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Wikipedia:Reference desk/Mathematics (view source)
Revision as of 06:17, 9 February 2022
, 06:17, 9 February 2022Imported page
<noinclude>{{Wikipedia:Reference desk/header|WP:RD/MA}}
[[Category:Non-talk pages that are automatically signed]]
[[Category:Pages automatically checked for incorrect links]]
[[Category:Wikipedia resources for researchers]]
[[Category:Wikipedia help forums]]
[[Category:Wikipedia reference desk|Mathematics]]
[[Category:Wikipedia help pages with dated sections]]</noinclude>
= February 2 =
== English ==
What is noun phrase <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Ibrahim shaaibu|Ibrahim shaaibu]] ([[User talk:Ibrahim shaaibu#top|talk]] • [[Special:Contributions/Ibrahim shaaibu|contribs]]) 01:30, 2 February 2022 (UTC)</span> <!--Autosigned by SineBot-->
:This is really a question for the [[WP:RDL|Language section]], but see our [[Noun phrase]] article. All noun phrases in the following lines are underlined:
::{{tq|Twinkle, twinkle, <u>little star</u>,}}
::{{tq|How I wonder what you are!}}
::{{tq|Up above <u>the world</u> so high,}}
::{{tq|Like <u>a diamond in the sky</u>.}}
:The noun phrase in the last line contains two embedded noun phrases: {{tq|<u>a diamond</u>}} and {{tq|<u>the sky</u>}}. --[[User talk:Lambiam|Lambiam]] 02:44, 2 February 2022 (UTC)
= February 3 =
== Definition of a topological space by neighborhoods ==
In the article on "[[Topological_space|topological space]]" the section Definition by Neighborhoods lists four axioms. I don't see how axiom 4 is independent.
Axiom 1 says:
If N is a neighborhood of point x, then x is an element of N.
Axiom 4 says:
If N is a neighborhood of point x, then N contains a neighborhood M of x such that N is a neighborhood of every point in M.
Doesn't 4 follow directly from 1? The singleton set containing only the point x is a subset of every neighborhood of x, so it always satisfies the conditions for neighborhood M in 4.
What am I missing here? Can you point me to a description of axiom 4 that I can understand without a deep background in topology? <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Bronco creek|Bronco creek]] ([[User talk:Bronco creek#top|talk]] • [[Special:Contributions/Bronco creek|contribs]]) 19:01, 3 February 2022 (UTC)</span> <!--Autosigned by SineBot-->
:What you're missing is that the singleton {''x''} might not be a neighborhood of ''x''. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:09, 3 February 2022 (UTC)
Thanks. That'll teach me to trust my intuition. <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Bronco creek|Bronco creek]] ([[User talk:Bronco creek#top|talk]] • [[Special:Contributions/Bronco creek|contribs]]) 19:54, 3 February 2022 (UTC)</span> <!--Autosigned by SineBot-->
:The key is not so much not to trust intuition, as to get a more trustworthy intuition.
:Intuition is fundamental to mathematics. You simply can't do anything nontrivial without it. But you have to ''develop'' it. And you have to keep testing it, looking for places where it doesn't serve you well, and be willing to change those.
:In this particular case, the intuition that might have served you is that a neighborhood is a set that contains a given point, plus a little margin around it. That ordinarily rules out singletons. Note that it doesn't ''always'' rule out singletons, but that's the base case you can have in mind, being aware of exceptions. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:19, 3 February 2022 (UTC)
:: {{tq|The key is not so much not to trust intuition, as to get a more trustworthy intuition.}} Words to live by. --[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 22:58, 3 February 2022 (UTC)
= February 7 =
== Height of Simplexes ==
I'm working my way through [[Simplex]] and have been unable to find exactly what the height of each simplex is. I'm defining the "height" of an n-simplex as the distance from one vertex to the center of the n-1 simplex made up of the other vertices. So the "Height" of a 2-simplex is sqrt(3)/2. and the "Height" of a 3-simplex is sqrt(6)/3 (according to [[Tetrahedron]]). I'm presuming that they tend to 0 as n goes to infinity, but I don't know if there is an easy formula.[[User:Naraht|Naraht]] ([[User talk:Naraht|talk]]) 09:37, 7 February 2022 (UTC)
: That obviously depends on the choice of a vertex, as in 2D example you may have three different heights of a triangle, depending on a vertex chosen as an apex. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 10:21, 7 February 2022 (UTC)
:I think we are assuming that it is the "standard simplex", which will be symmetric with respect to choice of vertex. See a discussion here: https://math.stackexchange.com/questions/1697870/height-of-n-simplex showing that this height is <math>\sqrt{\frac{n+1}{2n}}</math>. <s>Your answer is different I guess because you assumed the simplex edge length is 1? The typical construction of the standard simplex makes the edge length <math>\sqrt 2</math>.</s> [[User:Staecker|Staecker]] ([[User talk:Staecker|talk]]) 12:34, 7 February 2022 (UTC)
::The formula <math>\sqrt{\frac{n+1}{2n}}</math> works for edge length <math>1</math>; the height of a 1-dimensional simplex is the length of its single edge, so just put <math>n=1</math> and indeed <math>1</math> pops out. --[[User talk:Lambiam|Lambiam]] 13:07, 7 February 2022 (UTC)
:::Yes Lambiam is right- I popped it out wring! [[User:Staecker|Staecker]] ([[User talk:Staecker|talk]]) 14:39, 7 February 2022 (UTC)
::::So sqrts of 2/2, 3/4 , 4/6, 5/8, 6/10 , so 1, sqrt(3)/2, sqrt(4/6)= sqrt(6/9) = sqrt(6)/3, sqrt(5/8) = sqrt(10)/4, sqrt(6/10)=sqrt(3/5)=sqrt(15)/5 etc. which as n-> inf approaches sqrt(1/2) = sqrt(2)/2. Interesting.
:::::In contrast, the volume approaches <math>0</math> more than exponentially fast; see the last paragraph of {{section link|Simplex#Volume}}. The volume formula gives an alternative way to derive the height formula, using an obvious way of obtaining the volume of the regular <math>(n{+}1)</math>-simplex from that of its predecessor by integration over the height. --[[User talk:Lambiam|Lambiam]] 21:10, 7 February 2022 (UTC)
= February 8 =
== Multiplication methods ==
2 310(35) <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/41.114.170.234|41.114.170.234]] ([[User talk:41.114.170.234#top|talk]]) 19:51, 8 February 2022 (UTC)</small> <!--Autosigned by SineBot-->
:Do you have a question for us? You only posted some numbers. --[[User:Jayron32|<span style="color:#009">Jayron</span>]][[User talk:Jayron32|<b style="color:#090">''32''</b>]] 20:28, 8 February 2022 (UTC)
:2310 = 35 × 66. Is that somehow related? --[[User talk:Lambiam|Lambiam]] 21:16, 8 February 2022 (UTC)
= February 9 =
[[Category:Non-talk pages that are automatically signed]]
[[Category:Pages automatically checked for incorrect links]]
[[Category:Wikipedia resources for researchers]]
[[Category:Wikipedia help forums]]
[[Category:Wikipedia reference desk|Mathematics]]
[[Category:Wikipedia help pages with dated sections]]</noinclude>
= February 2 =
== English ==
What is noun phrase <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Ibrahim shaaibu|Ibrahim shaaibu]] ([[User talk:Ibrahim shaaibu#top|talk]] • [[Special:Contributions/Ibrahim shaaibu|contribs]]) 01:30, 2 February 2022 (UTC)</span> <!--Autosigned by SineBot-->
:This is really a question for the [[WP:RDL|Language section]], but see our [[Noun phrase]] article. All noun phrases in the following lines are underlined:
::{{tq|Twinkle, twinkle, <u>little star</u>,}}
::{{tq|How I wonder what you are!}}
::{{tq|Up above <u>the world</u> so high,}}
::{{tq|Like <u>a diamond in the sky</u>.}}
:The noun phrase in the last line contains two embedded noun phrases: {{tq|<u>a diamond</u>}} and {{tq|<u>the sky</u>}}. --[[User talk:Lambiam|Lambiam]] 02:44, 2 February 2022 (UTC)
= February 3 =
== Definition of a topological space by neighborhoods ==
In the article on "[[Topological_space|topological space]]" the section Definition by Neighborhoods lists four axioms. I don't see how axiom 4 is independent.
Axiom 1 says:
If N is a neighborhood of point x, then x is an element of N.
Axiom 4 says:
If N is a neighborhood of point x, then N contains a neighborhood M of x such that N is a neighborhood of every point in M.
Doesn't 4 follow directly from 1? The singleton set containing only the point x is a subset of every neighborhood of x, so it always satisfies the conditions for neighborhood M in 4.
What am I missing here? Can you point me to a description of axiom 4 that I can understand without a deep background in topology? <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Bronco creek|Bronco creek]] ([[User talk:Bronco creek#top|talk]] • [[Special:Contributions/Bronco creek|contribs]]) 19:01, 3 February 2022 (UTC)</span> <!--Autosigned by SineBot-->
:What you're missing is that the singleton {''x''} might not be a neighborhood of ''x''. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:09, 3 February 2022 (UTC)
Thanks. That'll teach me to trust my intuition. <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Bronco creek|Bronco creek]] ([[User talk:Bronco creek#top|talk]] • [[Special:Contributions/Bronco creek|contribs]]) 19:54, 3 February 2022 (UTC)</span> <!--Autosigned by SineBot-->
:The key is not so much not to trust intuition, as to get a more trustworthy intuition.
:Intuition is fundamental to mathematics. You simply can't do anything nontrivial without it. But you have to ''develop'' it. And you have to keep testing it, looking for places where it doesn't serve you well, and be willing to change those.
:In this particular case, the intuition that might have served you is that a neighborhood is a set that contains a given point, plus a little margin around it. That ordinarily rules out singletons. Note that it doesn't ''always'' rule out singletons, but that's the base case you can have in mind, being aware of exceptions. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:19, 3 February 2022 (UTC)
:: {{tq|The key is not so much not to trust intuition, as to get a more trustworthy intuition.}} Words to live by. --[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 22:58, 3 February 2022 (UTC)
= February 7 =
== Height of Simplexes ==
I'm working my way through [[Simplex]] and have been unable to find exactly what the height of each simplex is. I'm defining the "height" of an n-simplex as the distance from one vertex to the center of the n-1 simplex made up of the other vertices. So the "Height" of a 2-simplex is sqrt(3)/2. and the "Height" of a 3-simplex is sqrt(6)/3 (according to [[Tetrahedron]]). I'm presuming that they tend to 0 as n goes to infinity, but I don't know if there is an easy formula.[[User:Naraht|Naraht]] ([[User talk:Naraht|talk]]) 09:37, 7 February 2022 (UTC)
: That obviously depends on the choice of a vertex, as in 2D example you may have three different heights of a triangle, depending on a vertex chosen as an apex. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 10:21, 7 February 2022 (UTC)
:I think we are assuming that it is the "standard simplex", which will be symmetric with respect to choice of vertex. See a discussion here: https://math.stackexchange.com/questions/1697870/height-of-n-simplex showing that this height is <math>\sqrt{\frac{n+1}{2n}}</math>. <s>Your answer is different I guess because you assumed the simplex edge length is 1? The typical construction of the standard simplex makes the edge length <math>\sqrt 2</math>.</s> [[User:Staecker|Staecker]] ([[User talk:Staecker|talk]]) 12:34, 7 February 2022 (UTC)
::The formula <math>\sqrt{\frac{n+1}{2n}}</math> works for edge length <math>1</math>; the height of a 1-dimensional simplex is the length of its single edge, so just put <math>n=1</math> and indeed <math>1</math> pops out. --[[User talk:Lambiam|Lambiam]] 13:07, 7 February 2022 (UTC)
:::Yes Lambiam is right- I popped it out wring! [[User:Staecker|Staecker]] ([[User talk:Staecker|talk]]) 14:39, 7 February 2022 (UTC)
::::So sqrts of 2/2, 3/4 , 4/6, 5/8, 6/10 , so 1, sqrt(3)/2, sqrt(4/6)= sqrt(6/9) = sqrt(6)/3, sqrt(5/8) = sqrt(10)/4, sqrt(6/10)=sqrt(3/5)=sqrt(15)/5 etc. which as n-> inf approaches sqrt(1/2) = sqrt(2)/2. Interesting.
:::::In contrast, the volume approaches <math>0</math> more than exponentially fast; see the last paragraph of {{section link|Simplex#Volume}}. The volume formula gives an alternative way to derive the height formula, using an obvious way of obtaining the volume of the regular <math>(n{+}1)</math>-simplex from that of its predecessor by integration over the height. --[[User talk:Lambiam|Lambiam]] 21:10, 7 February 2022 (UTC)
= February 8 =
== Multiplication methods ==
2 310(35) <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/41.114.170.234|41.114.170.234]] ([[User talk:41.114.170.234#top|talk]]) 19:51, 8 February 2022 (UTC)</small> <!--Autosigned by SineBot-->
:Do you have a question for us? You only posted some numbers. --[[User:Jayron32|<span style="color:#009">Jayron</span>]][[User talk:Jayron32|<b style="color:#090">''32''</b>]] 20:28, 8 February 2022 (UTC)
:2310 = 35 × 66. Is that somehow related? --[[User talk:Lambiam|Lambiam]] 21:16, 8 February 2022 (UTC)
= February 9 =