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If the coefficients of the polynomial d_nx^n+d_(n-1)x^(n-1)+...+d_0=0 (1) are specified to be integers, then rational roots must have a numerator which is a factor of d_0 and ...

By a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge. For example, S = 1-1/2+1/3-1/4+1/5+... ...

In the usual diagram of inclusion homomorphisms, if the upper two maps are injective, then so are the other two. More formally, consider a space X which is expressible as the ...

In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, ...

A sphere is defined as the set of all points in three-dimensional Euclidean space R^3 that are located at a distance r (the "radius") from a given point (the "center"). Twice ...

A short mnemonic for remembering the first seven decimal digits of pi is "How I wish I could calculate pi" (C. Heckman, pers. comm., Feb. 3, 2005). Eight digits are given by ...

A quantitative measure of the simplicity of a geometric construction which reduces geometric constructions to five steps. It was devised by È. Lemoine. S_1 Place a ...

Analytic continuation (sometimes called simply "continuation") provides a way of extending the domain over which a complex function is defined. The most common application is ...

product_(k=1)^(infty)(1-x^k) = sum_(k=-infty)^(infty)(-1)^kx^(k(3k+1)/2) (1) = 1+sum_(k=1)^(infty)(-1)^k[x^(k(3k-1)/2)+x^(k(3k+1)/2)] (2) = (x)_infty (3) = ...

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an ...