|
The vast majority of problems with floating point numbers are simply due to people not understanding number bases. Only a tiny subset is caused by people not understanding floats proper. Consider the number 1/3. In ternary (base 3), you write that as 0.1, whereas in decimal (base 10) you write it as 0.3333... recurring. If you try to represent that number with a fixed number of decimal places, you have precision issues. E.g. decimal 0.333 converts to 0.02222220210 in ternary. Now, the thing with that example is that we treat decimal as a special, privileged representation, so we accept that 1/3 doesn't have a finite representation in decimal as an entirely natural fact of life, while we treat that same problem converting between decimal and binary as a fundamental deficiency of binary. Let's talk about why some numbers have finite representations while others don't. 53/100 is written as 0.53 in decimal. The general rule is that if the denominator is a power of ten, you just write the numerator, then put the decimal mark have as many digits from the right as the power of ten. If the denominator is not a power of ten, you make it one. 1/2 turns into 5/10, which is 5 with one decimal place, or 0.5. Obviously, you can't actually do this for 1/3. There's no integer `a` where 3a is a power of ten. The general rule here is that, if the denominator has any prime factors not present in the base, you don't have a finite representation. 4/25 = 16/100 = 0.16 has a finite representation, but 5/7, 13/3 don't. Now, because 3 is coprime with 10, no number with a finite ternary representation can have a finite decimal representation (and vice versa), and we're used to it. Where binary vs decimal becomes tricky and confuses people is that 10 = 2 * 5, so numbers that can be expressed with a power-of-two denominator have finite representations in both binary and decimal, so you can convert some numbers back and forth with no loss of precision. Numbers with a factor of 5 somewhere in their denominator can have finite decimal represntations (1/5 = 2/10 = 0.2), but can't have finite representations in binary. 0.1 = 1/10 = 1/(2 * 5) and you can't get rid of that five. And that's why everybody gets bitten by 0.1 seeming to be broken. |