How do programming languages handle huge number arithmetic

Question

For a computer working with a 64 bit processor, the largest number that it can handle would be 2⁶⁴ = 18,446,744,073,709,551,616. How does programming languages, say J

Answer 1

There are lots of specialized techniques for doing calculations on numbers larger than the register size. Some of them are outlined in this wikipedia article on arbitrary precision arithmetic

Low level languages, like C and C++, leave large number calculations to the library of your choice. One notable one is the GNU Multi-Precision library. High level languages like Python, and others, integrate this into the core of the language, so normal numbers and very large numbers are identical to the programmer.

Answer 2

Think of one machine-size number as a digit and apply the algorithm for multi-digit multiplication from primary school. Then you don't need to keep the whole numbers in registers, just the digits as they are worked on.

Answer 3

You assume the wrong thing. The biggest number it can handle in a single register is a 64-bits number. However, with some smart programming techniques, you could just combined a few dozens of those 64-bits numbers in a row to generate a huge 6400 bit number and use that to do more calculations. It's just not as fast as having the number fit in one register.

Even the old 8 and 16 bits processors used this trick, where they would just let the number overflow to other registers. It makes the math more complex but it doesn't put an end to the possibilities.

However, such high-precision math is extremely unusual. Even if you want to calculate the whole national debt of the USA and store the outcome in Zimbabwean Dollars, a 64-bits integer would still be big enough, I think. It's definitely big enough to contain the amount of my savings account, though.