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Two complex numbers x=a+ib and y=c+id are multiplied as follows: xy = (a+ib)(c+id) (1) = ac+ibc+iad-bd (2) = (ac-bd)+i(ad+bc). (3) In component form, ...

An improper use of the symbol sqrt(-1) for the imaginary unit leads to the apparent proof of a false statement. sqrt(-1) = sqrt(-1) (1) sqrt((-1)/1) = sqrt(1/(-1)) (2) ...

The complex structure of a point x=x_1,x_2 in the plane is defined by the linear map J:R^2->R^2 J(x_1,x_2)=(-x_2,x_1), (1) and corresponds to a counterclockwise rotation by ...

The composite number problem asks if for a given positive integer N there exist positive integers m and n such that N=mn. The complexity of the composite number problem was ...

Every finite group G of order greater than one possesses a finite series of subgroups, called a composition series, such that I<|H_s<|...<|H_2<|H_1<|G, where H_(i+1) is a ...

Let P be the principal (initial investment), r be the annual compounded rate, i^((n)) the "nominal rate," n be the number of times interest is compounded per year (i.e., the ...

Any computable function can be incorporated into a program using while-loops (i.e., "while something is true, do something else"). For-loops (which have a fixed iteration ...

A number which can be computed to any number of digits desired by a Turing machine. Surprisingly, most irrationals are not computable numbers!

A computation is an operation that begins with some initial conditions and gives an output which follows from a definite set of rules. The most common example are ...

Computation time (also called "running time") is the length of time required to perform a computational process. Representation a computation as a sequence of rule ...