If programming languages like FORTRAN can't work with more than a dozen decimal places, how can Wolfram Alpha return results with as many decimal places as desired? Does it use special algorithms, or is it just that they allow their computers to store numbers in a different number format, lots of bytes long?
The Wolfram Language internally uses several highly optimized number representations, but nevertheless provides a uniform interface for digit and precision manipulation, while allowing numerical analysts to study representation details when desired.
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While most compilers of languages such as FORTRAN, C, Java, etc... use the machine number representation, it is possible to build your own "type" (e.g., a class in Java) that will support the precision you would like because it is independent of the machine representation.
Does it use special algorithms, or is it just that they allow their computers to store numbers in a different number format, lots of bytes long?
Their computers are most likely just standard Intel/AMD server machines without any special numeric precision abilities. They use different representation that is defined programically.