Интересное про точную вещественную арифметику.
http://sigfpe.blogspot.com/2008/01/what-does-topology-have-to-do-with.html
Цитаты...For example, a common way to represent real numbers is as lists of binary digits. Any programming language that supports the notion of an infinite stream, like Haskell, can therefore represent an infinite stream of digits. If you're unhappy with the notion of an infinite stream, think functionally instead. Define a stream to be a function that takes as argument an integer n and returns the nth element. For example
third n | even n = 1
| odd n = 0can be interpreted as exactly representing the real number 1/3=0.010101..2. This is a perfectly well defined object that sits in a finite amount of memory.
...If we represent real numbers as streams of binary digits like above (maybe stored with an exponent or a separate integer part) then it's impossible to implement addition correctly.
...Even the standard decimal representation is non-unique because 1 and 0.999... recurring both represent the same real. This means we can write functions on our real representation that depend on the specific sequence we've chosen to represent a real number and hence can't be interpreted as functions of the real numbers. (Eg. consider the trivial function that just returns the first element of he sequence.) But it's still possible to write functions are well-defined in the sense that if two different representations of the same real are passed in as inputs, then the returned sequences represent the same real, even though the actual sequences may be different.
Очень интересно.
Цитаты...For example, a common way to represent real numbers is as lists of binary digits. Any programming language that supports the notion of an infinite stream, like Haskell, can therefore represent an infinite stream of digits. If you're unhappy with the notion of an infinite stream, think functionally instead. Define a stream to be a function that takes as argument an integer n and returns the nth element. For example
third n | even n = 1
| odd n = 0can be interpreted as exactly representing the real number 1/3=0.010101..2. This is a perfectly well defined object that sits in a finite amount of memory.
...If we represent real numbers as streams of binary digits like above (maybe stored with an exponent or a separate integer part) then it's impossible to implement addition correctly.
...Even the standard decimal representation is non-unique because 1 and 0.999... recurring both represent the same real. This means we can write functions on our real representation that depend on the specific sequence we've chosen to represent a real number and hence can't be interpreted as functions of the real numbers. (Eg. consider the trivial function that just returns the first element of he sequence.) But it's still possible to write functions are well-defined in the sense that if two different representations of the same real are passed in as inputs, then the returned sequences represent the same real, even though the actual sequences may be different.
Очень интересно.