Trigonometry Over Arbitrary Fields
Rational trigonometry makes it possible to work in almost any mathematical field (fields of characteristic '2' are excluded for technical reasons) whether finite or infinite. The real numbers are not considered a true algebraic field and rational numbers fulfil their role in relation to a linear continuum. Where the output of a calculation would be the root of a rational number (an algebraic number) it can be added as a discreet element (extending the field) and does not require further evaluation: all results having 'exact' expressions.
Over finite fields, the 'plane' is actually a torus, corresponding to the elements of the cartesian product of ordered pairs, with opposite edges identified. An individual 'point' corresponds to one of these elements and a 'line' (which now 'wraps around' this region) corresponds to an initial point plus all exact multiples of the 'vector' (say '2 over and 1 up') giving the line its direction or slope.
Read more about this topic: Rational Trigonometry
Famous quotes containing the words arbitrary and/or fields:
“A state that denies its citizens their basic rights becomes a danger to its neighbors as well: internal arbitrary rule will be reflected in arbitrary external relations. The suppression of public opinion, the abolition of public competition for power and its public exercise opens the way for the state power to arm itself in any way it sees fit.... A state that does not hesitate to lie to its own people will not hesitate to lie to other states.”
—Václav Havel (b. 1936)
“Smart lad, to slip betimes away
From fields where glory does not stay,
And early though the laurel grows
It withers quicker than the rose.”
—A.E. (Alfred Edward)