octonion
non-commutative, non-associative algebra of numbers with eight real components
| rdfs:label |
"octonion"@en
|
|
skos:alt |
"Cayley number"@en
"Cayley numbers"@en
"octave"@en
"octaves"@en
"octonions"@en
|
| schema:description |
"non-commutative, non-associative algebra of numbers with eight real components"@en
|
|
schema:date |
"2026-05-26T04:26:52+00:00"^^xsd:dateTime |
| schema:version | 2497650042 |
| wikibase:identifiers | 15 |
| wikibase:sitelinks | 36 |
| wikibase:statements | 38 |
| p:P10376 | wd:statement/Q743418-c3502535-4d3b-ae70-cc2a-df854db92c7e |
| p:P13591 | wd:statement/Q743418-99164630-147B-4C53-928F-8DFCAED6CBF8 |
| p:P1482 | wd:statement/Q743418-265f0edb-486f-b2b6-47f5-70d808d54076 |
| wd:statement/Q743418-3d2793d8-4f1f-6fdc-cbf4-a843770fac16 | |
| p:P155 | wd:statement/Q743418-6e65700e-4031-5869-d600-09711a6b88a0 |
| p:P156 | wd:statement/Q743418-c7a31b70-4d65-20d6-6e19-a176c455e7be |
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| p:P2163 | wd:statement/Q743418-8fd284d3-4afe-2582-b92c-e4639c747270 |
| p:P227 | wd:statement/Q743418-72f8bc03-4e6e-3de7-69f8-c5efe6d9c4c3 |
| p:P244 | wd:statement/Q743418-AE654345-E1EA-42DC-878E-A904C0DA1E66 |
| p:P2534 | wd:statement/Q743418-e98092be-4e38-50b9-03b0-9d7356bf2149 |
| p:P268 | wd:statement/Q743418-613d713f-469e-fac0-c0d5-47a6c27bddf0 |
| p:P269 | wd:statement/Q743418-11c0cc15-47a6-d3bb-3d45-b600125a4d3b |
| p:P279 | wd:statement/Q743418-5a52e529-49a6-234c-5293-26e2fa677d62 |
| wd:statement/Q743418-8a88b442-46d8-dca7-e482-d44787247d55 | |
| wd:statement/Q743418-a6e0c32d-458a-2cc3-f52d-c05bc377f1c7 | |
| wd:statement/Q743418-ae7a6177-42f0-e0b7-dcf2-47b162a6f045 | |
| wd:statement/Q743418-f9560e22-4223-54e6-aed4-910b8e8d40cf | |
| p:P2812 | wd:statement/Q743418-A2978209-AA93-4ED1-AF54-47B99C31FBFE |
| wd:statement/Q743418-E3F9C42F-6B18-42C6-8B90-EB8884AAE28B | |
| p:P31 | wd:statement/Q743418-565fa215-4265-062c-c02b-6f874dbb2bf9 |
| wd:statement/Q743418-70ef22c0-48d9-ca4b-952c-588f507c95a5 | |
| p:P361 | wd:statement/Q743418-4b7746c3-443b-2a1b-6fbd-ddef92ac3ea8 |
| p:P373 | wd:statement/Q743418-655630B4-A8A7-4071-8A0F-39B2B371C648 |
| p:P4215 | wd:statement/Q743418-E335E0D6-8FFB-4458-B7BC-73873DBFD648 |
| p:P527 | wd:statement/Q743418-0bc2cd71-40ef-901d-32cb-3841948948a2 |
| p:P5555 | wd:statement/Q743418-6c1ba2bb-46c6-b1ef-736a-fd39258f5de1 |
| p:P575 | wd:statement/Q743418-39ca2545-48cc-b7af-c474-7680bd45b7c5 |
| p:P61 | wd:statement/Q743418-74d79f63-4bfe-909f-482a-2841f547bf32 |
| p:P6104 | wd:statement/Q743418-1D166FCF-D1DD-4BB2-AC6B-46D4FBE38079 |
| p:P6366 | wd:statement/Q743418-D1C06324-B320-47BB-A4AC-9C27FD31A0E6 |
| p:P646 | wd:statement/Q743418-D14DD54E-08E1-4C1B-9DA0-8CD5D00BDAFE |
| p:P6781 | wd:statement/Q743418-c15b5d25-4822-b515-d1f7-408c9ae96dad |
| p:P7235 | wd:statement/Q743418-2fdc688d-442a-bffd-568b-94b77ef2e0b6 |
| wd:statement/Q743418-e4791195-4aa8-5ace-4e66-1715c5a8b38c | |
| p:P8189 | wd:statement/Q743418-CDD55727-871E-48D6-92E8-E82673D8ADF4 |
| p:P8885 | wd:statement/Q743418-86f0582b-4d69-a74d-0c2c-faac07fc5c1b |
| p:P910 | wd:statement/Q743418-2F62D871-29EE-4CAC-98EA-6E10A8CC7735 |
|
p:direct- |
<http:/ |
|
p:direct- |
<https:/ |
|
p:direct- |
<http:/ |
|
p:direct- |
<http:/ |
|
p:direct- |
<http:/ |
|
p:direct- |
<https:/ |
| wdt:P10376 | "mathematics/octonions" |
| wdt:P13591 | "concept/55f53b31-2e07-4558-b872-9767f8de88cc" |
| wdt:P1482 |
<https:/ |
|
<https:/ |
|
| wdt:P155 | quaternion |
| wdt:P156 | sedenion |
| wdt:P18 |
<http:/ |
| wdt:P2163 | "849971" |
| wdt:P227 | "4745179-8" |
| wdt:P244 | "sh2002008702" |
| wdt:P2534 | "<math class="mwe-math-element mwe-math-element-inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mi>𝕆</mi></mrow></mtd><mtd><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo stretchy="false">+</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><mo stretchy="false">⋯</mo><mo stretchy="false">+</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>7</mn></mrow></msub><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mn>7</mn></mrow></msub><mspace width="0.278em"></mspace><mo stretchy="false">∣</mo><mspace width="0.278em"></mspace><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo>,</mo><mo stretchy="false">…</mo><mo>,</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mn>7</mn></mrow></msub><mo stretchy="false">∈</mo><mrow data-mjx-texclass="ORD"><mi>ℝ</mi></mrow><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mtd><mtd><mo stretchy="false">=</mo><mo stretchy="false">−</mo><msub><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub><mo stretchy="false">+</mo><msub><mi>c</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></mrow></msub><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>c</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></mrow></msub></mtd><mtd><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN">{</mo><mtable columnspacing="1em" rowspacing="4pt" columnalign="left left"><mtr><mtd><mn>1</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mtext>even perm. of </mtext></mrow><mn>1</mn><mn>2</mn><mn>3</mn><mo>,</mo><mn>1</mn><mn>4</mn><mn>5</mn><mo>,</mo><mn>1</mn><mn>7</mn><mn>6</mn><mo>,</mo><mn>2</mn><mn>4</mn><mn>6</mn><mo>,</mo><mn>2</mn><mn>5</mn><mn>7</mn><mo>,</mo><mn>3</mn><mn>4</mn><mn>7</mn><mo>,</mo><mn>3</mn><mn>6</mn><mn>5</mn></mtd></mtr><mtr><mtd><mo stretchy="false">−</mo><mn>1</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mtext>odd perm. of </mtext></mrow><mn>1</mn><mn>2</mn><mn>3</mn><mo>,</mo><mn>1</mn><mn>4</mn><mn>5</mn><mo>,</mo><mn>1</mn><mn>7</mn><mn>6</mn><mo>,</mo><mn>2</mn><mn>4</mn><mn>6</mn><mo>,</mo><mn>2</mn><mn>5</mn><mn>7</mn><mo>,</mo><mn>3</mn><mn>4</mn><mn>7</mn><mo>,</mo><mn>3</mn><mn>6</mn><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mtext>otherwise</mtext></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow></mtd></mtr></mtable></mrow></mstyle></mrow></math>"^^http://www.w3.org/1998/Math/MathML |
| wdt:P268 | "15608111r" |
| wdt:P269 | "122706331" |
| wdt:P279 | hypercomplex number |
| wdt:P2812 | "CayleyNumber" |
| "Octonion" | |
| wdt:P31 | alternative algebra |
| octonion algebra | |
| wdt:P361 | octonion algebra |
| wdt:P373 | "Octonions" |
| wdt:P4215 | "octonion" |
| wdt:P527 | quaternion |
| wdt:P5555 |
<http:/ |
| wdt:P575 | "1843-01-01T00:00:00+00:00"^^xsd:dateTime |
| wdt:P61 | John T. Graves |
| wdt:P6104 | WikiProject Mathematics |
| wdt:P6366 | "2779181008" |
| wdt:P646 | "/m/0dkps" |
| wdt:P6781 | "Definition:Octonion" |
| wdt:P7235 | "<math class="mwe-math-element mwe-math-element-inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub></mstyle></mrow></math>"^^http://www.w3.org/1998/Math/MathML |
| "<math class="mwe-math-element mwe-math-element-inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mstyle></mrow></math>"^^http://www.w3.org/1998/Math/MathML | |
| wdt:P8189 | "987007558916605171" |
| wdt:P8885 | "팔원수" |
| wdt:P910 | Category:Octonions |
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