Binary, Floats, and Modern Computers

Question

I have been reading a lot about floats and computer-processed floating-point operations. The biggest question I see when reading about them is why are they so inaccurate? I understand this is because binary cannot accurately represent all real numbers, so the numbers are rounded to the 'best' approximation.

My question is, knowing this, why do we still use binary as the base for computer operations? Surely using a larger base number than 2 would increase the accuracy of floating-point operations exponentially, would it not?

What are the advantages of using a binary number system for computers as opposed to another base, and has another base ever been tried? Or is it even possible?

Answer 1

First of all: You cannot represent all real numbers even when using say, base 100. But you already know this. Anyway, this means: Inaccuracy will always arise due to 'not being able to represent all real numbers'.

Now lets talk about "what can higher bases bring to you when doing math?": Higher bases bring exactly 'nothing' in terms of precision. Why?

If you want to use base 4, then a 16 digit base 4 number provides exactly 4¹⁶ different values.

But you can get the same number of different values from a 32 digit base 2 number (2³² = 4¹⁶).

As another answer already said: Transistors can either be on or off. So your newly designed base 4 registers need to be an abstraction over (base 2) ON/OFF 'bits'. This means: Use two 'bits' to represent a base 4 digit. But you'll still get exactly 2^N levels by spending N 'bits' (or N/2 base-4 digits). You can only get better accuracy by spending more bits, not by increasing the base. Which base you 'imagine/abstract' your numbers to be in (e.g. like how printf can print these base-2 numbers in base-10) is really just a matter of abstraction and not precision.