Geometrical Representations

  • Walter Ledermann
Part of the Library of Mathematics book series (LIMA)


We have already remarked that the complex number x + iy may be thought of as an ordered pair (x, y) of real numbers. As the reader knows from co-ordinate geometry, such a pair of real numbers may be regarded as the coordinates of a point in a plane. We are thus led to the following geometrical interpretation of complex numbers: choose a plane and furnish it with Cartesian rectangular axes Ox, Oy. Let the number x+iy be represented by the point P=(x,y). In this way we can plot complex numbers as points of a plane (see Fig. 2). This plane (or more correctly such a plane) is often called the Argana plane or Argana diagram or simply the complex plane. Any algebraical proposition between complex numbers can be translated into a geometrical relation between the corresponding points of the Argand plane, and conversely, any relation between points of a plane can, in principle, be regarded as a relation between complex numbers. The reader should, however, keep the logical situation clearly in mind: we have defined complex numbers as algebraical objects obeying certain laws of composition.


Complex Number Geometrical Representation Polar Form Algebraical Object Logical Situation 
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© Walter Ledermann 1962

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  • Walter Ledermann

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