Furthermore, the first (PC1 x PC2) and second Cartesian planes
(PC1 x PC3), which described the variability of the data in further detail (63.68% and 56.36%, respectively), enabled the placement of the clusters in relation to the different quality parameters of the milk.
Hence the horizontal plane contains the Re(x) ([equivalent to] G) and Im(x) ([equivalent to] H) axes, thus forming the Argand plane, whilst the vertical plane given by GA or GB represents the Cartesian plane for each surface.
For the specific example considered here, Figures 6 and 7 show one real root and a pair of complex conjugate roots located respectively at (G, H) or (Re(x), Im(x)) = (1, 0), (-2, 1) and (-2, -1) which accords fully with the solution for y = [x.sup.3] + 3[x.sup.2] + x - 5, given by x = 1, -2 [+ or -] i, described at the end of the section entitled Cubic polynomials in the Cartesian plane.
Figure 8(a) shows another instance where the complex conjugate roots are not evident in the Cartesian plane. The shape of this cubic differs from that shown in Figure 1 in that there are no turning points, the PoI is positively inclined to the x-axis, and the function is monotonically increasing with increasing x.
This also leads to the n-th roots of unity, although the location of these n-roots relative to the Cartesian plane is commonly misunderstood (see Stroud (1986, Programme 2, Theory of Equations); he (incorrectly) places the complex conjugate roots in the Cartesian plane at one of the turning points of a cubic equation.) In fact, Equations (1) and (2) are essentially the same, since the possibility of complex solutions should most definitely be entertained for Equation (2).
The aim of this paper is to demonstrate visually the connection between the reduced polynomial y = [x.sup.n] - 1 in the Cartesian plane and the resulting n-roots which invariably appear in the Argand plane.
A simple plot of this equation in the Cartesian plane is shown in Figure 2.
It is understood that the GA and GB planes represent the Cartesian plane for each surface.
This confirms the GA (or GB) plane is in fact the Cartesian plane. It is observed that in the GA plane the locus of points represented by B = 0 is in fact a straight 'nodal' line coincident with the G-axis.
For instance, by extending the solution space of Equation (2) to now include complex numbers, it is evident that the original x-y plot in the Cartesian plane is merely a two-dimensional 'slice' of a much more general three-dimensional surface.
The beauty of this method is that it always places the roots in the complex Argand plane, which is orthogonal to the Cartesian plane
in which the quadratic equation was originally presented.