2.2.2 Argand plane

. . Argand plane A complex number z iy is uniquely determined by an ordered pair of real numbers x y The numbers − i , and − i are equivalent to and respectively. In this way we are able to associate a complex number z iy with a point x y in a coordinate plane. If we consider x axis as real axis and y axis as imaginary axis to represent a complex number, then the xy -plane is called complex plane or Argand plane. It is named after the Swiss mathematician Jean Argand ( – ). A complex number is represented not only by a point, but also by a position vector pointing from the origin to the point. The number, the point, and the vector will all be denoted by the same letter z . As usual we identify all vectors which can be obtained from each other by parallel displacements. In this chapter,  denotes the set of all complex numbers. Geometrically, a complex number can be viewed as either a point in  or a vector in the Argand plane. Fig. . Fig. . Fig. . Illustration . Here are some complex numbers: + i , i , , −− i , cos and + i . Some of them are plotted in Argand plane. O O β Re Im Complex number by a position vector pointing from the origin to the point β β Re Im Complex number as a point β Re Im Complex number as a vector O - - The diagram below shows k z for Re Im k = k=