The important point is that a TM is capable to calculate everything that is considered to be decidable. It is _the_ computation machine in theoretical CS and different models (eg lambda calculus) are equivalent in computational power. Implementations are a proof-of-concept to show/visualize their power and can be seen as compilers for basic languages.
Their exact mathematical description differs from source to source but the interesting thing is that just a small program (the TM) is capable of calculating everything (efficiency is not the question here).
Their exact mathematical description differs from source to source but the interesting thing is that just a small program (the TM) is capable of calculating everything (efficiency is not the question here).