Method of successive substitution Method of successive substitution In modular arithmetic, the method of successive substitution is a method of solving problems of simultaneous congruences by using the definition of the congruence equation. It is commonly applied in cases where the conditions of the Chinese remainder theorem are not satisfied. There is also an unrelated numerical-analysis method of successive substitution, a randomized algorithm used for root finding, an application of fixed-point iteration. The method of successive substitution is also known as back substitution. Consider the simple set of simultaneous congruences Now, for "x" ≡ 3 (mod 4) to be true, "x" = 3 +
How many angels can dance on the head of a pin? about how many could perch at the same time on the head of a pin"". Dorothy L. Sayers argued that the question was "simply a debating exercise" and that the answer "usually adjudged correct" was stated as, "Angels are pure intelligences, not material, but limited, so that they have location in space, but not extension." Sayers compares the question to that of how many people's thoughts can be concentrated upon a particular pin at the same time. She concludes that infinitely many angels can be located on the head of a pin, since they do not occupy any space there: