increases to positive angles for ccw rotations, whereas in navigation (bearing, heading) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations. Uniqueness of polar coordinates Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point (r, φ) can be expressed with an infinite number of different polar coordinates and , where n is an arbitrary integer. Moreover, the pole itself can be expressed as (0, φ) for any angle φ. Where a unique representation is needed for any point besides the pole, it is usual to limit r to positive numbers () and φ to either the interval or the interval , which in radians are or . Another convention, in reference to the usual codomain of the arctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to . In all cases a unique azimuth for the pole (r = 0) must be chosen, e.g., φ = 0. Converting between polar and Cartesian coordinates The polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine: The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−, ] by: where hypot is the Pythagorean sum and atan2 is a common variation on the arctangent function defined as If r is calculated first as above, then this formula for φ may be stated more simply using the arccosine function: Complex numbers Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number z can be represented in rectangular form as where i is the imaginary unit, or can alternatively be written in polar form as and from there, by Euler's formula, as where e is Euler's number, and φ, expressed in radians, is the principal value of the complex number function arg