This is a free lesson from our course in Algebra II
In this section you'll explore the concepts from plotting complex numbers in the complex plane. You will also find
here some examples with solution, and the instructor explains all that with the
help of audio, video presentation and in own hand writing. A complex number z
can be expressed as:
z = a
+ ib
where, a and b are real numbers and i =
1
is the imaginary unit.This complex
number z = a
+ ib, can be associated with the ordered pair (a, b), and this is associated with the point whose x coordinate is a, and whose y coordinate is b.
(More text below video...)
(Continued from above) This allows you to associate every complex number
with a point in the xy plane. Similarly, every point in the xy plane is associated with an ordered pair of numbers (a,b), and you can associate this ordered
pair with the complex number z = a
+ ib. Thus, every complex number is associated with a point in the xy plane, and every point in the xy plane is associated with a complex number. This allows you to represent every complex number as a point in the xy plane, and viceversa. When you do this, you call the xy plane the complex plane.
On this plane the real numbers are plotted along the horizontal axis (the
axis of real) and pure imaginaries are plotted along the vertical axis (the axis
of imaginaries). Other points in the plane must represent numbers that are part
real and part imaginary i.e. complex numbers.
E.g. To plot the point 3 + 2i (number made up of the real number 3 and
the imaginary number 2i), measure along the real axis in a positive direction
3 units, turn through one right angle and measure 2 units up, parallel the imaginary
axis as shown in figure below.
The other way to represent complex number can be in the polar form. For example, plot the complex number z = x+iy =
r (cos
+ isin).
Here r =
(x^{2}+y^{2}), cos =x/r, sin =y/r
and tan
= y/x. It can be plotted as shown in figure below.
A plane in which we represent a complex number geometrically is known as argand
plane and points plotted on the argand plane is known as argand diagram.
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