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i think it is our educational system that is weird. we teach students from arithmetic to differential equations assuming they are working with Real Numbers and

i think it is our educational system that is weird. we teach students from arithmetic to differential equations assuming they are working with Real Numbers and never really talking about what that means or what a Field is. Then we present computation as something that can easily be done with Real numbers. Which is a lie because we don't even know if Pi plus E is rational or not. Then we present computers as tools of computation, handwaving away the fact that Computer floating point math is not using real numbers, and is not a Field nor anything even remotely like a Field. Floats are not even closed under basic arithmetic operations. And god help someone who learned Geometry on graph paper then tries to apply it to a computer with Floats because Floats cannot even represent a uniform grid in space, they vary in resolution by definition depending on distance from origin. So in the end by trying to gloss over the details of how Real Numbers work, and how Computers work, we have actually misrepresented both Real Numbers and Computers to generations of people, for going on 80 years now. And we keep having to have these articles, or "your calculator is wrong" videos go viral every few months/years because people were not taught basic facts about Reals and Computers.

galdosdi 3y ago
Great points, let me riff on that though -- what's really "weird" (but totally understandable in the historical context of underpowered computers until recently) is that we think of floats as the default representation of a real at all, when more reasonable modern default choices for all purposes outside of high performance applications would either be a rational composed of bigints, or a bigdecimal, or maybe a generating function for the coefficients of a Taylor series expansion or something like that if you really want to get nuts. In any case you can have as much precision as you can afford, although for anything but rationals it'll be an approximation always, and the question is just how many digits you care for. A lot of choices but the only one that's really absurd is the actual default of floats which are terrible for every purpose except their original raison detre, which was performance in a time when computers were so slow that buying a floating point unit to put in your computer was a normal desirable upgrade for a PC owner (and still crucial for high thoroughput computing, video games, graphics rendering, ML, etc) I do remember being taught the difference between rationals and irrationals in high school. Another aside... The really crazy difference is between the definable and non definable numbers. The first which includes everything from pi to every busy beaver number is countably finite just like the integers, only the nondefinables (numbers whose shortest mathematical definition would not be finite or not exist) give the reals their uncountably infinite nature
- gugagore 3y ago
  > when more reasonable modern default choices for all purposes outside of high performance applications would either be a rational composed of bigints, or a bigdecimal, or maybe a generating function for the coefficients of a Taylor series expansion or something like that if you really want to get nuts. Rationals of bigints doesn't work in general. First of all, you still have to do approximations if you take e.g. a square root or a sine. And even if you have an algorithm that is closed under rational numbers (like the Simplex method for solving linear programs), the number of digits you need in your integers blows up unless you have a tiny problem. My rough sketch is that inv(A) involves det(A) in the denominator, and the number of digits in det(A) is exponential in nature (n! if A is n-by-n).
  - galdosdi 3y ago
    Well, yeah, nothing works in general, which is why I prevaricated amongst several ideas -- but the key point is that if something has to be a default, floats work worst of all if accuracy or precision are your goals (but again, make sense for performance and performance alone -- but we really should be making accuracy the default and performance something you have to opt in to these days) (Rational of bigints also has the advantage of being simple to understand, but I'd happily endorse pretty much anything but floats, even bigdecimals.)

galdosdi 3y ago

Great points, let me riff on that though -- what's really "weird" (but totally understandable in the historical context of underpowered computers until recently) is that we think of floats as the default representation of a real at all, when more reasonable modern default choices for all purposes outside of high performance applications would either be a rational composed of bigints, or a bigdecimal, or maybe a generating function for the coefficients of a Taylor series expansion or something like that if you really want to get nuts. In any case you can have as much precision as you can afford, although for anything but rationals it'll be an approximation always, and the question is just how many digits you care for. A lot of choices but the only one that's really absurd is the actual default of floats which are terrible for every purpose except their original raison detre, which was performance in a time when computers were so slow that buying a floating point unit to put in your computer was a normal desirable upgrade for a PC owner (and still crucial for high thoroughput computing, video games, graphics rendering, ML, etc) I do remember being taught the difference between rationals and irrationals in high school. Another aside... The really crazy difference is between the definable and non definable numbers. The first which includes everything from pi to every busy beaver number is countably finite just like the integers, only the nondefinables (numbers whose shortest mathematical definition would not be finite or not exist) give the reals their uncountably infinite nature

gugagore 3y ago
> when more reasonable modern default choices for all purposes outside of high performance applications would either be a rational composed of bigints, or a bigdecimal, or maybe a generating function for the coefficients of a Taylor series expansion or something like that if you really want to get nuts. Rationals of bigints doesn't work in general. First of all, you still have to do approximations if you take e.g. a square root or a sine. And even if you have an algorithm that is closed under rational numbers (like the Simplex method for solving linear programs), the number of digits you need in your integers blows up unless you have a tiny problem. My rough sketch is that inv(A) involves det(A) in the denominator, and the number of digits in det(A) is exponential in nature (n! if A is n-by-n).
- galdosdi 3y ago
  Well, yeah, nothing works in general, which is why I prevaricated amongst several ideas -- but the key point is that if something has to be a default, floats work worst of all if accuracy or precision are your goals (but again, make sense for performance and performance alone -- but we really should be making accuracy the default and performance something you have to opt in to these days) (Rational of bigints also has the advantage of being simple to understand, but I'd happily endorse pretty much anything but floats, even bigdecimals.)