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.
Huh. If that’s the case, I totally missed it. Integrals sounded a lot simpler to me before I had to actually solve them, too, and that’s where I assumed OP was coming from. A /s would have helped.
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.
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
I did know about hyperreals, which is why I went with just “questionable”. IIRC you lose things like commutativity and associativity of arithmetic when you include extra numbers in the real line, and I feel like numbers should really have those.
Maybe that’s just my opinion though. Should I edit it?
I do know the meme, but your use of it and your earlier statement of “I get the feeling you haven’t solved many”; does not really convey irony, but rather, an elitist attitude that doesn’t leave an openness for discussion. Without openness for discussion on a forum, there is little opportunity for an openness to learn.
So discussion may have occurred on the topic, by way of someone proving you wrong; but that leaves you as the only one learning, without offering the same opportunities to casual commentors.
The “irony” is that a cheeky comment went over your head, you began a closed-format lecture, and then you later tried to use irony yourself in another response (expecting to receive the same understanding that you yourself missed earlier).
So now maybe, we both have learned something. But it was through “ackchyually’s” and “one-upping”, rather than through openness. Be kind, be patient, be understanding, be welcoming… that helps nurture honest discourse
It’s merely a slightly longer sum, so what’s the problem?
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.
OP was surely joking in the first post
Huh. If that’s the case, I totally missed it. Integrals sounded a lot simpler to me before I had to actually solve them, too, and that’s where I assumed OP was coming from. A /s would have helped.
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.
Yes, they are right about that being the general concept. I only take issue with the implication that it’s equally simple.
It mentions x^5
Just a polynomial is the easy case, though. Once you start adding other operations, then it gets spicy.
See folks? Meeting dry humor with dry humor—this is the way.
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.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 did know about hyperreals, which is why I went with just “questionable”.
IIRC you lose things like commutativity and associativity of arithmetic when you include extra numbers in the real line, and I feel like numbers should really have those.Maybe that’s just my opinion though. Should I edit it?Edit: I remembered very wrong. Fixing.
Nerd. Just shh away and be quiet
Dude. Be better
What did I do wrong here?
Demeaning without educating. Lacks opportunity for discussion and learning.
There was self-parodying irony here you might have missed. KnowYourMeme
I feel like discussion opportunities were present. In fact, a discussion about hyperreals did start, and I learned something in the process.
I do know the meme, but your use of it and your earlier statement of “I get the feeling you haven’t solved many”; does not really convey irony, but rather, an elitist attitude that doesn’t leave an openness for discussion. Without openness for discussion on a forum, there is little opportunity for an openness to learn.
So discussion may have occurred on the topic, by way of someone proving you wrong; but that leaves you as the only one learning, without offering the same opportunities to casual commentors.
The “irony” is that a cheeky comment went over your head, you began a closed-format lecture, and then you later tried to use irony yourself in another response (expecting to receive the same understanding that you yourself missed earlier).
So now maybe, we both have learned something. But it was through “ackchyually’s” and “one-upping”, rather than through openness. Be kind, be patient, be understanding, be welcoming… that helps nurture honest discourse