lorem ipsum (pets)

Pets, needing attention, save you from metastable stagnation. (A famous example of metastable stagnation is referred to by P.B Wodehouse in his remark that he was sitting in his easy chair, trying to summon the energy to get up and go to bed.)
keywords: life, processes

degree angle measure versus radian angle measure

The nature of the angle measure we are dealing with is determined, strictly speaking, not by the angle, but by the function, but the two types of function (that might be separately denoted by, say, degree-sin(x) and radian-sin(x)) are, for the sake of convenience, conflated, and therefore we have to resort to a voice-over to indicate which angle-measure is being employed. Typically, angles in degree-measure are expressed as raw decimal real numbers, whereas angles in radian-measure are expressed in terms of something involving pi (coinciding, of course, at the special value 0), however this is mere convention, not strictly a mathematical rule or result. For example, radian-cos(pi) = -1, but degree-cos(pi) is approximately 1.

the various meanings of ‘counterexample’

1. an instance contradicting a purposed general proposition; Wikipedia has an entry on this notion. Here it is.
2. an example showing that omission of a certain part of the hypothesis of a certain theorem results in a statement that is not a theorem; Wikipedia is fond of this usage. Here is the link to an example of such.
3. a non-fatal anomaly left unaddressed by a stylized fact; Wikipedia has an entry on the notion of ‘stylized fact’. Here it is.
keywords: Logic, Mathematics, terminology, wide-sense, narrow-sense