Advanced Geometry : How to graph complex numbers

Study concepts, example questions & explanations for Advanced Geometry

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Example Questions

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Example Question #1 : How To Graph Complex Numbers

Coordinate_pair_1

Point A represents a complex number.  Its position is given by which of the following expressions?

Possible Answers:

Correct answer:

Explanation:

Complex numbers can be represented on the coordinate plane by mapping the real part to the x-axis and the imaginary part to the y-axis.  For example, the expression  can be represented graphically by the point .

Here, we are given the graph and asked to write the corresponding expression.

 not only correctly identifies the x-coordinate with the real part and the y-coordinate with the imaginary part of the complex number, it also includes the necessary 

 correctly identifies the x-coordinate with the real part and the y-coordinate with the imaginary part of the complex number, but fails to include the necessary .

 misidentifies the y-coordinate with the real part and the x-coordinate with the imaginary part of the complex number.

 misidentifies the y-coordinate with the real part and the x-coordinate with the imaginary part of the complex number.  It also fails to include the necessary .

Example Question #7 : Graphing

Which of the following graphs represents the expression ?

Possible Answers:

Coordinate_pair_4

Complex numbers cannot be represented on a coordinate plane.

Coordinate_pair_3

Coordinate_pair_2

Coordinate_pair_5

Correct answer:

Coordinate_pair_4

Explanation:

Complex numbers can be represented on the coordinate plane by mapping the real part to the x-axis and the imaginary part to the y-axis.  For example, the expression  can be represented graphically by the point .

Here, we are given the complex number  and asked to graph it.  We will represent the real part, , on the x-axis, and the imaginary part, , on the y-axis.  Note that the coefficient of  is ; this is what we will graph on the y-axis.  The correct coordinates are .

 

Example Question #1 : How To Graph Complex Numbers

Give the -intercept(s) of the parabola with equation . Round to the nearest tenth, if applicable.

Possible Answers:

The parabola has no  -intercept.

Correct answer:

The parabola has no  -intercept.

Explanation:

The -coordinate(s) of the -intercept(s) are the real solution(s) to the equation . We can use the quadratic formula to find any solutions, setting  - the coefficients of the expression.

An examination of the discriminant , however, proves this unnecessary.

The discriminant being negative, there are no real solutions, so the parabola has no  -intercepts.

Example Question #2 : How To Graph Complex Numbers

In which quadrant does the complex number    lie?

Possible Answers:

-axis

Correct answer:

Explanation:

When plotting a complex number, we use a set of real-imaginary axes in which the x-axis is represented by the real component of the complex number, and the y-axis is represented by the imaginary component of the complex number. The real component is    and the imaginary component is  ,  so this is the equivalent of plotting the point    on a set of Cartesian axes.  Plotting the complex number on a set of real-imaginary axes, we move    to the left in the x-direction and    up in the y-direction, which puts us in the second quadrant, or in terms of Roman numerals:

Example Question #3 : How To Graph Complex Numbers

In which quadrant does the complex number    lie?

Possible Answers:

Correct answer:

Explanation:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point    on a set of Cartesian axes.  We move    units right in the x direction, and    units down in the y direction, which puts us in the fourth quadrant, or in terms of Roman numerals:

Example Question #4 : How To Graph Complex Numbers

In which quadrant does the complex number    lie?

Possible Answers:

Correct answer:

Explanation:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point    on a set of Cartesian axes.  We move    units left of the origin in the x direction, and    units down from the origin in the y direction, which puts us in the third quadrant, or in terms of Roman numerals:

Example Question #5 : How To Graph Complex Numbers

In which quadrant does the complex number    lie?

Possible Answers:

Correct answer:

Explanation:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point    on a set of Cartesian axes.  We move    units right of the origin in the x direction, and    units up from the origin in the y direction, which puts us in the first quadrant, or in terms of Roman numerals:

Example Question #2 : How To Graph Complex Numbers

In the complex plane, what number does this point represent?

Complex grid

Possible Answers:

Correct answer:

Explanation:

In the complex plane, the x-axis represents the real component of the complex number, and the y-axis represents the imaginary part. The point shown is (8,3) so the real part is 8 and the imaginary part is 3, or 8+3i.

Example Question #7 : How To Graph Complex Numbers

The graph below represents which complex number? 

2 3ia

Possible Answers:

Correct answer:

Explanation:

For the answer we must know that the x-axis is the real axis and the y-axis is the imaginary axis. We can see that we have gone 2 spaces in the x-axis or real direction and -3 spaces in the y-axis or imaginary direction to give us the answer 2-3i. 

Example Question #8 : How To Graph Complex Numbers

Find,

 

Possible Answers:

Correct answer:

Explanation:

The definition of Absolute Value on a coordinate plane is the distance from the origin to the point. 

When graphing this complex number, you would go 3 spaces right (real axis is the x-axis) and 4 spaces down (the imaginary axis is the y-axis). 

This forms a right triangle with legs of 3 and 4. 

To solve, plug in each directional value into the Pythagorean Theorem. 

 

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