Question

I know this is a bit hypothetical but I am wondering why no language I know does it.

For example, you want to store 1/3. Give the programmer an option to specify it as 1/3, and store 1 and 3. Something like

struct float {
    int numerator;
    int denominator;
};

Rational number arithmetic becomes really easy and considerably more accurate!

This would solve so many problems related to the precision and storage limitations of floating point numbers, and I dont see it introducing any new problems as well!

Hence my question: Why aren’t rational numbers implemented and stored as fractions with zero loss of information?

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As Joe asked, and others might also point out, I do not mean this to replace existing system, but to complement it.

Q: How do you store pi?

A: So many times, I am just storing 1/3 and not pi. pi can be stored the old way, and 1/3 in the new way.

Answer 1

The reason they are not stored this way by default is that the range of valid values that can fit in a fixed set of bits is smaller. Your float class can store numbers between 1/MAXINT and MAXINT (plus or minus). A C/C++ float can represent numbers between 1E+37 and 1E-37 (plus or minus). In other words, a standard float can represent values 26 orders of magnitude bigger and 26 orders of magnitude smaller then yours despite taking half the number of bits. In general, it’s more convenient to be able to represent very large and very small values than to be perfectly precise. This is especially true since rounding tends to give us the right answers with small fractions like 1/3. In g++, the following gives 1:

std::cout << ((1.0/3.0) * 3.0) << std::endl; 

Remember that types in C++ have a fixed size in bits. Thus a datatype in 32 bits has at most MAX_UINT values. If you change the way it is represented, you’re just changing which values can be precisely represented, not increasing them. You can’t cram more in, and thus can’t be “more precise”. You trade being able to represent 1/3 precisely for not being able to represent other values precisely, like 5.4235E+25.

It is true that your float can represent values more precisely between 1E-9 and 1E+9 (assuming 32 bit ints) but at a cost of being completely unable to represent values outside of this range. Worse, while the standard float always has 6 digits of precision, your float would have precision that varied depending on how close to zero the values were. (And note that you are using twice the bits that float does.)

(I’m assuming 32 bit ints. Same argument applies for 64 bit ints.)

Edit: Also note that most data people use floats for is not precise anyway. If you are reading data off of a sensor, you’ve already got imprecision, so being about to “perfectly” represent the value is pointless. If you are using a float in any sort of computing context, it’s not going to matter. There is no point in perfectly describing ‘1/3’ if your purpose is to display a bit of text 1/3rd of the way across the screen.

The only people who really need perfect precision are mathematicians, and they generally have software that gives them this. Very few others need precision beyond what double gives.