Graphing Complex Numbers
A complex number is the sum of a real number and an imaginary number; that is, a complex number is of the form \(x+iy\) and is usually represented by \(z\).
Here, both \(x\) and \(y\) are real numbers.
- \(x\) is called the real part.
- \(y\) is called the imaginary part.
- \(iy\) is an imaginary number.
In this mini-lesson, we will explore the world of graphing complex numbers on a complex plane. We will learn plotting complex numbers on a complex plane, complex plane grapher, graphing imaginary numbers, and discover other interesting aspects of it.
You can check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
What Is a Complex Plane?
Every complex number can be represented by a point in the XY-plane.
The complex number \(x+iy\) indicates the point \((x,y)\) in the XY-plane.
The plane where a complex number is assigned to each of its points is called a complex plane.
A complex plane is also called an argand plane.
How to Plot Complex Numbers as Points on a Complex Plane?
Graphing Complex Numbers on a Complex Plane Grapher
In this section, you will learn plotting complex numbers on a complex plane.
Steps to Plot Complex Numbers
Follow the steps mentioned below to plot complex numbers on the complex plane.
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Determine the real part and imaginary part of the given complex number. For example, for \(z=x+iy\), the real part is \(x\) and the imaginary part is \(y\).
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Form an ordered pair where the first element is the real part and the second element is the imaginary part. For example, for \(z=x+iy\), the ordered pair is \((x,y)\)
- Plot the point \((x,y)\) on the plane.
Examples
Let us consider the complex number \(z=3+4i\).
The real part is 3 and the imaginary part is 4.
So, the ordered pair is (3, 4).
The complex number \(z=3+4i\) is represented in the graph below.
- We can plot real numbers on a complex plane.
- On the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
- The conjugate of a complex number \(z=x+iy\) is the reflection of the point \((x,y)\) about the \(x\)-axis on the complex plane.
- Multiplying complex numbers with its conjugate, gives a simple real number value.
Solved Examples
| Example 1 |
Ms Dolma asked her students to classify the following complex numbers on the basis of the quadrant in which they lie.
Can you classify them?
Solution
Let's find the points corresponding to each complex number.
\[ \begin{aligned} A &=3+7i\;\;\;\;\;\;\; \rightarrow (3,7)\\[0.2cm] B&=6-i\;\;\;\;\;\;\;\; \rightarrow (6,-1)\\[0.2cm] C&=-2-4i\;\;\; \rightarrow (-2,-4)\\[0.2cm] D&=-5+2i\;\;\; \rightarrow (-5, 2)\end{aligned} \]
Let's plot the given complex numbers on a complex plane.
So, the given complex numbers can be classified as:
| Quadrant 1 | 3+7i |
|---|---|
| Quadrant 2 | -5+2i |
| Quadrant 3 | -2-4i |
| Quadrant 4 | 6-i |
| So, the complex numbers are classified. |
| Example 2 |
Jenny says to Jolly that the points \(2-i, i\), and \(2+3i\) form the vertices of a right-angled triangle.
Do you think she is correct? Give reasons for your answer.
Solution
Let us assume the given points to be:
\[ \begin{aligned} A &=2-i\;\;\;\; \rightarrow (2,-1)\\[0.2cm] B&=i\;\;\;\;\;\;\;\;\;\; \rightarrow (0,1)\\[0.2cm] C&=2+3i\;\;\; \rightarrow (2,3)\\[0.2cm] \end{aligned} \]
We will find the distance between every two points using the distance formula.
\[\begin{aligned} AB&\! =\! \sqrt{(0-2)^2+(1-(-1)^2}\!\\&=\! \sqrt{(-2)^2+(2)^2}\!=\! \sqrt{4+4}\!=\! \sqrt{8}\\ BC &\!=\! \sqrt{(2-0)^2+(3-1)^2}\!\\&=\! \sqrt{(2)^2+(2)^2}\!=\! \sqrt{4+4}\!=\! \sqrt{8}\\ CA &\!=\! \sqrt{(2-2)^2+(3-(-1))^2}\! \\&=\! \sqrt{0^2+4^2}\!=\! \sqrt{16}\!=\!4\end{aligned} \]
Now that we know the lengths of all three sides,
\[ \begin{aligned} AB^2+BC^2 &= CA^2 \\[0.3cm] (\sqrt{8})^2 +(\sqrt{8})^2 &= 4^2 \\[0.3cm] 8+8&=16\\[0.3cm] 16&=16 \end{aligned} \]
Thus, \(A, B,\) and \(C\) satisfy Pythagoras theorem.
So, \(\Delta ABC\) is a right-angled triangle.
We can prove the same by marking all the coordinates on a graph:
| \( \therefore\) The given points form a right-angled triangle |
| Example 3 |
The town of Lotto is mapped on a complex plane as shown.
The chocolate house is located at the point \(3+7i\) and the cake factory is located at the point \(-1-3i\).
The main entrance of the town is located halfway between the chocolate house and the cake factory.
Can you calculate the point of the main entrance?
Solution
The complex numbers \(3+7i\) and \(-1-3i\) correspond to the points \((3, 7)\) and \((-1, -3)\) on the complex plane.
To calculate the point of the main entrance, we have to calculate the midpoint of (3, 7) and (-1, -3).
Let \(x_{1}=3\), \(y_{1}=7\), \(x_{2}=-1\), and \(y_{2}=-3\)
The coordinates of the main entrance are calculated as:
\(\begin{align}\!\!\left(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\right)\!\!&=\!\!\left(\dfrac{3+(-1)}{2}, \dfrac{7+(-3)}{2}\right)\\&=\!\!\left(\dfrac{2}{2}, \dfrac{4}{2}\right)\\&=\!\left(1, 2\right)\end{align}\)
| \(\therefore\) The main entrance is located at the point \(1+2i\). |
| 1. | The diameter of a circle has endpoints 2-3i and -6+5i. |
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| Can you find the coordinates of the center of this circle? |
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The mini-lesson targeted the fascinating concept of Graphing Complex Numbers. The math journey around Graphing Complex Numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.
We hope you learned plotting complex numbers on a complex plane and graphing imaginary numbers in this lesson on graphing complex numbers.
About Cuemath
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FAQs on Graphing complex numbers
1. What is a complex plane used for?
A complex plane is used to plot complex numbers on a graph.
2. How do you plot a complex number on a complex plane?
Follow the steps mentioned below to plot complex numbers on a complex plane.
- Determine the real part and imaginary part of the given complex number. For example, for \(z=x+iy\), the real part is \(x\) and the imaginary part is \(y\).
- Form an ordered pair where the first element is the real part and the second element is the imaginary part. For example, for \(z=x+iy\), the ordered pair is \((x,y)\)
- Plot the point \((x,y)\) on the plane.
3. What is a complex graph?
A complex graph is a graph where complex numbers are represented.
4. How do you graph complex numbers?
Follow the steps mentioned below to plot complex numbers on a complex plane.
- Determine the real part and imaginary part of the given complex number. For example, for \(z=x+iy\), the real part is \(x\) and the imaginary part is \(y\).
- Form an ordered pair where the first element is the real part and the second element is the imaginary part. For example, for \(z=x+iy\), the ordered pair is \((x,y)\)
- Plot the point \((x,y)\) on the plane.
5. Where are complex numbers used in real life?
Complex numbers are used to solve quadratic equations. For example, the solution of \(x^2+1=0\) is \(z=i\).
6. What is Z* in complex numbers?
\(z*\) in complex numbers is the conjugate of the complex number \(z=x+iy\) given by \(z*=x-iy\).
7. How do you graph i on a complex plane?
The number \(i\) corresponds to point \(0,1\) on the graph.
8. Where is the real part of the complex number plotted on the graph?
The real part of the complex number is plotted on the horizontal axis in the graph.
9. Where is the imaginary part of the complex number plotted on the graph?
The imaginary part of the complex number is plotted on the vertical axis in the graph.
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