applied to x + iy. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the and angle notations: For the operations of multiplication, division, exponentiation, and root extraction of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation: Multiplication Division Exponentiation (De Moivre's formula) Root Extraction (Principal root) Polar equation of a curve The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r. Note that, in contrast to Cartesian coordinates, the independent variable φ is the second entry in the ordered pair. Different forms of symmetry can be deduced from the equation of a polar function r: If the curve will be symmetrical about the horizontal (0°/180°) ray; If it will be symmetric about the vertical (90°/270°) ray: If it will be rotationally symmetric by α clockwise and counterclockwise about the pole. Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid. For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve. Circle The general equation for a circle with a center at and radius a is This can be simplified in various ways, to conform to more specific cases, such as the equation for a circle with a center at the pole and radius a. When or the origin lies on the circle, the equation becomes In the general case, the equation can be solved for , giving The solution with a minus sign in front of the square root gives the same curve. Line Radial lines (those running through the pole) are represented by the equation where is the angle of elevation of the line; that is, , where is the slope of the line in the Cartesian coordinate system. The