4 ms·

> If they don't specify then the best definition is the commonly used intuitive one, under which it's indeed possible to sum infinite positive integers, resulti

> If they don't specify then the best definition is the commonly used intuitive one, under which it's indeed possible to sum infinite positive integers, resulting in positive infinity. I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete. I also still claim that the result cannot be positive infinity due to the definition of addition on integers. The result of addition of integers must be another integer and positive infinity is not an integer. I do agree with the article however, that one can take the limit of a well-defined infinite series; but the limit is not the sum, only the bound that will never be exceeded no matter how long you are able to continue adding numbers for. I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected.

andrepd 10y ago
>I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected. This not something you can agree or disagree on. You can make physical calculations with this result and get a prediction that you can measure and confirm. This result is sound.
- Dylan16807 10y ago
  You can definitely agree or disagree on whether a definition should be rejected. If I tell you I want to replace 'five' with 'fish', even though my new system can be used to calculate things, you should tell me it's a terrible idea. In this case you might find the -1/12 useful, but have the opinion that they really should not be using '=' as a shorthand for what they're doing with the zeta function.
eximius 10y ago
Reread the parent comment. In the hyperreals, the reals are extended by infintesmals and positive and negative infinity. Two things happen there: 1. You no longer have +:Z -> Z, you have +:R'->R', which means that plus can be closed under the hyperreals. Also, yes, you can't literally compute an infinite sum but among any crowd that has likely taken calculus 1, you can place implied limits. :P (which, arent actually needed in the hyperreals because it HAS infinity, but whatever)
Dylan16807 10y ago
>I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete. That same argument gives you Zeno's paradox. You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. It doesn't matter if physically spending O(n) time on something is impossible when you only need O(1).
- jsprogrammer 10y ago
  >That same argument gives you Zeno's paradox. I believe Zeno's paradox is on the rationals, not integers. >You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. There may be closed solutions for finite summation patterns, but infinite summation patterns of integers have no solution (by definition).

andrepd 10y ago

>I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected. This not something you can agree or disagree on. You can make physical calculations with this result and get a prediction that you can measure and confirm. This result is sound.

Dylan16807 10y ago
You can definitely agree or disagree on whether a definition should be rejected. If I tell you I want to replace 'five' with 'fish', even though my new system can be used to calculate things, you should tell me it's a terrible idea. In this case you might find the -1/12 useful, but have the opinion that they really should not be using '=' as a shorthand for what they're doing with the zeta function.

eximius 10y ago

Reread the parent comment. In the hyperreals, the reals are extended by infintesmals and positive and negative infinity. Two things happen there: 1. You no longer have +:Z -> Z, you have +:R'->R', which means that plus can be closed under the hyperreals. Also, yes, you can't literally compute an infinite sum but among any crowd that has likely taken calculus 1, you can place implied limits. :P (which, arent actually needed in the hyperreals because it HAS infinity, but whatever)

Dylan16807 10y ago

>I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete. That same argument gives you Zeno's paradox. You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. It doesn't matter if physically spending O(n) time on something is impossible when you only need O(1).

jsprogrammer 10y ago
>That same argument gives you Zeno's paradox. I believe Zeno's paradox is on the rationals, not integers. >You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. There may be closed solutions for finite summation patterns, but infinite summation patterns of integers have no solution (by definition).