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The process of adding one more partial quotient to a finite continued fraction is in many ways analogous to this process of "punching a hole" in an interval of real numbers. The size of the "hole" is inversely proportional to the next partial denominator chosen – if the next partial denominator is 1, the gap between successive convergents is ...
This lets us maintain an invariant relation at every step: q × m + r = n, where q is the partially-constructed quotient (above the division bracket) and r the partially-constructed remainder (bottom number below the division bracket).
In mathematics education at the primary school level, chunking (sometimes also called the partial quotients method) is an elementary approach for solving simple division questions by repeated subtraction. It is also known as the hangman method with the addition of a line separating the divisor, dividend, and partial quotients. [1]
For example, if 45 eggs are to be put into 12-egg cartons, then after the first 3 cartons have been filled there are 9 eggs remaining, which only partially fill the 4th carton. The answer to the question "How many cartons are needed to fit 45 eggs?" is 4 cartons, since = + rounds up to 4.
The quotient ring / gives the field of real numbers. [12] This construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by the axiom of choice. It turns out that the maximal ideal respects the order on . Hence the resulting field is an ordered field.
In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme. Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit ...
Forming a partial setoid from a type and a PER is analogous to forming subsets and quotients in classical set-theoretic mathematics. The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive ...
The quotient of a K3 surface by a fixed point free involution is an Enriques surface, and all Enriques surfaces in characteristic other than 2 can be constructed like this. For example, if S is the K3 surface w 4 + x 4 + y 4 + z 4 = 0 and T is the order 4 automorphism taking (w,x,y,z) to (w,ix,–y,–iz) then T 2 has eight fixed points.