Thank you for this expansion. I was about to rabbit hole on how it could be that ratio-based trig (and what is that?) is more accurate than modern calculations.
Re: rationals, I mean there's an infinite number of rationals available arbitrarily near any other rational, that has to mean they are good enough for all practical purposes, right?
that has to mean they are good enough for all practical purposes, right?
For practical purposes, they’re bad. Denominators tend to explode when you do a few operations (for example 11/123 + 3/17 = 556/2091), and it’s not easy to spot whether you can simplify results. 12/123 + 3/17 = 191/697, for example.
You can counteract things by ‘rounding’ to fractions with denominators below a given limit (say 1000) but then, you likely are better of with reckoning with a fixed denominator that you then do not have to store with each number, allowing you to increase the maximal denominator.
For example (https://en.wikipedia.org/wiki/Farey_sequence), there are 965 rational fractions in [0,1] with denominator at most 10 (https://oeis.org/A005728/list), so storing one requires just under 10 bits. If you use the fractions n/964 for 0 ≤ n ≤ 964 as your representable numbers, arithmetic becomes easier.
That "density" is how Euclid defined the irrational real numbers in terms of the rationals; his definition, cast into modern language by Dedekind, is what we normally use today.
Comments
Thank you for this expansion. I was about to rabbit hole on how it could be that ratio-based trig (and what is that?) is more accurate than modern calculations.
Re: rationals, I mean there's an infinite number of rationals available arbitrarily near any other rational, that has to mean they are good enough for all practical purposes, right?
For practical purposes, they’re bad. Denominators tend to explode when you do a few operations (for example 11/123 + 3/17 = 556/2091), and it’s not easy to spot whether you can simplify results. 12/123 + 3/17 = 191/697, for example.
You can counteract things by ‘rounding’ to fractions with denominators below a given limit (say 1000) but then, you likely are better of with reckoning with a fixed denominator that you then do not have to store with each number, allowing you to increase the maximal denominator.
For example (https://en.wikipedia.org/wiki/Farey_sequence), there are 965 rational fractions in [0,1] with denominator at most 10 (https://oeis.org/A005728/list), so storing one requires just under 10 bits. If you use the fractions n/964 for 0 ≤ n ≤ 964 as your representable numbers, arithmetic becomes easier.
That "density" is how Euclid defined the irrational real numbers in terms of the rationals; his definition, cast into modern language by Dedekind, is what we normally use today.