Technology Guide--The Technology Inside Matters
Significance Arithmetic

Significance arithmetic, a basis for Mathematica's high-precision arithmetic, is a powerful technique that offers many advantages over fixed precision (such as the floating-point or integer arithmetic used by purely numerical technical software).

It not only keeps track of numerical results, but also uses error propagation to track their accuracy. In this way, numerical computations can carry accuracy and precision information with them, returning in the end a numerical quantity together with its estimated uncertainty.

This means that with Mathematica, unlike other software, you always get both a numerical result and the knowledge of to how many digits it is correct.

These two simulations of light bouncing among cylindrical mirrors show the importance of using significance-arithmetic. Compare the machine-precision (purple) and significance-arithmetic (yellow) paths. After fewer than 20 reflections, the purple path is highly inaccurate. For the orange path, Mathematica was started with 120 digits of precision. Because Mathematica uses significance arithmetic throughout, it was able to report automatically that 62 digits of precision remained at the end.

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