Question Corner and Discussion Area

What kind of relationship can be made between geometry and Imaginary Numbers?The set of complex numbers form a plane; that is, the complex number a + bi corresponds to a point with coordinates (a,b).

Therefore, every complex number is a point on a plane. Equations that single out certain complex numbers over others correspond to various geometric figures. For example, the set of complex numbers whose magnitude is 1 forms a circle. The set of complex numbers whose imaginary part is 17 forms a line. And so on.

In higher dimensions, if you take equations involving several complex
variables, the solution sets are geometric objects of various dimensions.
For example, an equation such as z^2 = w^3 + 1 (where z and
w are complex numbers) describes an interesting
surface sitting inside 4-dimensional space. Understanding the
geometric properties of surfaces like these, and their higher-dimensional
analogues, is the aim of an important field of mathematics
known as *Algebraic Geometry*.

You can also study the geometry of shapes given by equations involving purely real variables. However, it turns out that there is a far greater richness of structure in the complex case (where imaginary numbers are allowed), and many more important theorems that are true, than in the case of objects defined by equations involving real-only variables.

I don't know if this exactly answers what you were asking or not, but I hope it does.

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Navigation Panel:

Go backward to More Complex Number Questions

Go up to Question Corner Index

Go forward to Patterns in the Towers of Hanoi Solution

Switch to graphical version (better pictures & formulas)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network Home Page