Graphing Complex Numbers: Videos & Practice Problems

To plot complex numbers on a complex plane, identify the real and imaginary parts. The real part corresponds to the horizontal axis, while the imaginary part corresponds to the vertical axis. For example, the complex number $3+2i$ is plotted by moving 3 units right and 2 units up. Similarly, $2-4i$ is plotted by moving 2 units right and 4 units down. Understanding this process is essential for visualizing complex numbers effectively.

How To Plot Complex Numbers

Understanding how to plot complex numbers on a graph is essential for visualizing their properties. Complex numbers are expressed in the form x + yi, where x represents the real part and y represents the imaginary part. To plot these numbers, we utilize a complex plane, which consists of a real axis (horizontal) and an imaginary axis (vertical).

When plotting a complex number, the real part indicates the position along the real axis, while the imaginary part indicates the position along the imaginary axis. For example, to plot the complex number 3 + 2i, you would move 3 units to the right on the real axis and 2 units up on the imaginary axis, placing the point at the intersection of these movements.

In another example, for the complex number 2 - 4i, the real part is 2, so you move 2 units to the right. The imaginary part is -4, indicating a movement of 4 units down. Thus, the point is plotted accordingly. If you encounter a complex number like 3i, it can be rewritten as 0 + 3i, showing that the real part is 0 and the imaginary part is 3. This means you stay at the origin on the real axis and move 3 units up on the imaginary axis.

In summary, plotting complex numbers involves identifying the real and imaginary components and translating them into movements on the complex plane. This method is straightforward and mirrors the process of plotting points on a standard x-y graph, making it an accessible concept for students to grasp.

Plot the complex number $z=2-4i$ on the graph below.

• A.
• B.
• C.
• D.

Answer: (B)

Plot the complex number $z=-i$ on the graph below.

• A.
• B.
• C.
• D.

Answer: (C)

Plot the complex number $z=5$ on the graph below.

• A.
• B.
• C.
• D.

Answer: (A)

Here’s what students ask on this topic:

How do you plot complex numbers on a complex plane?

To plot complex numbers on a complex plane, you need to understand the format of complex numbers, which is $x+yi$. Here, $x$ is the real part and $y$ is the imaginary part. The complex plane consists of a real axis (horizontal) and an imaginary axis (vertical). To plot a complex number, move $x$ units along the real axis and $y$ units along the imaginary axis. For example, for the complex number $3+2i$, move 3 units right on the real axis and 2 units up on the imaginary axis. This process is similar to plotting points on a standard x-y graph.

What is the difference between a complex plane and a standard x-y graph?

The main difference between a complex plane and a standard x-y graph lies in the axes. A standard x-y graph has an x-axis and a y-axis, both representing real numbers. In contrast, a complex plane has a real axis (horizontal) and an imaginary axis (vertical). The real axis corresponds to the real part of a complex number, while the imaginary axis corresponds to the imaginary part. When plotting complex numbers, the real part determines movement along the real axis, and the imaginary part determines movement along the imaginary axis. This allows complex numbers to be represented visually, similar to plotting points on a standard graph.

How do you plot a complex number with no real part?

When plotting a complex number with no real part, such as $\mathrm{3i}$, you can consider it as $0+3i$. This means the real part is zero, and the imaginary part is 3. On the complex plane, you start at the origin and move 3 units up along the imaginary axis. Since there is no real part, you do not move left or right on the real axis. This results in a point located directly on the imaginary axis, at the position corresponding to the imaginary part's value.

What is the significance of the real and imaginary axes in a complex plane?

The real and imaginary axes in a complex plane are crucial for visualizing complex numbers. The real axis represents the real part of a complex number, while the imaginary axis represents the imaginary part. This allows complex numbers to be plotted as points on the plane, with the real part determining horizontal movement and the imaginary part determining vertical movement. This visualization helps in understanding operations involving complex numbers, such as addition, subtraction, and multiplication, by providing a geometric interpretation. The complex plane thus serves as a valuable tool for analyzing and comprehending the behavior of complex numbers.

Can complex numbers be plotted similarly to points on a standard graph?

Yes, complex numbers can be plotted similarly to points on a standard graph, but with a slight modification. Instead of using an x-axis and y-axis, a complex plane uses a real axis and an imaginary axis. The real part of the complex number determines movement along the real axis, while the imaginary part determines movement along the imaginary axis. This is analogous to plotting points on a standard graph, where the x-coordinate and y-coordinate determine horizontal and vertical positions, respectively. This similarity makes plotting complex numbers straightforward once you understand the role of the real and imaginary parts.