An Argand diagram, named after French mathematician Jean-Robert Argand, is a graphical representation of complex numbers in the complex plane. It is a two-dimensional coordinate system in which the real part of a complex number is represented by the horizontal axis, and the imaginary part is represented by the vertical axis.
A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. The real part, a, is the horizontal coordinate on the Argand diagram, and the imaginary part, b, is the vertical coordinate.
For example, the complex number 3 + 4i can be plotted on an Argand diagram at the point (3,4). This point is 3 units to the right of the origin and 4 units above the origin. The origin, (0,0), is the point where the real and imaginary axes intersect.
One useful feature of the Argand diagram is that it allows us to visualize the geometric representation of complex numbers. Every complex number corresponds to a point on the diagram, and the distance between two points represents the magnitude of their difference. This distance is called the Euclidean distance, and it is given by the formula:
distance = ?((a2 – b2) + (c2 – d2))
where (a,b) and (c,d) are the coordinates of the two points.
For example, consider the complex numbers 2 + 3i and -1 + 4i. These two numbers can be plotted on the Argand diagram at the points (2,3) and (-1,4), respectively. The distance between these two points is:
distance = ?((2 – (-1))2 + (3 – 4)2)
= ?((3)2 + (-1)2)
= ?(9 + 1)
= ?10
This distance represents the magnitude of the difference between the two complex numbers.
Now let’s look at some examples of complex numbers plotted on the Argand diagram.
Example 1: The complex number 2 + 3i is plotted at the point (2,3).
Example 2: The complex number -1 + 4i is plotted at the point (-1,4).
Example 3: The complex number -2 – 3i is plotted at the point (-2,-3).
Example 4: The complex number 4 – 5i is plotted at the point (4,-5).
Example 5: The complex number 0 + 5i is plotted at the point (0,5).
Now let’s test your knowledge with a quiz.
Quiz:
- What is an Argand diagram used for? a) Plotting complex numbers b) Solving equations c) Graphing functions
- What does the real part of a complex number correspond to on an Argand diagram? a) The horizontal axis b) The vertical axis c) The origin
- What does the imaginary part of a complex number correspond to on an Argand diagram? a) The horizontal axis b) The vertical axis c) The origin
- What is the formula for the Euclidean distance between two points on an Argand diagram? a) distance = ?(a2 + b2) b) distance = ?((a2 – b2) + (c2 – d2)) c) distance = a2 + b2
