MATH1110 Lecture Notes - Lecture 6: Global Positioning System, Analytic Geometry, Bisection
18 Jun 2018
School
Department
Course
Professor
MATH1110: Mathematics 1
Complex Numbers (IV): Regions in the
Complex Plane
Complex equations and inequalities can be used to represent many dif-
ferent kinds of regions, geometric figures and curves in the complex plane.
Alternatively, there are geometrical interpretations to many complex equa-
tions and inequalities. The following examples are just a small sample of
what is possible.
Example 1. Im(z)≥0
Thinking geometrically, we see that this equation will describe the half
plane above the real axis in the complex plane since all of those complex
numbers (and only those complex numbers) have an imaginary part which is
positive.
While it is not necessary in this case, sometimes it is useful to work
algebraically. To this end let z=x+iy and hence let the axes in the complex
plane be labelled xand y(as is the convention in coordinate geometry).
Then on substituting in the complex equation/inequality we can produce an
equation / inequality in xand ythat may be more familiar to us. In this
example we obtain
Im(z)≥0
Im(x+iy)≥0
y≥0
Page 1
Document Summary
Complex equations and inequalities can be used to represent many dif- ferent kinds of regions, geometric gures and curves in the complex plane. Alternatively, there are geometrical interpretations to many complex equa- tions and inequalities. The following examples are just a small sample of what is possible. Thinking geometrically, we see that this equation will describe the half plane above the real axis in the complex plane since all of those complex numbers (and only those complex numbers) have an imaginary part which is positive. While it is not necessary in this case, sometimes it is useful to work algebraically. To this end let z = x+iy and hence let the axes in the complex plane be labelled x and y (as is the convention in coordinate geometry). Then on substituting in the complex equation/inequality we can produce an equation / inequality in x and y that may be more familiar to us. Im(x + iy) 0 y 0.
