV: In this quadrant, the real part is positive and the imaginary part is negative ([tex]\( a > 0 \)[/tex] and [tex]\( b < 0 \)[/tex]).
Since for [tex]\( 6 - 8i \)[/tex]:
- The real part [tex]\( a = 6 \)[/tex] is positive.
- The imaginary part [tex]\( b = -8 \)[/tex] is negative.
Therefore, the complex number [tex]\( 6 - 8i \)[/tex] is located in Quadrant IV.
So, the correct answer is:
IV.
