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Geometry and Imaginary Numbers

Asked by B. Delbecq on November 24, 1996:
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.

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