An Argand diagram is a visual representation used to plot complex numbers. It's essentially a modified Cartesian coordinate system where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.
To sketch an Argand diagram and plot a complex number, follow these steps:
- Draw the Axes:
- Draw a horizontal line. This is the real axis. Label it appropriately (e.g., "Real Axis" or "Re").
- Draw a vertical line perpendicular to the real axis, intersecting at the origin (0,0). This is the imaginary axis. Label it (e.g., "Imaginary Axis" or "Im").
- Mark the origin (0) where the axes cross.
- Scale the Axes:
- Add a numerical scale to both axes, similar to a standard graph. Positive numbers go to the right on the real axis and upwards on the imaginary axis. Negative numbers go to the left on the real axis and downwards on the imaginary axis.
- Plot the Complex Number (a + bi):
- A complex number is typically written in the form a + bi, where a is the real part and b is the imaginary part.
- Locate the value of the real part (a) on the real (horizontal) axis.
- Locate the value of the imaginary part (b) on the imaginary (vertical) axis. Remember, as mentioned in the reference regarding "2 + i", the imaginary component being "one" means it has "1i". So, for a + bi, you look for the value b on the imaginary axis.
- Imagine drawing a vertical line from the point a on the real axis and a horizontal line from the point b on the imaginary axis.
- The point where these two lines intersect represents the complex number a + bi.
- Mark the Point:
- Place a dot or a cross at the intersection point.
- Often, a line (a vector) is drawn from the origin (0,0) to this point to represent the complex number visually.
Example: Plotting 2 + i
Let's sketch an Argand diagram and plot the complex number 2 + i (which is equivalent to 2 + 1i), as discussed in the reference.
- Here, the real part (a) is 2.
- The imaginary part (b) is 1 (since it's 1i).
Using the steps above:
- Draw the real and imaginary axes.
- Find 2 on the real axis.
- Find 1 on the imaginary axis.
- The point where the vertical line from 2 on the real axis meets the horizontal line from 1 on the imaginary axis is the location of the complex number 2 + i.
This process allows you to visualize any complex number as a point (or a vector) in a two-dimensional plane.
Summary Table
| Component | Value for 2 + i | Axis | Location on Argand Diagram |
|---|---|---|---|
| Real Part (a) | 2 | Real Axis | Horizontal position at 2 |
| Imaginary Part (b) | 1 | Imaginary Axis | Vertical position at 1 |
By following these steps, you can effectively sketch an Argand diagram and plot any complex number.
