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The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations.
Begin with a cube. Divide every face of the cube into nine squares in a similar manner to a Rubik's Cube. This sub-divides the cube into 27 smaller cubes. Remove the smaller cube in the middle of each face, and remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ...
God's algorithm is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, [1] but which can also be applied to other combinatorial puzzles and mathematical games. [2] It refers to any algorithm which produces a solution having the fewest possible moves (i.e., the solver should not require any more than this number).
The resulting algorithm for solving Pell's equation is more efficient than the continued fraction method, though it still takes more than polynomial time. Under the assumption of the generalized Riemann hypothesis , it can be shown to take time exp O ( log N ⋅ log log N ) , {\displaystyle \exp O\left({\sqrt {\log N\cdot \log ...
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The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.