Operations in the Complex Plane (Grades 11-12)

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Operations in the Complex Plane

1. 
Graph [math](1-3i) - (4+5i)[/math] on the complex plane.
Graph 10x10



2. 
Add [math]z_1 = -2+5i[/math] and [math]z_2= 3+i[/math] graphically.
Graph 10x10



3. 
Find the value of [math](9-3i) - (4 - 5i)[/math] graphically.
Graph 10x10



4. 
Find the value of [math](-3+i) + (7-3i)[/math] graphically.
Graph 10x10



5. 
One method for adding or subtracting complex numbers in the complex plane is to look at the numbers as vectors, and then add or subtract these vectors as is done with vectors in the real plane. A vector is defined as something having both a direction and a magnitude. Why can complex numbers in the complex plane be represented as vectors? Choose all correct answers.
  1. They can't really, but it's a useful tool to use when adding or subtracting.
  2. Because, as seen in polar form, they have a magnitude, [math]r[/math], and direction, [math]theta[/math].
  3. Because they are represented by two "coordinates", a real and imaginary value, which is similar to the component form of a vector.
  4. Because complex numbers and vectors are identical.
6. 
André is working on a math problem where he has to subtract two complex numbers graphically. They are [math]z_1 = 3-8i[/math] and [math]z_2 = -5+i[/math]. If he is using the translation method, and is subtracting [math]z_1[/math] from [math]z_2[/math], which complex number is considered the starting point, and then how is it translated?
  1. [math]z_1[/math] is the starting point, and is translated 5 units left and 1 unit up.
  2. [math]z_1[/math] is the starting point, and is translated 5 units right and 1 unit down.
  3. [math]z_2[/math] is the starting point, and is translated 3 units right and 8 units down.
  4. [math]z_2[/math] is the starting point, and is translated 3 units left and 8 units up.
7. 
Let x represent the real axis and y the imaginary axis in the graph below. Given the complex number [math]2+4i[/math], represented by C, which of the following represents its conjugate?
Coordinate Plane - 5x5 - With Dots
  1. H
  2. D
  3. B
  4. Its conjugate is not represented by any given value on this graph.
8. 
Let x represent the real axis and y the imaginary axis in the graph below. Given the complex number [math]3i[/math], represented by A, which of the following represents its conjugate?
Coordinate Plane - 5x5 - With Dots
  1. E
  2. F
  3. G
  4. Its conjugate is not represented by any given value on this graph.
9. 
In the graph below, let x represent the real axis and y the imaginary axis. Given the complex number [math]-3[/math], represented by the point E on the graph, which of the following represents its conjugate?
Coordinate Plane - 5x5 - With Dots
  1. G
  2. E
  3. F
  4. Its conjugate is not represented by any given value on this graph.
10. 
The following graph represents the complex plane, where x is the real axis and y the imaginary axis. If [math]z = -7+6i[/math], represented by the letter A in the graph, what does letter B represent in relation to [math]z?[/math]
Coordinate Plane - 10x10 - With Dots
  1. No special relation.
  2. The complex conjugate, [math]bar{z}[/math].
  3. The inverse, [math]z^{-1][/math].
  4. The negative, [math]-z[/math].

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