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"approximating" is the right term. I hate to wordsmith things like this to death, but in speaking about maths, I think it's important to use precise language.

"approximating" is the right term. I hate to wordsmith things like this to death, but in speaking about maths, I think it's important to use precise language. To me, "computing" implies an exact calculation. As does "solving". From the article, they did neither, but instead only got practically "close enough." You are right, the headline leads one to conclude something that is not true.

betterunix2 8y ago
"To me, "computing" implies an exact calculation" You are not thinking like a computer scientist: https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number "Computing" typically means finding an approximation up to an arbitrary degree of precision. As for whether or not that counts as "solving," it is a matter of whether or not you will allow lambda expressions in a solution. Think of this: does the quadratic formula count as a solution to quadratic equations? x^2-2=0 has +/-sqrt(2) as a solution, but only if you allow "sqrt" as part of a "solution." If you wanted a decimal representation of the roots you would have to accept an approximation and you would be using some square root finding algorithm. If you are OK with that situation, why not allow a quintic formula that involves lambda expressions? What makes radicals so special? Remember, all the Abel-Ruffini theorem says is that there is no quintic formula involving only arithmetic and radicals -- it leaves open the possibility of another kind of formula for the quintic.
ummonk 8y ago
Computing is an algorithm which can produce the value to any given precision (I.e. bounds size). Approximating is an algorithm which eventually converges to the true value but you never know when you’ve actually gotten close enough. For this reason the computable numbers are a subset of the approximately numbers.

betterunix2 8y ago

"To me, "computing" implies an exact calculation" You are not thinking like a computer scientist: https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number "Computing" typically means finding an approximation up to an arbitrary degree of precision. As for whether or not that counts as "solving," it is a matter of whether or not you will allow lambda expressions in a solution. Think of this: does the quadratic formula count as a solution to quadratic equations? x^2-2=0 has +/-sqrt(2) as a solution, but only if you allow "sqrt" as part of a "solution." If you wanted a decimal representation of the roots you would have to accept an approximation and you would be using some square root finding algorithm. If you are OK with that situation, why not allow a quintic formula that involves lambda expressions? What makes radicals so special? Remember, all the Abel-Ruffini theorem says is that there is no quintic formula involving only arithmetic and radicals -- it leaves open the possibility of another kind of formula for the quintic.

ummonk 8y ago

Computing is an algorithm which can produce the value to any given precision (I.e. bounds size). Approximating is an algorithm which eventually converges to the true value but you never know when you’ve actually gotten close enough. For this reason the computable numbers are a subset of the approximately numbers.