Graph in the Complex Number Plane - Algebra-2

1. Fundamental Concepts

The complex number plane (also known as the Argand plane) is a coordinate system used to graph complex numbers. Similar to the Cartesian plane, it has two axes, but with distinct meanings:
- The horizontal axis represents the real part of the complex number (corresponding to a in ).
- The vertical axis represents the imaginary part of the complex number (corresponding to b in , where the axis is labeled with i).
A complex number is graphed as a point in this plane, where the first coordinate is the real part and the second is the coefficient of the imaginary unit i.

Coordinate mapping: Each complex number $a + bi$ is uniquely mapped to a point $(a, b)$ in the complex plane. For example:
- A real number a (i.e., ) maps to on the horizontal (real) axis.
- A pure imaginary number bi (i.e., ) maps to on the vertical (imaginary) axis.
Visual representation: The graph visually shows the relationship between the real and imaginary components of a complex number, making it easier to understand the structure of complex numbers.

Example 1: Graph the complex number in the complex plane.
Solution: The real part is 5 and the imaginary part coefficient is 3, so it is plotted as the point .
Example 2: Graph the complex number in the complex plane.
Solution: can be written as , so the real part is 0 and the imaginary part coefficient is . It is plotted as the point .
Example 3: Graph the complex number in the complex plane.
Solution: can be written as , so the real part is and the imaginary part coefficient is 0. It is plotted as the point .

Step 1: Identify components: For a given complex number , separate the real part a and the coefficient of the imaginary part b (note the sign of b).
Step 2: Locate coordinates: In the complex plane, move a units along the horizontal (real) axis (right if , left if ) and b units along the vertical (imaginary) axis (up if , down if ).
Step 3: Plot the point: Mark the intersection point of the two movements as the graph of the complex number.