What is meant by a machine number in the Mathematica documentation? What is the difference between machine-precision and fixed-point precision? What is arbitrary precision?
This is not an answer. But I don't believe we should close this question as "easily found in the documentation".
Numerics in Mathematica is an extremely complicated and mostly undocumented subject, where several mathematical concepts run up against each other in subtle and non-trivial ways. I have been thinking for some time that we ought to address this properly. Here is an outline for how I thought this could be approached.
There are three main headings, each containing enough material for several answers:
The formalist's view: floating-point numbers as rationals
- Decimal vs. binary digits
- IEEE issues:
NaN; rounding modes; LAPACK vs. C definition of
Mathematica's view: floating-point numbers as distributions
- The nature of the distribution: interval arithmetic versus Gaussian error propagation
- Significance arithmetic and error propagation
Practicalities: floating-point numbers as a model of the reals
- Dealing with numerically unstable functions
- Adaptive-precision evaluation;
PossibleZeroQand associated system options
Anyone should feel free to add to these lists of topics in case I forgot anything. There are answers covering some of them already, but a lot of it is not widely known. I propose that, as a collaborative effort, we could address this question comprehensively (it's too much work for me to do by myself). This thread seems like a golden opportunity to do so.