Currently I am using XaoS to view fractals. However, when zoomed in too much, it gets pixelized and I can't zoom in any more.
How can I use this program, or some other program, in a way which allows me to zoom in forever in a fractal?
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This is probably better suited to Math, but in this form, the question actually does apply to computers as well in a way.
Computers represent numbers with bits. Fractals are created with real numbers (numbers with fractions/decimal places). Unlike integers which are nice and straight forward, real numbers are a little trickier to implement in computers. Regardless, just like integers have limits on the range of number that can be represented (n number of bits allows for 0-2n positive numbers), floating point numbers also have limits to the numbers that can be represented.
Because of the limits, there is only so much detail that a fractal viewer can show. At some point, the program can no longer calculate precise enough numbers. As a simplification, lets say that the maximum precision that the program can represent is 10 decimal places. After that, numbers just get rounded to the nearest 10 decimal places, regardless of the calculation.
Fractals start out fairly simple, but quickly explode to real numbers with many decimal places after just a few levels of zoom. That is why when you zoom in for a bit, the program can no longer provide the precision (number of decimal places) required to provide accurate, detailed edges to the fractal and ends up just rounding it out both mathematically, which depending on the program may result in pixelation or rounded, non-detailed/fractal edges (some programs just take it a step further and limit the amount you can zoom in the first place).
The limitations depend on both the hardware and software. You can improve the quality by using a system with 64-bit hardware, lots of memory, and software that can take advantage of them.