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Magazine Chapter 12: Problem 31
Show the given number, its negative, and its conjugate on the same coordinate system. $$-3-5 j$$
Short Answer
Expert verified
Original: (-3, -5), Negative: (3, 5), Conjugate: (-3, 5).
Step by step solution
01
Identify the Given Complex Number
The complex number given is
-3-5j
. In the coordinate system, this will correspond to the point
(-3, -5)
on the complex plane, where
-3
is the real part and
-5
is the imaginary part.
02
Find the Negative of the Complex Number
To find the negative of a complex number
-3-5j
, you multiply both the real and imaginary parts by
-1
. This gives
-(-3) + (-1)(-5)j = 3 + 5j
, which corresponds to the point
(3, 5)
on the coordinate system.
03
Find the Conjugate of the Complex Number
The conjugate of a complex number
-3-5j
is obtained by changing the sign of the imaginary part. Thus, the conjugate is
-3 + 5j
, which corresponds to the point
(-3, 5)
on the complex plane.
04
Plot the Points on the Coordinate System
On the complex plane, plot the points
(-3, -5)
for the original number
-3-5j
,
(3, 5)
for the negative
3 + 5j
, and
(-3, 5)
for the conjugate
-3 + 5j
. The real axis corresponds to
x ext{-axis}
and the imaginary axis to the
y ext{-axis}
.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Plane
The complex plane, also known as the Argand plane, is a unique coordinate system used for representing complex numbers visually. It resembles the Cartesian coordinate system but has a distinct purpose. The horizontal axis (real axis) denotes the real part of a complex number, while the vertical axis (imaginary axis) represents the imaginary part. This system allows for the plotting of complex numbers as points using two coordinates: one for the real part and one for the imaginary part.
- Each complex number corresponds to a unique point on the plane.
- This visualization helps in understanding operations on complex numbers, like addition and multiplication.
- Distances and angles in this plane have significant meanings related to complex number operations.
Imaginary Part
The imaginary part of a complex number is a crucial component that dictates its vertical position on the complex plane. In a complex number expressed as (-3-5j), the -5j represents the imaginary part. This value is multiplied by the imaginary unit 'i' (or 'j' in certain contexts), which is defined as \\( i^2 = -1 \).
- In the coordinate (-3, -5), -5 is the imaginary part plotting along the y-axis.
- The sign of the imaginary part determines if the point is plotted above or below the real axis.
- Changing the sign of the imaginary part results in its conjugate, affecting its position relative to the real axis.
Conjugate of a Complex Number
The conjugate of a complex number is obtained by changing the sign of the imaginary part. For instance, given the complex number
(-3-5j), its conjugate is
(-3+5j). On the complex plane, this change results in a reflection across the real axis.
- The conjugate helps in various calculations, like dividing complex numbers.
- Multiplying a complex number by its conjugate yields a real number.
- Graphically, this is seen as a mirrored position with respect to the x-axis.
Coordinate System
A coordinate system is fundamental in graphing complex numbers and understanding their structure. The real and imaginary parts of a complex number are plotted as coordinates on a plane. For example, the complex number
(-3-5j)
translates to the point
(-3, -5)
on this system.
- The real part determines positioning along the x-axis.
- The imaginary part corresponds to movement along the y-axis.
- This creates a unique pairing of coordinates for each complex number.
- The system helps in visualizing operations, such as addition and multiplication, as geometric transformations.
