, is modified to become , and the Taylor series for the sine of an angle becomes:The capitalized function is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while is the traditional function on pure numbers which assumes its argument is in radians. can be denoted if it is clear that the complete form is meant. SI can be considered relative to this framework as a natural unit system where the equation is assumed to hold, or similarly, . This radian convention allows the omission of in mathematical formulas. A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use. Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal. For example, the Boost units library defines angle units with a plane_angle dimension, and Mathematica's unit system similarly considers angles to have an angle dimension. Conversions Between degrees As stated, one radian is equal to . Thus, to convert from radians to degrees, multiply by . For example: Conversely, to convert from degrees to radians, multiply by . For example: Radians can be converted to turns (one turn is the angle corresponding to a revolution) by dividing the number of radians by 2. Between gradians radians equals one turn, which is by definition 400 gradians (400 gons or 400g). To convert from radians to gradians multiply by , and to convert from gradians to radians multiply by . For example, Usage Mathematics In calculus and most other branches of mathematics beyond practical geometry, angles are measured in radians. This is because radians have a mathematical naturalness that leads to a more elegant formulation of some important results. Results in analysis involving trigonometric functions can be elegantly stated when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit formula which is the basis of many other identities in mathematics, including Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the differential equation , the evaluation of the integral and so on). In all such cases, it is found that the arguments to the functions are most naturally