 This idea extends to other operations such as multiplication and division, as with fields. A particularly useful case of such a generalization is given by the use of complex numbers in Cartesian geometry. When \(x\) and \(y\) range over the field of reals, \(x^2 +y^2 =1\) describes the ordinary Euclidean circle in two dimensions, but when the variables range over the complex numbers this equation describes the complex counterpart of the circle, visualizable as a two-dimensional surface embedded in four real dimensions (regarding the complex plane as having two real dimensions). Or if the variables range over the integers mod 7, which form a field under the usual arithmetic operations mod 7, the circle consists of eight points, namely \((\pm 1, 0), (0, \pm 1)\), and \((\pm 2, \pm 2)\). Certain theorems about the Euclidean circle provable purely algebraically remain provable about these other kinds of circles because all the equations on which the proof depends continue to hold in these other fields, for example the theorem that a line intersects a circle in at most two points.