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SEO Keyword summary for en.wikipedia.org/wiki/quaternion
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be quaternions
Focus keyword
Short and long tail
Short Tail Keywords quaternions displaystyle mathbf |
long Tail Keywords (2 words) real numbers complex numbers mathbb h hdisplaystyle mathbb vector part |
long Tail Keywords (3 words) hdisplaystyle mathbb h 1 i j j and k p and q j dmathbf k j d k b i c |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/quaternion page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
quaternion wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia displaystyle amathbf mathbf ain mathbb abmathbf cmathbf dmathbf operatorname cong ijk qabmathbf bmathbf left right beginaligneda muqquad endaligned lambda alambda frac qquad beginalignedmathbf mumathbf qqmathbf beginalignedat endalignedat rvec rin vec vin beginalignedr mur cdot times murvec leftfrac vcdot vrightendaligned tilde qtfrac qmathbf lvert qrvert sqrt qqsqrt alpha leftalpha rightlvert pqrvert prvert det beginpmatrixaibidcidcaibendpmatrixa dpqlvert qfrac qlvert qleftvert qright mathrm pcdot qtextstyle pqqptextstyle pqqp ptimes tfrac beginalignedpptextsptextv muqqtextsqtextvendaligned pqpqtextspqtextvptextsqtextsptextvcdot qtextvptextsqtextvqtextsptextvptextvtimes qtextv beginbmatrixphantom abicdicdiabiendbmatrix aibisigma cisigma disigma isigma sigma mapsto quad beginbmatrixadicbicbiphantom adiendbmatrix beginalignedleftbeginarrayrrrrabcdbadccdabdcbaendarrayrightaleftbeginarrayrrrr endarrayrightbleftbeginarrayrrrr endarrayright cleftbeginarrayrrrr endarrayrightdleftbeginarrayrrrr endarrayrightendaligned beginalignedleftbeginarrayrrrradbcdacbbcadcbdaendarrayrightaleftbeginarrayrrrr amapsto bmapsto icmapsto jdmapsto abi cdimathbf abicdileftrightarrow abcd vphantom abimapsto abq colon qqsvec qqslvert qvrvert qvqsvec qvfrac qvvec qslvert xvec vpm leftsqrt bigl rbigr vvec vsqrt textstyle vvphantom expqsum infty qnnealeftcos sin lnqln arccos qqehat nvarphi qleftcosvarphi hat nsinvarphi varphi aqcosvarphi nmathbf nqsinvarphi qexp qxqxehat nxvarphi qxleftcosxvarphi nsinxvarphi dtextgpqlvert lnp dtextgpqarccos textso msigma nsigma mqquad mneq rprime wrw prime rsigma rmathbf mathcal edit wikidata wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/quaternion
Mobile rendering
Mobile optimizations
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No flash detected !
Mobile improvement
Marketing / lead generation for en.wikipedia.org/wiki/quaternion
Social Media
Facebook shares | Facebook likes | ||
Facebook comments | Tweets | ||
Google +1 |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
quaternion found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
Sitemap found?
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
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en.wikipedia.org wikimedia foundation inc
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w view history
httpsenwikipediaorgwindexphptitlequaternionoldid1247511773
page information
printable version
shoemake ken
kravchenko vladislav
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wiki read
quaternion disambiguation
cayley graph
wikimedia commons
mathematics
number system
complex numbers
hamilton william rowan
mechanics
threedimensional space
blackboard bold
commutative
real numbers
pure mathematics
applied mathematics
quaternions and spatial rotation
threedimensional computer graphics
computer vision
magnetic resonance imaging
crystallographic texture
euler angles
rotation matrices
associative algebra
composition algebras
division algebras
division rings
domain
clifford algebra
classified
frobenius theorem
subrings
isomorphism
hurwitzs theorem
nonassociative
octonions
normed division algebra
sedenions
zero divisors
unit quaternions
group
3sphere
spin3
su2
universal cover
so3
quaternion group
history of quaternions
brougham bridge
dublin
eulers foursquare identity
olinde rodrigues
parameterization of general rotations by four parameters
carl friedrich gauss
points
plane
space
ferdinand georg frobenius
royal irish academy
royal canal
hamilton walk
dunsink observatory
john t graves
london edinburgh and dublin philosophical magazine and journal of science
hamiltons work
geometric
algebraic
hamilton we
tait peter guthrie
kinematics
maxwells equations
quaternion society
hypercomplex
vector analysis
josiah willard gibbs
heaviside oliver
hermann von helmholtz
physics
matrices
gimbal lock
computer graphics
robotics
nuclear magnetic resonance
control theory
signal processing
attitude control
bioinformatics
molecular dynamics
computer simulations
orbital mechanics
number theory
quadratic forms
quantum mechanics
spin
spinors
plate trick
rotation groups
expression
symbols
scalar
field
vector space
basis
identity element
center
distributive property
commutative
associative
multiplication table
multiplicative identity
noncommutative ring
polynomial equation
unit sphere
distributive law
dot product
cross product
conjugate
involution
inverse
square root
norm
tensor of q
tensor
imaginary unit
determinant
metric topology
continuous
versors
polar decomposition
reciprocal
reciprocal
dimension
banach algebra
abelian group
group ring
quotient ring
ideal
additive inverse
commutator
elliptic geometry
hanson
represented as matrices
matrix multiplication
injective
ring homomorphism
matrix ring
matrix trace
conjugate transpose
isomorphism
pauli matrices
transpose
lagranges foursquare theorem
combinatorial design
hurwitz quaternions
euclidean algorithm
cayleydickson construction
sphere
antipodal points
image
subalgebra
homomorphism
normed algebras
union
pencil of planes
real line
complex planes
ring
2sphere
quaternionic analysis
complex variable
mandelbrot set
julia fractals
angle
power
geodesic distance
great arc
rotation operator vector space
conjugation
elements that are conjugate to a given element
rotations in 4dimensional euclidean space
lie group
double covering
matrices
point group
icosahedral group
binary icosahedral group
unitary
integers
halfintegers
lattice
regular 24 cell
schlfli symbol
tetrahedron
regular 600 cell
icosians
regular icosahedron
octahedron
disphenoidal 288cell
quaternion algebra
spinor three dimensions
isomorphic
bivectors
magnitudes
orientations
rotors
homogeneous coordinates
pseudovector
conformal model
pseudoeuclidean spaces
minkowski space
special relativity
lorentz boosts
gamma matrices
geometric algebra
brauer group
central simple algebra
brauer equivalent
artinwedderburn theorem
simple algebras
extension fields
clerk maxwell
w thompson lord kelvin
silberstein ludwik
conversion between quaternions and euler angles
dual quaternions
dualcomplex numbers
exterior algebra
hurwitz quaternion order
hyperbolic quaternions
lnrt sphere
quaternionic manifold
quaternionic matrix
quaternionic polytope
quaternionic projective space
slerp
splitquaternions
tesseract
joachim lambek
national council of teachers of mathematics
porteous ian r
another variable
isbn
springerverlag
bibcode
s2cid
wilhelm blaschke
siggraph
tomb raider
pmid
philosophical magazine
trinity college dublin
oclc
cambridge university press
springer
arxiv
morgan kaufmann
lccn
university of dublin
charles jasper joly
longmans green co
a treatise on electricity and magnetism
encyclopdia britannica
postscript
macfarlane alexander
chisholm hugh
encyclopdia britannica
du val patrick
a history of vector analysis
princeton university press
conway john horton
hazewinkel michiel
american mathematical society
doran chris jl
holt rinehart and winston
encyclopedia of mathematics
ems press
nui maynooth
defence research and development canada
john baez
paulson lawrence c
classification
definable numbers
natural numbers
rational numbers
constructible numbers
algebraic numbers
closedform numbers
periods
computable numbers
settheoretically definable numbers
gaussian integers
gaussian rationals
eisenstein integers
splitcomplex numbers
splitoctonions
bicomplex numbers
biquaternions
bioctonions
dual numbers
sedenions
splitbiquaternions
multicomplex numbers
algebra of physical space
spacetime algebra
planebased geometric algebra
infinities
infinitesimals
cardinal numbers
extended natural numbers
extended real numbers
projective
extended complex numbers
hyperreal numbers
levicivita field
ordinal numbers
supernatural numbers
surreal numbers
superreal numbers
irrational numbers
fuzzy numbers
transcendental numbers
padic numbers
padic solenoids
profinite integers
normal numbers
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www.kopkatalogs.lv
olduli.nli.org.il
foundation.wikimedia.org
foundation.wikimedia.org
www.wikimediafoundation.org
SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
Item | Factor | Pointers | |
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PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 65% | A title should reflect the contents of a site. This site has a 50 % match | |
Title Length | 30% | Limit your title to anywhere between 40 and 70 characters. Your title was 23 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
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Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 534 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 3 level 1 folders and 6 folders above or in the first level of navigation. | |
Headings | 62% | Headers should reflect the contents of a site. This site has a 27 % match | |
Links | 8% | Link anchors should to some degree reflect the contents of a site. This site has a 4 % match | |
Image alt tags | 11% | Image alt tags should to some degree reflect the contents of a site. This site has a 4 % match | |
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Html ratio | 30% | Try to keep the html / text ratio as low as possible. More html means longer loading times. Layout should be handled in a serpate css file | |
Image descriptions | 93% | 93.085106382979 % of all images have been described via the "alt" attribute. Describing images with relevant text may lead to better results in the search engines. | |
Page errors | 100% | Pages with no errors display significantly faster on most browsers. We detected 0 errors and warnings | |
WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 12313 words | |
Server response time | 100% | A fast server speeds up a website. This server responds 89.04% faster then average | |
Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 506 inline style declarations ( <a style="color:green">) with a size of 15188 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
How would you like to have SEO advice for all your pages ?? Start your SEO Dashboard and optimize your website!
en.wikipedia.org images and descriptions
143 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
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https://wikimedia.org/api/rest_v1/media/math/render/svg/acc831004353757280ba962a1ca33071de4380c6 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {i\,j\,k} &=-1.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fd108c7242311c4bc42466350d341f446effd678 height: height attribute not set width: width attribute not set description: {\displaystyle 0=\mathbf {i} \,q-q\,\mathbf {i} =2c\,\mathbf {ij} +2d\,\mathbf {ik} =2c\,\mathbf {k} -2d\,\mathbf {j} ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/587e5ee002a35af32e975540cc1d82f1c393222c height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{alignedat}{4}&a_{1}a_{2}&&+a_{1}b_{2}\mathbf {i} &&+a_{1}c_{2}\mathbf {j} &&+a_{1}d_{2}\mathbf {k} \\{}+{}&b_{1}a_{2}\mathbf {i} &&+b_{1}b_{2}\mathbf {i} ^{2}&&+b_{1}c_{2}\mathbf {ij} &&+b_{1}d_{2}\mathbf {ik} \\{}+{}&c_{1}a_{2}\mathbf {j} &&+c_{1}b_{2}\mathbf {ji} &&+c_{1}c_{2}\mathbf {j} ^{2}&&+c_{1}d_{2}\mathbf {jk} \\{}+{}&d_{1}a_{2}\mathbf {k} &&+d_{1}b_{2}\mathbf {ki} &&+d_{1}c_{2}\mathbf {kj} &&+d_{1}d_{2}\mathbf {k} ^{2}\end{alignedat}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3335e723c66797cd6e8f918c7a8d4142b7894a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{alignedat}{4}&a_{1}a_{2}&&-b_{1}b_{2}&&-c_{1}c_{2}&&-d_{1}d_{2}\\{}+{}(&a_{1}b_{2}&&+b_{1}a_{2}&&+c_{1}d_{2}&&-d_{1}c_{2})\mathbf {i} \\{}+{}(&a_{1}c_{2}&&-b_{1}d_{2}&&+c_{1}a_{2}&&+d_{1}b_{2})\mathbf {j} \\{}+{}(&a_{1}d_{2}&&+b_{1}c_{2}&&-c_{1}b_{2}&&+d_{1}a_{2})\mathbf {k} \end{alignedat}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{3}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2a63723119958c1edf60b19931a1c0e8dc270721 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {q} =(r,\,{\vec {v}}),\ \mathbf {q} \in \mathbb {h} ,\ r\in \mathbb {r} ,\ {\vec {v}}\in \mathbb {r} ^{3},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/af3b95c2bc8ca03f3f3eba649fd30351e7d69a19 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}(r_{1},\,{\vec {v}}_{1})+(r_{2},\,{\vec {v}}_{2})&=(r_{1}+r_{2},\,{\vec {v}}_{1}+{\vec {v}}_{2}),\\[5mu](r_{1},\,{\vec {v}}_{1})(r_{2},\,{\vec {v}}_{2})&=(r_{1}r_{2}-{\vec {v}}_{1}\cdot {\vec {v}}_{2},\,r_{1}{\vec {v}}_{2}+r_{2}{\vec {v}}_{1}+{\vec {v}}_{1}\times {\vec {v}}_{2}),\\[5mu](r,\,{\vec {v}})^{-1}&=\left({\frac {r}{r^{2}+{\vec {v}}\cdot {\vec {v}}}},\ {\frac {-{\vec {v}}}{r^{2}+{\vec {v}}\cdot {\vec {v}}}}\right),\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/25742624c37b8e9e12571c0c4c14a3971058891c height: height attribute not set width: width attribute not set description: {\displaystyle {}\cdot {}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703 height: height attribute not set width: width attribute not set description: {\displaystyle \times } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/708ac96c1741cb088eddaa1fa8629ae45402d755 height: height attribute not set width: width attribute not set description: {\displaystyle q^{*}=a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9799511dfd2e51ec2827f6cdab2b75ed8cd1595e height: height attribute not set width: width attribute not set description: {\displaystyle {\tilde {q}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0c4a22837b277a3fbb4fce958b3859e793e455 height: height attribute not set width: width attribute not set description: {\displaystyle q^{*}=-{\tfrac {1}{2}}(q+\mathbf {i} \,q\,\mathbf {i} +\mathbf {j} \,q\,\mathbf {j} +\mathbf {k} \,q\,\mathbf {k} ).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/79bff63c2c0287d00cd638996d95c7452cf0167d height: height attribute not set width: width attribute not set description: {\displaystyle \lvert q\rvert ={\sqrt {qq^{*}}}={\sqrt {q^{*}q}}={\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{4}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/db538c966b99d537d09ff11090432ea627569cd4 height: height attribute not set width: width attribute not set description: {\displaystyle \lvert \alpha q\rvert =\left|\alpha \right|\,\lvert q\rvert .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d944a047b1b2554a3305c081598c1f9e987bd3c6 height: height attribute not set width: width attribute not set description: {\displaystyle \lvert pq\rvert =\lvert p\rvert \,\lvert q\rvert } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9432958fe8d23609d1ffb50314f509409eb09462 height: height attribute not set width: width attribute not set description: {\displaystyle \det {\begin{pmatrix}a+ib&id+c\\id-c&a-ib\end{pmatrix}}=a^{2}+b^{2}+c^{2}+d^{2},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc34f13d3a6b44a1aee897f3ba570a1efb0d802 height: height attribute not set width: width attribute not set description: {\displaystyle d(p,q)=\lvert p-q\rvert .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/84a7b1bc090820afe2834c251afc3005b5c222ca height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {u} q={\frac {q}{\lvert q\rvert }}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c71d4f268efe4169216fb2e743be350597f289d6 height: height attribute not set width: width attribute not set description: {\displaystyle q=\lvert q\rvert \cdot \mathbf {u} q,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d height: height attribute not set width: width attribute not set description: {\displaystyle q} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c792b8b703da26f580674036636ab548cb5723 height: height attribute not set width: width attribute not set description: {\displaystyle q^{*}/\left\vert q\right\|^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/511d5e9ee6ef17a8d1800767a82de4207e6f6191 height: height attribute not set width: width attribute not set description: {\displaystyle q^{-1}={\frac {q^{*}}{\lvert q\rvert ^{2}}}.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/cayley_graph_q8.svg/220px-cayley_graph_q8.svg.png height: 220 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {c} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {o} } |
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https://upload.wikimedia.org/wikipedia/commons/thumb/0/02/quaternion-multiplication-cayley-3d-with-legend.png/220px-quaternion-multiplication-cayley-3d-with-legend.png height: 165 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3ca1ed22f8ae5680236bb417a05d2e38886ae height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} [\mathrm {q} _{8}]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee56242c98ca2b1e828702ebadcb314876e0035 height: height attribute not set width: width attribute not set description: {\displaystyle \mathrm {q} _{8}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ae613b45e932009c0ba932442199b5e65b9bffc9 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{3};} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{3},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6441b2f0008410fe825d3d46ddc11c823429f4 height: height attribute not set width: width attribute not set description: {\displaystyle p\cdot q=b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc3f138ca902921efcc2bc3c7b468171925b656 height: height attribute not set width: width attribute not set description: {\displaystyle p\cdot q=\textstyle {\frac {1}{2}}(p^{*}q+q^{*}p)=\textstyle {\frac {1}{2}}(pq^{*}+qp^{*}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb71a6d026b97a69d27be9b2f803a3e49646cfb height: height attribute not set width: width attribute not set description: {\displaystyle p\times q=(c_{1}d_{2}-d_{1}c_{2})\mathbf {i} +(d_{1}b_{2}-b_{1}d_{2})\mathbf {j} +(b_{1}c_{2}-c_{1}b_{2})\mathbf {k} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2b00d42142debab04bb87f40904c9b1b5e0cb96b height: height attribute not set width: width attribute not set description: {\displaystyle p\times q=\textstyle {\tfrac {1}{2}}(pq-qp).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c45178772ffab33a9936c42a9df2be320f5d22b8 height: height attribute not set width: width attribute not set description: {\displaystyle [p,q]=2p\times q.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab7ef853cfa04d68fc3cd8add199f463e3eb1c8 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}p&=p_{\text{s}}+p_{\text{v}},\\[5mu]q&=q_{\text{s}}+q_{\text{v}},\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/972c92325dab28b3c8ba0bce039019f3dff4c78b height: height attribute not set width: width attribute not set description: {\displaystyle pq=(pq)_{\text{s}}+(pq)_{\text{v}}=(p_{\text{s}}q_{\text{s}}-p_{\text{v}}\cdot q_{\text{v}})+(p_{\text{s}}q_{\text{v}}+q_{\text{s}}p_{\text{v}}+p_{\text{v}}\times q_{\text{v}}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{3}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/004243df8d7aa9545f0de583e4ae030af4bc08fe height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{bmatrix}{\phantom {-}}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf12559748efda4130ab7f119da487da68c8cad height: height attribute not set width: width attribute not set description: {\displaystyle ai+bi\sigma _{3}+ci\sigma _{2}+di\sigma _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/157e31db2e8b7f8f9aa2295fad7aed9db3638035 height: height attribute not set width: width attribute not set description: {\displaystyle \pm i\sigma _{3}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/209b6c96156e18df0d881fc365d3b14fc7619be1 height: height attribute not set width: width attribute not set description: {\displaystyle \pm i\sigma _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/99c82a9c4d81cdb56c7e61d19302aa844f265fa5 height: height attribute not set width: width attribute not set description: {\displaystyle \pm i\sigma _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd1d0bd1a7b18a273a7c1adbc00ef74f2a13b69 height: height attribute not set width: width attribute not set description: {\displaystyle \pm \sigma _{1}\sigma _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/96a45e59139d09e686aa9ddf7a65b245ad7b8d41 height: height attribute not set width: width attribute not set description: {\displaystyle \pm \sigma _{3}\sigma _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a062a21b9e71179661e08423370f44b6058cb5fe height: height attribute not set width: width attribute not set description: {\displaystyle \pm \sigma _{2}\sigma _{3}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c0af8a12e268387f8c9eeaf6ed5a2e4cfd713e91 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i\;j\;k} =\sigma _{1}\sigma _{2}\sigma _{3}\sigma _{1}\sigma _{2}\sigma _{3}=-1.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/82cefdfa947ffbae7e82c2981e7a39131b3bf146 height: height attribute not set width: width attribute not set description: {\displaystyle 1\mapsto \mathbf {i} ,\quad \mathbf {i} \mapsto -i\sigma _{1}=-\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -i\sigma _{2}=-\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -i\sigma _{3}=-\sigma _{1}\sigma _{2},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f1412da754e8b7f9bae246d9446dfe91176dd1bf height: height attribute not set width: width attribute not set description: {\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} \mapsto {\begin{bmatrix}a-di&-c-bi\\c-bi&{\phantom {-}}a+di\end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f963faf6a3e52166e4f4421c4cbb7f5b111699 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\left[{\begin{array}{rrrr}a&-b&-c&-d\\b&a&-d&c\\c&d&a&-b\\d&-c&b&a\end{array}}\right]&=a\left[{\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]+b\left[{\begin{array}{rrrr}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{array}}\right]\\[10mu]&\qquad +c\left[{\begin{array}{rrrr}0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{array}}\right]+d\left[{\begin{array}{rrrr}0&0&0&-1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{array}}\right].\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5cdb3a03c5a1b9df69d9244469a9344df49978ad height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\left[{\begin{array}{rrrr}a&d&-b&-c\\-d&a&c&-b\\b&-c&a&-d\\c&b&d&a\end{array}}\right]&=a\left[{\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]+b\left[{\begin{array}{rrrr}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{array}}\right]\\[10mu]&\qquad +c\left[{\begin{array}{rrrr}0&0&0&-1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{array}}\right]+d\left[{\begin{array}{rrrr}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{array}}\right].\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a862a6ce296ade620aa708d384521a46d6c06037 height: height attribute not set width: width attribute not set description: {\displaystyle \{a\mapsto 1,\,b\mapsto i,\,c\mapsto j,\,d\mapsto k\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {c} ^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/19d96e72562ed8ee65be996e5c984e238ddcb85e height: height attribute not set width: width attribute not set description: {\displaystyle (a+bi)1+(c+di)\mathbf {j} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/250f48c76c690baa42415c524ec31a9c0c7825e9 height: height attribute not set width: width attribute not set description: {\displaystyle (a+bi,\,c+di)\leftrightarrow (a,\,b,\,c,\,d).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {c} ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2fff161699a746d453fc9110f35c86a9fbcb4164 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}a^{2}-b^{2}-c^{2}-d^{2}&=-1,{\vphantom {x^{|}}}\\[3mu]2ab&=0,\\[3mu]2ac&=0,\\[3mu]2ad&=0.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4a20511dde2ccc49ee007f85f928f47fd1f36285 height: height attribute not set width: width attribute not set description: {\displaystyle a+bi\mapsto a+bq.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1d178e5ac94e706fdb8d8733d567b7c087b23195 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {h} ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/80340ec51617aad6d38eaa91cd47e6fdfb2b1621 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {h} \colon } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c00f7a6daf5d48c7f9d6eadc3ee53403eb51f98 height: height attribute not set width: width attribute not set description: {\displaystyle q=q_{s}+{\vec {q}}_{v}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b26c47b075c757d64e351f71a0f4d00b6ed3aa height: height attribute not set width: width attribute not set description: {\displaystyle q=q_{s}+\lvert {\vec {q}}_{v}\rvert \cdot \mathbf {u} {\vec {q}}_{v}=q_{s}+\|{\vec {q}}_{v}\|\,{\frac {{\vec {q}}_{v}}{\|{\vec {q}}_{v}\|}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bdac5fe01487eab56d4f07a5ecaddf8a5dde2842 height: height attribute not set width: width attribute not set description: {\displaystyle q_{s}+\lvert q\rvert \cdot \mathbf {u} q} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23909fad68ad27b3e861672f14077358621e8e34 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {u} {\vec {q}}_{v}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e8fb0dae06ab3c3270122900c2a2bfd0357ac209 height: height attribute not set width: width attribute not set description: {\displaystyle a+bi\mapsto a+b\mathbf {u} {\vec {q}}_{v}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55e9cc29f39f5db36c403acbea3b69e101282383 height: height attribute not set width: width attribute not set description: {\displaystyle q_{s}+\lvert {\vec {q}}_{v}\rvert i} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/df4691c64f3da26687c8a3b9a93847823ac9ea81 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {q} =(r,\,{\vec {v}})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2bcb54ae9eeaa87583450660b0cb3648b093bb58 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\mathbf {q} }}=(x,\,{\vec {y}})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fea9bf2756e436fdd7582c7b70126588aaffd42c height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\mathbf {q} }}^{\,2}=(x,\,{\vec {y}})^{2}=\mathbf {q} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2a49b0fba4dbd5a5cb46c04c017f8ac2a8ee3de9 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\mathbf {q} }}={\sqrt {(r,\,{\vec {v}})}}=\pm \left({\sqrt {{\tfrac {1}{2}}{\bigl (}{\|\mathbf {q} \|+r}{\bigr )}}},\ {\frac {\vec {v}}{\|{\vec {v}}\|}}{\sqrt {{\tfrac {1}{2}}{\bigl (}{\|\mathbf {q} \|-r}{\bigr )}}}\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b60bad03052e64185d3e3bb73707c93392a192b6 height: height attribute not set width: width attribute not set description: {\textstyle \|{\vec {v}}\|={\sqrt {{\vec {v}}\cdot {\vec {v}}}}={\sqrt {-{\vec {v}}{\vphantom {v}}^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753 height: height attribute not set width: width attribute not set description: {\displaystyle {\vec {v}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5df705929c720712faa5500967512a610dfae0 height: height attribute not set width: width attribute not set description: {\textstyle \|\mathbf {q} \|={\sqrt {\mathbf {q} ^{*}\mathbf {q} }}=r^{2}+\|{\vec {v}}\|^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7be005a326b7ac3fe4c24bca391369f44c4c2876 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {q} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9397e0c83526c381d6dfbfe5009898fdf8bb48a7 height: height attribute not set width: width attribute not set description: {\textstyle {\vec {v}}/\|{\vec {v}}\|} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/158a339669f24367975e5fc87e0c6bf6a0517201 height: height attribute not set width: width attribute not set description: {\displaystyle \{0\}\times s^{2}{\bigl (}{\sqrt {-r}}{\bigr )}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3681ea2827b4db07689854249f0b3abb47095d19 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {-r}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/quaternion_julia_x%3d-0%2c75_y%3d-0%2c14.jpg/220px-quaternion_julia_x%3d-0%2c75_y%3d-0%2c14.jpg height: 127 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2c87d169e8d42b260e1035501eec55541ce206e6 height: height attribute not set width: width attribute not set description: {\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} =a+\mathbf {v} ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2b86a224c99d4900c693a1c353cf11e77589a7d0 height: height attribute not set width: width attribute not set description: {\displaystyle \exp(q)=\sum _{n=0}^{\infty }{\frac {q^{n}}{n!}}=e^{a}\left(\cos \|\mathbf {v} \|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\sin \|\mathbf {v} \|\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/60a21d39b4a86784735d723fa15a6cf161c5e2f2 height: height attribute not set width: width attribute not set description: {\displaystyle \ln(q)=\ln \|q\|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\arccos {\frac {a}{\|q\|}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d10dc2aa59c608b3e3393eac2f5ba875f78643c1 height: height attribute not set width: width attribute not set description: {\displaystyle q=\|q\|e^{{\hat {n}}\varphi }=\|q\|\left(\cos(\varphi )+{\hat {n}}\sin(\varphi )\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e height: height attribute not set width: width attribute not set description: {\displaystyle \varphi } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7e3f2b6b74ed01db9de286b12800fc93bd7d64 height: height attribute not set width: width attribute not set description: {\displaystyle a=\|q\|\cos(\varphi )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d125dccc556f5c8b0bf98a4f3847590b3f353bd4 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b5179e2f11afc0ee8712b030ace9078e5acbc1b0 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {v} ={\hat {n}}\|\mathbf {v} \|={\hat {n}}\|q\|\sin(\varphi ).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3db72b8b435f97ca53aa820773f3ddde9764e113 height: height attribute not set width: width attribute not set description: {\displaystyle q=\exp {({\hat {n}}\varphi )}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3655120c695e09b84b4fc6cc766950fbfaf9ba height: height attribute not set width: width attribute not set description: {\displaystyle q^{x}=\|q\|^{x}e^{{\hat {n}}x\varphi }=\|q\|^{x}\left(\cos(x\varphi )+{\hat {n}}\,\sin(x\varphi )\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1f1e6ce56253b44d6a847e264f4363c253d85c height: height attribute not set width: width attribute not set description: {\displaystyle d_{\text{g}}(p,q)=\lvert \ln(p^{-1}q)\rvert .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe762bbe2b556f76f109393da2c3feb7d28abe94 height: height attribute not set width: width attribute not set description: {\displaystyle d_{\text{g}}(p,q)=\arccos(2(p\cdot q)^{2}-1).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/83f067485af5efb7fd272a8ea6467a6660d85951 height: height attribute not set width: width attribute not set description: {\displaystyle {\text{so}}(3,\mathbb {r} )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1339d724f8ca701c44e265b7d8bb5f811f8b96d5 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {cl} _{3,0}^{+}(\mathbb {r} )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a939301535d01bb585a0dedfcbd22f611d27331d height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {cl} _{3,0}(\mathbb {r} ).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f90fd7dccbc9d67ac5014160bda21eb5b4d93ef7 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=1,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c955aec13267d08431e0cba1c3fb7c7badfd1068 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma _{m}\sigma _{n}=-\sigma _{n}\sigma _{m}\qquad (m\neq n).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8266db74ad9449f62b6669bf6e570e28639dac36 height: height attribute not set width: width attribute not set description: {\displaystyle r^{\prime }=-w\,r\,w.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/46ebdc0a4b1149c9e8699a8ef7f069df2208b03b height: height attribute not set width: width attribute not set description: {\displaystyle r^{\prime \prime }=\sigma _{2}\sigma _{1}\,r\,\sigma _{1}\sigma _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/53db13c149bb524cc7364f956df6ab5e8496ca1f height: height attribute not set width: width attribute not set description: {\displaystyle r^{\prime \prime }=-\mathbf {k} \,r\,\mathbf {k} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b70621dfe37f00efef2a1dec074cc4f6fefb2ae3 height: height attribute not set width: width attribute not set description: {\displaystyle 1\mapsto 1,\quad \mathbf {i} \mapsto -\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -\sigma _{1}\sigma _{2},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d034bbc96465b01864c61099925b9780ee73f102 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i\,j\,k} =-1.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/acd427d3aafa49899a7d4c5884bacce3414992f6 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {cl} _{4,0}(\mathbb {r} ),} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/wikibooks-logo-en-noslogan.svg/40px-wikibooks-logo-en-noslogan.svg.png height: 40 width: 40 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/wikiquote-logo.svg/34px-wikiquote-logo.svg.png height: 40 width: 34 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/wiktionary-logo-en-v2.svg/40px-wiktionary-logo-en-v2.svg.png height: 40 width: 40 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/commons-logo.svg/12px-commons-logo.svg.png height: 16 width: 12 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {n} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {z} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {q} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {a} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {s} } |
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http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
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http://en.wikipedia.org/w/resources/assets/poweredby_mediawiki.svg height: 31 width: 88 description: powered by mediawiki |
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