ACKTUALLY neither. It’s most simply thought of as a limit of progressively longer sums. Infinitesimals help people understand but they’re kind of logically questionable.
Actually that last point isn’t quite right, in the 1960s Robinson proved that the set of hyperreals were logically consistent if and only if the reals were.
This put to rest the age-long speculation that the hyperreals were questionable.
This speculation is a pain in the ass since it means that we primarily use limits when talking about this sort of thing.
Which is fine, but infinitesimals are the coolest shit ever
No judgement, but you should know it’s not that simple. You can’t just pull out your calculator and add together an uncountably infinite collection of values one-by-one.
I mean, you could add together a finite subset of the values, which turns out to be the only practical way fairly often because a symbolic solution is too hard to find. You don’t get the actual answer that way, though, just an approximation.
The actual symbolic approaches to integrals are very algebra-heavy and they often require more than one whiteboard to solve by hand. Blackpenredpen “math for fun” on YouTube if you want to see it done at peak performance.
I mean they’re right, Leibniz used a modified s for summa, sum. And an integral is just a sum, an infinite sum over infinitesimal summands, but a sum nevertheless.
It’s merely a slightly longer sum, so what’s the problem?
Longer? Or infinitesimal?
ACKTUALLY neither. It’s most simply thought of as a limit of progressively longer sums. Infinitesimals help people understand
but they’re kind of logically questionable.Nerd. Just shh away and be quiet
Actually that last point isn’t quite right, in the 1960s Robinson proved that the set of hyperreals were logically consistent if and only if the reals were.
This put to rest the age-long speculation that the hyperreals were questionable.
This speculation is a pain in the ass since it means that we primarily use limits when talking about this sort of thing.
Which is fine, but infinitesimals are the coolest shit ever
I get the feeling you haven’t solved many.
What a curious and needlessly judgmental reply!
No judgement, but you should know it’s not that simple. You can’t just pull out your calculator and add together an uncountably infinite collection of values one-by-one.
I mean, you could add together a finite subset of the values, which turns out to be the only practical way fairly often because a symbolic solution is too hard to find. You don’t get the actual answer that way, though, just an approximation.
The actual symbolic approaches to integrals are very algebra-heavy and they often require more than one whiteboard to solve by hand. Blackpenredpen “math for fun” on YouTube if you want to see it done at peak performance.
I mean they’re right, Leibniz used a modified s for summa, sum. And an integral is just a sum, an infinite sum over infinitesimal summands, but a sum nevertheless.