A core activity in exploratory experimental mathematics is recognition of numbers: going backwards from a number to find out how it can be generated.
Mathematica provides tools for recognizing many classes of numbers, including a number of original algorithms.
Element — test whether a number is in a given class
Rationalize — find a rational approximation
RootApproximant — find an approximating algebraic number
LatticeReduce — find reduced basis vectors in a lattice
FunctionExpand — reduce an exact numeric expression to simpler functions
