Mixing
Can this “new” notation for Maclaurin expansions be useful? [closed]
$begingroup$
Here is a notation for Maclaurin expansions that I made up:
begin{align*}
sin(x) &= sum {alt(+); frac{x^n}{n!}; text{ n odd $in$ $mathbb{N}_0$} } \
cos(x) &= sum {alt(+); frac{x^n}{n!}; text{ n even $in$ $mathbb{N}_0$} } \
e^x &= sum {+; frac{x^n}{n!}; text{ n $in$ $mathbb{N}_0$} }\
end{align*}
Could this notation be useful?
The $alt(+)$ is indicative of an alternating sign that starts at $+$. Similarly we could have $alt(-)$ that starts at $-$.
Edit:
I have come up with the following notation that might make it easier to read from left to right, that is a bit more compact, and which looks a bit more like the notation that is already in use:
begin{align*}
sin(x) &= sum_{text{odd } n = 1}^{infty}alt(+) frac{x^n}{n!} \
cos(x) &= sum_{text{even } space n = 0}^{infty}alt(+) frac{x^n}{n!} \
e^x &= sum_{forall space n = 0}^{infty}frac{x^n}{n!}\
end{align*}
Here are some properties that I thought might be useful:
begin{align*}
alt(+)alt(+) = +\
alt(+)alt(-) = - \
end{align*}
begin{align*}
e^x = -alt(-)(sin(x) + cos(x)) \
end{align*}
This might be useful in the following way:
begin{align*}
sin(jx) &= j.alt(+)sin(x) \
cos(jx) &= alt(+)cos(x) \
\
e^{jx} &= -alt(-)(sin(jx) + cos(jx)) \
e^{jx} &= -alt(-)(j.alt(+)sin(x) + alt(+)cos(x)) \
e^{jx} &= -alt(-)alt(+)(jsin(x) + cos(x)) \
e^{jx} &= -(-1)(jsin(x) + cos(x)) \
e^{jx} &= jsin(x) + cos(x) \
e^{jx} &= cos(x) + jsin(x) \
end{align*}
notation
asked Jan 21 at 19:22
GustavGustav
1469
$endgroup$
closed as primarily opinion-based by Lord Shark the Unknown, Dietrich Burde, Randall, T. Bongers, Noah Schweber Jan 21 at 19:54
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 25 more comments
$begingroup$
Here is a notation for Maclaurin expansions that I made up:
begin{align*}
sin(x) &= sum {alt(+); frac{x^n}{n!}; text{ n odd $in$ $mathbb{N}_0$} } \
cos(x) &= sum {alt(+); frac{x^n}{n!}; text{ n even $in$ $mathbb{N}_0$} } \
e^x &= sum {+; frac{x^n}{n!}; text{ n $in$ $mathbb{N}_0$} }\
end{align*}
Could this notation be useful?
The $alt(+)$ is indicative of an alternating sign that starts at $+$. Similarly we could have $alt(-)$ that starts at $-$.
Edit:
I have come up with the following notation that might make it easier to read from left to right, that is a bit more compact, and which looks a bit more like the notation that is already in use:
begin{align*}
sin(x) &= sum_{text{odd } n = 1}^{infty}alt(+) frac{x^n}{n!} \
cos(x) &= sum_{text{even } space n = 0}^{infty}alt(+) frac{x^n}{n!} \
e^x &= sum_{forall space n = 0}^{infty}frac{x^n}{n!}\
end{align*}
Here are some properties that I thought might be useful:
begin{align*}
alt(+)alt(+) = +\
alt(+)alt(-) = - \
end{align*}
begin{align*}
e^x = -alt(-)(sin(x) + cos(x)) \
end{align*}
This might be useful in the following way:
begin{align*}
sin(jx) &= j.alt(+)sin(x) \
cos(jx) &= alt(+)cos(x) \
\
e^{jx} &= -alt(-)(sin(jx) + cos(jx)) \
e^{jx} &= -alt(-)(j.alt(+)sin(x) + alt(+)cos(x)) \
e^{jx} &= -alt(-)alt(+)(jsin(x) + cos(x)) \
e^{jx} &= -(-1)(jsin(x) + cos(x)) \
e^{jx} &= jsin(x) + cos(x) \
e^{jx} &= cos(x) + jsin(x) \
end{align*}
notation
asked Jan 21 at 19:22
GustavGustav
1469
$endgroup$
closed as primarily opinion-based by Lord Shark the Unknown, Dietrich Burde, Randall, T. Bongers, Noah Schweber Jan 21 at 19:54
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Inventing new symbols led to the overwhelming (and in my eyes desastrous) flooding of the unicode set with gazillions of mostly useless emojis
$endgroup$
– Hagen von Eitzen
Jan 21 at 19:29
1
$begingroup$
Your alternative notation for infinite series is horrendous! The traditional notation is simple and clear. (Also, your series for $sin$ should start at $n=0$, not $n=1$. You will no doubt claim that this error actually strengthens your case, but I'm not buying it.)
$endgroup$
– TonyK
Jan 21 at 19:37
1
$begingroup$
Good notation is quickly understood in a concise manner. Having to unpack your notation for something as simple as sin(x) is already pretty difficult, and it conflicts with broadly used notation already. This post seems more of a complaint/proposal rather than an actual question, and I'm voting to close as off-topic.
$endgroup$
– T. Bongers
Jan 21 at 19:49
1
$begingroup$
Honestly, it was pretty hard to read the rest of the post after "You see, calculus is quite complicated, but in its pure mathematical state its horrendous. If we had to solve calculus problems with actual mathematics, it would be utterly painful," @Gustav. I think some self-examination might lead you to a different conclusion, since this is starting to feel like "It's not immediately clear to me, so it's bad."
$endgroup$
– T. Bongers
Jan 21 at 19:54
1
$begingroup$
There is a lot of unsubstantiated opinion here. Just one immediate observation: note that using your notation to find the MacLaurin series of a sum of two functions, given their MacLaurin series, is meaningfully harder than via the usual notation (for example, note the "sign-play" in e.g. $e^x+cos(x)$). Quite frankly I don't think this question can be made appropriate for MSE while keeping its spirit - there's too much presupposition in "What are good reasons for recycling and reusing the same symbols past their clear conveying ability?" (emphasis mine).
$endgroup$
– Noah Schweber
Jan 21 at 19:59
|
show 25 more comments
$begingroup$
Here is a notation for Maclaurin expansions that I made up:
begin{align*}
sin(x) &= sum {alt(+); frac{x^n}{n!}; text{ n odd $in$ $mathbb{N}_0$} } \
cos(x) &= sum {alt(+); frac{x^n}{n!}; text{ n even $in$ $mathbb{N}_0$} } \
e^x &= sum {+; frac{x^n}{n!}; text{ n $in$ $mathbb{N}_0$} }\
end{align*}
Could this notation be useful?
The $alt(+)$ is indicative of an alternating sign that starts at $+$. Similarly we could have $alt(-)$ that starts at $-$.
Edit:
I have come up with the following notation that might make it easier to read from left to right, that is a bit more compact, and which looks a bit more like the notation that is already in use:
begin{align*}
sin(x) &= sum_{text{odd } n = 1}^{infty}alt(+) frac{x^n}{n!} \
cos(x) &= sum_{text{even } space n = 0}^{infty}alt(+) frac{x^n}{n!} \
e^x &= sum_{forall space n = 0}^{infty}frac{x^n}{n!}\
end{align*}
Here are some properties that I thought might be useful:
begin{align*}
alt(+)alt(+) = +\
alt(+)alt(-) = - \
end{align*}
begin{align*}
e^x = -alt(-)(sin(x) + cos(x)) \
end{align*}
This might be useful in the following way:
begin{align*}
sin(jx) &= j.alt(+)sin(x) \
cos(jx) &= alt(+)cos(x) \
\
e^{jx} &= -alt(-)(sin(jx) + cos(jx)) \
e^{jx} &= -alt(-)(j.alt(+)sin(x) + alt(+)cos(x)) \
e^{jx} &= -alt(-)alt(+)(jsin(x) + cos(x)) \
e^{jx} &= -(-1)(jsin(x) + cos(x)) \
e^{jx} &= jsin(x) + cos(x) \
e^{jx} &= cos(x) + jsin(x) \
end{align*}
notation
asked Jan 21 at 19:22
GustavGustav
1469
$endgroup$
Here is a notation for Maclaurin expansions that I made up:
begin{align*}
sin(x) &= sum {alt(+); frac{x^n}{n!}; text{ n odd $in$ $mathbb{N}_0$} } \
cos(x) &= sum {alt(+); frac{x^n}{n!}; text{ n even $in$ $mathbb{N}_0$} } \
e^x &= sum {+; frac{x^n}{n!}; text{ n $in$ $mathbb{N}_0$} }\
end{align*}
Could this notation be useful?
The $alt(+)$ is indicative of an alternating sign that starts at $+$. Similarly we could have $alt(-)$ that starts at $-$.
Edit:
I have come up with the following notation that might make it easier to read from left to right, that is a bit more compact, and which looks a bit more like the notation that is already in use:
begin{align*}
sin(x) &= sum_{text{odd } n = 1}^{infty}alt(+) frac{x^n}{n!} \
cos(x) &= sum_{text{even } space n = 0}^{infty}alt(+) frac{x^n}{n!} \
e^x &= sum_{forall space n = 0}^{infty}frac{x^n}{n!}\
end{align*}
Here are some properties that I thought might be useful:
begin{align*}
alt(+)alt(+) = +\
alt(+)alt(-) = - \
end{align*}
begin{align*}
e^x = -alt(-)(sin(x) + cos(x)) \
end{align*}
This might be useful in the following way:
begin{align*}
sin(jx) &= j.alt(+)sin(x) \
cos(jx) &= alt(+)cos(x) \
\
e^{jx} &= -alt(-)(sin(jx) + cos(jx)) \
e^{jx} &= -alt(-)(j.alt(+)sin(x) + alt(+)cos(x)) \
e^{jx} &= -alt(-)alt(+)(jsin(x) + cos(x)) \
e^{jx} &= -(-1)(jsin(x) + cos(x)) \
e^{jx} &= jsin(x) + cos(x) \
e^{jx} &= cos(x) + jsin(x) \
end{align*}
notation
notation
asked Jan 21 at 19:22
GustavGustav
1469
asked Jan 21 at 19:22
GustavGustav
1469
edited Jan 22 at 16:30
Gustav
asked Jan 21 at 19:22
GustavGustav
1469
asked Jan 21 at 19:22
GustavGustav
1469
asked Jan 21 at 19:22
GustavGustav
1469
1469
closed as primarily opinion-based by Lord Shark the Unknown, Dietrich Burde, Randall, T. Bongers, Noah Schweber Jan 21 at 19:54
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as primarily opinion-based by Lord Shark the Unknown, Dietrich Burde, Randall, T. Bongers, Noah Schweber Jan 21 at 19:54
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Inventing new symbols led to the overwhelming (and in my eyes desastrous) flooding of the unicode set with gazillions of mostly useless emojis
$endgroup$
– Hagen von Eitzen
Jan 21 at 19:29
1
$begingroup$
Your alternative notation for infinite series is horrendous! The traditional notation is simple and clear. (Also, your series for $sin$ should start at $n=0$, not $n=1$. You will no doubt claim that this error actually strengthens your case, but I'm not buying it.)
$endgroup$
– TonyK
Jan 21 at 19:37
1
$begingroup$
Good notation is quickly understood in a concise manner. Having to unpack your notation for something as simple as sin(x) is already pretty difficult, and it conflicts with broadly used notation already. This post seems more of a complaint/proposal rather than an actual question, and I'm voting to close as off-topic.
$endgroup$
– T. Bongers
Jan 21 at 19:49
1
$begingroup$
Honestly, it was pretty hard to read the rest of the post after "You see, calculus is quite complicated, but in its pure mathematical state its horrendous. If we had to solve calculus problems with actual mathematics, it would be utterly painful," @Gustav. I think some self-examination might lead you to a different conclusion, since this is starting to feel like "It's not immediately clear to me, so it's bad."
$endgroup$
– T. Bongers
Jan 21 at 19:54
1
$begingroup$
There is a lot of unsubstantiated opinion here. Just one immediate observation: note that using your notation to find the MacLaurin series of a sum of two functions, given their MacLaurin series, is meaningfully harder than via the usual notation (for example, note the "sign-play" in e.g. $e^x+cos(x)$). Quite frankly I don't think this question can be made appropriate for MSE while keeping its spirit - there's too much presupposition in "What are good reasons for recycling and reusing the same symbols past their clear conveying ability?" (emphasis mine).
$endgroup$
– Noah Schweber
Jan 21 at 19:59
|
show 25 more comments
2
$begingroup$
Inventing new symbols led to the overwhelming (and in my eyes desastrous) flooding of the unicode set with gazillions of mostly useless emojis
$endgroup$
– Hagen von Eitzen
Jan 21 at 19:29
1
$begingroup$
Your alternative notation for infinite series is horrendous! The traditional notation is simple and clear. (Also, your series for $sin$ should start at $n=0$, not $n=1$. You will no doubt claim that this error actually strengthens your case, but I'm not buying it.)
$endgroup$
– TonyK
Jan 21 at 19:37
1
$begingroup$
Good notation is quickly understood in a concise manner. Having to unpack your notation for something as simple as sin(x) is already pretty difficult, and it conflicts with broadly used notation already. This post seems more of a complaint/proposal rather than an actual question, and I'm voting to close as off-topic.
$endgroup$
– T. Bongers
Jan 21 at 19:49
1
$begingroup$
Honestly, it was pretty hard to read the rest of the post after "You see, calculus is quite complicated, but in its pure mathematical state its horrendous. If we had to solve calculus problems with actual mathematics, it would be utterly painful," @Gustav. I think some self-examination might lead you to a different conclusion, since this is starting to feel like "It's not immediately clear to me, so it's bad."
$endgroup$
– T. Bongers
Jan 21 at 19:54
1
$begingroup$
There is a lot of unsubstantiated opinion here. Just one immediate observation: note that using your notation to find the MacLaurin series of a sum of two functions, given their MacLaurin series, is meaningfully harder than via the usual notation (for example, note the "sign-play" in e.g. $e^x+cos(x)$). Quite frankly I don't think this question can be made appropriate for MSE while keeping its spirit - there's too much presupposition in "What are good reasons for recycling and reusing the same symbols past their clear conveying ability?" (emphasis mine).
$endgroup$
– Noah Schweber
Jan 21 at 19:59
2
2
$begingroup$
Inventing new symbols led to the overwhelming (and in my eyes desastrous) flooding of the unicode set with gazillions of mostly useless emojis
$endgroup$
– Hagen von Eitzen
Jan 21 at 19:29
$begingroup$
Inventing new symbols led to the overwhelming (and in my eyes desastrous) flooding of the unicode set with gazillions of mostly useless emojis
$endgroup$
– Hagen von Eitzen
Jan 21 at 19:29
1
1
$begingroup$
Your alternative notation for infinite series is horrendous! The traditional notation is simple and clear. (Also, your series for $sin$ should start at $n=0$, not $n=1$. You will no doubt claim that this error actually strengthens your case, but I'm not buying it.)
$endgroup$
– TonyK
Jan 21 at 19:37
$begingroup$
Your alternative notation for infinite series is horrendous! The traditional notation is simple and clear. (Also, your series for $sin$ should start at $n=0$, not $n=1$. You will no doubt claim that this error actually strengthens your case, but I'm not buying it.)
$endgroup$
– TonyK
Jan 21 at 19:37
1
1
$begingroup$
Good notation is quickly understood in a concise manner. Having to unpack your notation for something as simple as sin(x) is already pretty difficult, and it conflicts with broadly used notation already. This post seems more of a complaint/proposal rather than an actual question, and I'm voting to close as off-topic.
$endgroup$
– T. Bongers
Jan 21 at 19:49
$begingroup$
Good notation is quickly understood in a concise manner. Having to unpack your notation for something as simple as sin(x) is already pretty difficult, and it conflicts with broadly used notation already. This post seems more of a complaint/proposal rather than an actual question, and I'm voting to close as off-topic.
$endgroup$
– T. Bongers
Jan 21 at 19:49
1
1
$begingroup$
Honestly, it was pretty hard to read the rest of the post after "You see, calculus is quite complicated, but in its pure mathematical state its horrendous. If we had to solve calculus problems with actual mathematics, it would be utterly painful," @Gustav. I think some self-examination might lead you to a different conclusion, since this is starting to feel like "It's not immediately clear to me, so it's bad."
$endgroup$
– T. Bongers
Jan 21 at 19:54
$begingroup$
Honestly, it was pretty hard to read the rest of the post after "You see, calculus is quite complicated, but in its pure mathematical state its horrendous. If we had to solve calculus problems with actual mathematics, it would be utterly painful," @Gustav. I think some self-examination might lead you to a different conclusion, since this is starting to feel like "It's not immediately clear to me, so it's bad."
$endgroup$
– T. Bongers
Jan 21 at 19:54
1
1
$begingroup$
There is a lot of unsubstantiated opinion here. Just one immediate observation: note that using your notation to find the MacLaurin series of a sum of two functions, given their MacLaurin series, is meaningfully harder than via the usual notation (for example, note the "sign-play" in e.g. $e^x+cos(x)$). Quite frankly I don't think this question can be made appropriate for MSE while keeping its spirit - there's too much presupposition in "What are good reasons for recycling and reusing the same symbols past their clear conveying ability?" (emphasis mine).
$endgroup$
– Noah Schweber
Jan 21 at 19:59
$begingroup$
There is a lot of unsubstantiated opinion here. Just one immediate observation: note that using your notation to find the MacLaurin series of a sum of two functions, given their MacLaurin series, is meaningfully harder than via the usual notation (for example, note the "sign-play" in e.g. $e^x+cos(x)$). Quite frankly I don't think this question can be made appropriate for MSE while keeping its spirit - there's too much presupposition in "What are good reasons for recycling and reusing the same symbols past their clear conveying ability?" (emphasis mine).
$endgroup$
– Noah Schweber
Jan 21 at 19:59
|
show 25 more comments
1 Answer
1
active
oldest
votes
$begingroup$
You've asked several different questions about the history and purpose of particular mathematical notations. They have different answers.
For single symbols used essentially as adjectives, like $f^*$, $hat f$ and $f'$ the meaning depends on the context, and it should. There are not enough of them to cover all the possible different uses where a short annotation is really useful.
In your second example you are pointing out one of the shortcomings of the $Sigma$ notation for sums. It is indeed sometimes hard to unpack. I almost always write my sums with ellipses:
$$
sin(x) = x - frac{x^2}{2} + frac{x^3}{3 times 2} - cdots
$$
with as many terms as I need to see the pattern.
Your two descriptions of integrals are not those of Leibniz and Newton respectively. The first is (essentially) Leibniz. The second is a formal definition of the integral as a limit of Riemann sums. Both are necessary and useful - the first when you are using integrals in a problem in math or physics, the second when you are actually defining integrals and proving their properties.
will finish later ...
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
$endgroup$
$begingroup$
I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
$endgroup$
– Gustav
Jan 21 at 19:45
$begingroup$
Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
$endgroup$
– Gustav
Jan 21 at 20:16
1
$begingroup$
Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
$endgroup$
– Ethan Bolker
Jan 21 at 21:07
$begingroup$
I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
$endgroup$
– Gustav
Jan 22 at 14:46
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You've asked several different questions about the history and purpose of particular mathematical notations. They have different answers.
For single symbols used essentially as adjectives, like $f^*$, $hat f$ and $f'$ the meaning depends on the context, and it should. There are not enough of them to cover all the possible different uses where a short annotation is really useful.
In your second example you are pointing out one of the shortcomings of the $Sigma$ notation for sums. It is indeed sometimes hard to unpack. I almost always write my sums with ellipses:
$$
sin(x) = x - frac{x^2}{2} + frac{x^3}{3 times 2} - cdots
$$
with as many terms as I need to see the pattern.
Your two descriptions of integrals are not those of Leibniz and Newton respectively. The first is (essentially) Leibniz. The second is a formal definition of the integral as a limit of Riemann sums. Both are necessary and useful - the first when you are using integrals in a problem in math or physics, the second when you are actually defining integrals and proving their properties.
will finish later ...
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
$endgroup$
$begingroup$
I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
$endgroup$
– Gustav
Jan 21 at 19:45
$begingroup$
Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
$endgroup$
– Gustav
Jan 21 at 20:16
1
$begingroup$
Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
$endgroup$
– Ethan Bolker
Jan 21 at 21:07
$begingroup$
I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
$endgroup$
– Gustav
Jan 22 at 14:46
add a comment |
$begingroup$
You've asked several different questions about the history and purpose of particular mathematical notations. They have different answers.
For single symbols used essentially as adjectives, like $f^*$, $hat f$ and $f'$ the meaning depends on the context, and it should. There are not enough of them to cover all the possible different uses where a short annotation is really useful.
In your second example you are pointing out one of the shortcomings of the $Sigma$ notation for sums. It is indeed sometimes hard to unpack. I almost always write my sums with ellipses:
$$
sin(x) = x - frac{x^2}{2} + frac{x^3}{3 times 2} - cdots
$$
with as many terms as I need to see the pattern.
Your two descriptions of integrals are not those of Leibniz and Newton respectively. The first is (essentially) Leibniz. The second is a formal definition of the integral as a limit of Riemann sums. Both are necessary and useful - the first when you are using integrals in a problem in math or physics, the second when you are actually defining integrals and proving their properties.
will finish later ...
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
$endgroup$
$begingroup$
I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
$endgroup$
– Gustav
Jan 21 at 19:45
$begingroup$
Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
$endgroup$
– Gustav
Jan 21 at 20:16
1
$begingroup$
Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
$endgroup$
– Ethan Bolker
Jan 21 at 21:07
$begingroup$
I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
$endgroup$
– Gustav
Jan 22 at 14:46
add a comment |
$begingroup$
You've asked several different questions about the history and purpose of particular mathematical notations. They have different answers.
For single symbols used essentially as adjectives, like $f^*$, $hat f$ and $f'$ the meaning depends on the context, and it should. There are not enough of them to cover all the possible different uses where a short annotation is really useful.
In your second example you are pointing out one of the shortcomings of the $Sigma$ notation for sums. It is indeed sometimes hard to unpack. I almost always write my sums with ellipses:
$$
sin(x) = x - frac{x^2}{2} + frac{x^3}{3 times 2} - cdots
$$
with as many terms as I need to see the pattern.
Your two descriptions of integrals are not those of Leibniz and Newton respectively. The first is (essentially) Leibniz. The second is a formal definition of the integral as a limit of Riemann sums. Both are necessary and useful - the first when you are using integrals in a problem in math or physics, the second when you are actually defining integrals and proving their properties.
will finish later ...
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
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You've asked several different questions about the history and purpose of particular mathematical notations. They have different answers.
For single symbols used essentially as adjectives, like $f^*$, $hat f$ and $f'$ the meaning depends on the context, and it should. There are not enough of them to cover all the possible different uses where a short annotation is really useful.
In your second example you are pointing out one of the shortcomings of the $Sigma$ notation for sums. It is indeed sometimes hard to unpack. I almost always write my sums with ellipses:
$$
sin(x) = x - frac{x^2}{2} + frac{x^3}{3 times 2} - cdots
$$
with as many terms as I need to see the pattern.
Your two descriptions of integrals are not those of Leibniz and Newton respectively. The first is (essentially) Leibniz. The second is a formal definition of the integral as a limit of Riemann sums. Both are necessary and useful - the first when you are using integrals in a problem in math or physics, the second when you are actually defining integrals and proving their properties.
will finish later ...
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
answered Jan 21 at 19:38
Ethan BolkerEthan Bolker
44.8k553120
44.8k553120
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I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
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– Gustav
Jan 21 at 19:45
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Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
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– Gustav
Jan 21 at 20:16
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Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
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– Ethan Bolker
Jan 21 at 21:07
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I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
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– Gustav
Jan 22 at 14:46
add a comment |
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I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
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– Gustav
Jan 21 at 19:45
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Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
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– Gustav
Jan 21 at 20:16
1
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Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
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– Ethan Bolker
Jan 21 at 21:07
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I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
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– Gustav
Jan 22 at 14:46
$begingroup$
I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
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– Gustav
Jan 21 at 19:45
$begingroup$
I am not saying that the Riemann sum notation is not necessary. I am suggesting that in most contexts, it is better to use the Liebniz notation, which is actually representative of that. In a similar fashion I am suggesting we develop shorthand notations, or better structures, to more clearly convey ideas. This is the whole point of symbols.
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– Gustav
Jan 21 at 19:45
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Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
$endgroup$
– Gustav
Jan 21 at 20:16
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Hi Ethan. Thank you for your response. I would like to delete this question to better formulate it. Could you delete your answer so that I can delete the question?
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– Gustav
Jan 21 at 20:16
1
1
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Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
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– Ethan Bolker
Jan 21 at 21:07
$begingroup$
Actually, I'd rather not delete my answer so that you can delete the question. I'm not the only one who has invested time trying to answer it - and you and others can learn a lot about the nature of notation and how to ask a question by looking at this one, now closed. In principle you can edit the question in hopes of reopening - please don't, since that will make all the responses moot. Do ask more questions one at a time, respectfully, whenever you need the kind of help this site can offer.
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– Ethan Bolker
Jan 21 at 21:07
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I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
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– Gustav
Jan 22 at 14:46
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I respect your decision. However, I do not see how anyone will benefit from the question since it has been put on hold.
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– Gustav
Jan 22 at 14:46
add a comment |
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Inventing new symbols led to the overwhelming (and in my eyes desastrous) flooding of the unicode set with gazillions of mostly useless emojis
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– Hagen von Eitzen
Jan 21 at 19:29
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Your alternative notation for infinite series is horrendous! The traditional notation is simple and clear. (Also, your series for $sin$ should start at $n=0$, not $n=1$. You will no doubt claim that this error actually strengthens your case, but I'm not buying it.)
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– TonyK
Jan 21 at 19:37
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Good notation is quickly understood in a concise manner. Having to unpack your notation for something as simple as sin(x) is already pretty difficult, and it conflicts with broadly used notation already. This post seems more of a complaint/proposal rather than an actual question, and I'm voting to close as off-topic.
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– T. Bongers
Jan 21 at 19:49
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Honestly, it was pretty hard to read the rest of the post after "You see, calculus is quite complicated, but in its pure mathematical state its horrendous. If we had to solve calculus problems with actual mathematics, it would be utterly painful," @Gustav. I think some self-examination might lead you to a different conclusion, since this is starting to feel like "It's not immediately clear to me, so it's bad."
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– T. Bongers
Jan 21 at 19:54
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There is a lot of unsubstantiated opinion here. Just one immediate observation: note that using your notation to find the MacLaurin series of a sum of two functions, given their MacLaurin series, is meaningfully harder than via the usual notation (for example, note the "sign-play" in e.g. $e^x+cos(x)$). Quite frankly I don't think this question can be made appropriate for MSE while keeping its spirit - there's too much presupposition in "What are good reasons for recycling and reusing the same symbols past their clear conveying ability?" (emphasis mine).
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– Noah Schweber
Jan 21 at 19:59