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| In [[abstract algebra]], '''sedenions''' form a 16-[[dimension of a vector space|dimensional]] non-associative [[algebra over a field|algebra]] over the [[real number|reals]] obtained by applying the [[Cayley–Dickson construction]] to the [[octonions]]. The set of '''sedenions''' is denoted by <math>\mathbb{S}</math>.
| | Greetings I would like to start with letting you know the author's name - Su I presently live in Nebraska on altering it and that I do not plan Hiring is what I really do. To perform domino is anything she really likes performing<br><br>be a rapper, [http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Ddigital-text&field-keywords=how+to+become+a+singer%2C+B00JLYQZD2 websites], |
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| The term "sedenion" is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the [[biquaternion]]s, or the algebra of 4 by 4 matrices over the reals, or that studied by {{harvtxt|Smith |1995}}.
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| == Cayley–Dickson Sedenions ==
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| ===Arithmetic===
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| Like (Cayley–Dickson) [[octonion]]s, [[multiplication]] of Cayley–Dickson sedenions is neither [[commutative]] nor [[associative]].
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| But in contrast to the octonions, the sedenions do not even have the property of being [[alternative algebra|alternative]].
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| They do, however, have the property of [[power associativity]], which can be stated as for any element <var>x</var> of <math>\mathbb{S}</math>, the power <math>x^n</math> is well-defined. They are also [[Flexible identity|flexible]].
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| Every sedenion is a real [[linear combination]] of the unit sedenions 1, <var>e</var><sub>1</sub>, <var>e</var><sub>2</sub>, <var>e</var><sub>3</sub>, ..., and <var>e</var><sub>15</sub>,
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| which form a basis of the [[vector space]] of sedenions.
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| The sedenions have a multiplicative [[identity element]] 1 and multiplicative inverses, but they are not a [[division algebra]]. This is because they have [[zero divisors]]; this means that two non-zero numbers can be multiplied to obtain a zero result: a trivial example is (<var>e</var><sub>3</sub> + <var>e</var><sub>10</sub>)×(<var>e</var><sub>6</sub> − <var>e</var><sub>15</sub>). All [[hypercomplex number]] systems based on the Cayley–Dickson construction from sedenions on contain zero divisors.
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| The [[multiplication table]] of these unit sedenions follows:
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| {| class="wikitable" style="text-align: center;"
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| |-
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| ! ×
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| ! 1
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| ! <var>e</var><sub>1</sub>
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| ! <var>e</var><sub>2</sub>
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| ! <var>e</var><sub>3</sub>
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| ! <var>e</var><sub>4</sub>
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| ! <var>e</var><sub>5</sub>
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| ! <var>e</var><sub>6</sub>
| |
| ! <var>e</var><sub>7</sub>
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| ! <var>e</var><sub>8</sub>
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| ! <var>e</var><sub>9</sub>
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| ! <var>e</var><sub>10</sub>
| |
| ! <var>e</var><sub>11</sub>
| |
| ! <var>e</var><sub>12</sub>
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| ! <var>e</var><sub>13</sub>
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| ! <var>e</var><sub>14</sub>
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| ! <var>e</var><sub>15</sub>
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| |-
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| ! 1
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| | 1
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| | <var>e</var><sub>1</sub>
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| | <var>e</var><sub>2</sub>
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| | <var>e</var><sub>3</sub>
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| | <var>e</var><sub>4</sub>
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| | <var>e</var><sub>5</sub>
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| | <var>e</var><sub>6</sub>
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| | <var>e</var><sub>7</sub>
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| | <var>e</var><sub>8</sub>
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| | <var>e</var><sub>9</sub>
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| | <var>e</var><sub>10</sub>
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| | <var>e</var><sub>11</sub>
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| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
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| | <var>e</var><sub>14</sub>
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| | <var>e</var><sub>15</sub>
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| |-
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| ! <var>e</var><sub>1</sub>
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| | <var>e</var><sub>1</sub>
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| | −1
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| | <var>e</var><sub>3</sub>
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| | −<var>e</var><sub>2</sub>
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| | <var>e</var><sub>5</sub>
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| | −<var>e</var><sub>4</sub>
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| | −<var>e</var><sub>7</sub>
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| | <var>e</var><sub>6</sub>
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| | <var>e</var><sub>9</sub>
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| | −<var>e</var><sub>8</sub>
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| | −<var>e</var><sub>11</sub>
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| | <var>e</var><sub>10</sub>
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| | −<var>e</var><sub>13</sub>
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| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>15</sub>
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| | −<var>e</var><sub>14</sub>
| |
| |-
| |
| ! <var>e</var><sub>2</sub>
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| | <var>e</var><sub>2</sub>
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| | −<var>e</var><sub>3</sub>
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| | −1
| |
| | <var>e</var><sub>1</sub>
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| | <var>e</var><sub>6</sub>
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| | <var>e</var><sub>7</sub>
| |
| | −<var>e</var><sub>4</sub>
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| | −<var>e</var><sub>5</sub>
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| | <var>e</var><sub>10</sub>
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| | <var>e</var><sub>11</sub>
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| | −<var>e</var><sub>8</sub>
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| | −<var>e</var><sub>9</sub>
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| | −<var>e</var><sub>14</sub>
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| | −<var>e</var><sub>15</sub>
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| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
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| |-
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| ! <var>e</var><sub>3</sub>
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| | <var>e</var><sub>3</sub>
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| | <var>e</var><sub>2</sub>
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| | −<var>e</var><sub>1</sub>
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| | −1
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| | <var>e</var><sub>7</sub>
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| | −<var>e</var><sub>6</sub>
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| | <var>e</var><sub>5</sub>
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| | −<var>e</var><sub>4</sub>
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| | <var>e</var><sub>11</sub>
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| | −<var>e</var><sub>10</sub>
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| | <var>e</var><sub>9</sub>
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| | −<var>e</var><sub>8</sub>
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| | −<var>e</var><sub>15</sub>
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| | <var>e</var><sub>14</sub>
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| | −<var>e</var><sub>13</sub>
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| | <var>e</var><sub>12</sub>
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| |-
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| ! <var>e</var><sub>4</sub>
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| | <var>e</var><sub>4</sub>
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| | −<var>e</var><sub>5</sub>
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| | −<var>e</var><sub>6</sub>
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| | −<var>e</var><sub>7</sub>
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| | −1
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| | <var>e</var><sub>1</sub>
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| | <var>e</var><sub>2</sub>
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| | <var>e</var><sub>3</sub>
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| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
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| | <var>e</var><sub>14</sub>
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| | <var>e</var><sub>15</sub>
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| | −<var>e</var><sub>8</sub>
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| | −<var>e</var><sub>9</sub>
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| | −<var>e</var><sub>10</sub>
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| | −<var>e</var><sub>11</sub>
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| |-
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| ! <var>e</var><sub>5</sub>
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| | <var>e</var><sub>5</sub>
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| | <var>e</var><sub>4</sub>
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| | −<var>e</var><sub>7</sub>
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| | <var>e</var><sub>6</sub>
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| | −<var>e</var><sub>1</sub>
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| | −1
| |
| | −<var>e</var><sub>3</sub>
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| | <var>e</var><sub>2</sub>
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| | <var>e</var><sub>13</sub>
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| | −<var>e</var><sub>12</sub>
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| | <var>e</var><sub>15</sub>
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| | −<var>e</var><sub>14</sub>
| |
| | <var>e</var><sub>9</sub>
| |
| | −<var>e</var><sub>8</sub>
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| | <var>e</var><sub>11</sub>
| |
| | −<var>e</var><sub>10</sub>
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| |-
| |
| ! <var>e</var><sub>6</sub>
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| | <var>e</var><sub>6</sub>
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| | <var>e</var><sub>7</sub>
| |
| | <var>e</var><sub>4</sub>
| |
| | −<var>e</var><sub>5</sub>
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| | −<var>e</var><sub>2</sub>
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| | <var>e</var><sub>3</sub>
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| | −1
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| | −<var>e</var><sub>1</sub>
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| | <var>e</var><sub>14</sub>
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| | −<var>e</var><sub>15</sub>
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| | −<var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
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| | <var>e</var><sub>10</sub>
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| | −<var>e</var><sub>11</sub>
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| | −<var>e</var><sub>8</sub>
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| | <var>e</var><sub>9</sub>
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| |-
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| ! <var>e</var><sub>7</sub>
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| | <var>e</var><sub>7</sub>
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| | −<var>e</var><sub>6</sub>
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| | <var>e</var><sub>5</sub>
| |
| | <var>e</var><sub>4</sub>
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| | −<var>e</var><sub>3</sub>
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| | −<var>e</var><sub>2</sub>
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| | <var>e</var><sub>1</sub>
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| | −1
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| | <var>e</var><sub>15</sub>
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| | <var>e</var><sub>14</sub>
| |
| | −<var>e</var><sub>13</sub>
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| | −<var>e</var><sub>12</sub>
| |
| | <var>e</var><sub>11</sub>
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| | <var>e</var><sub>10</sub>
| |
| | −<var>e</var><sub>9</sub>
| |
| | −<var>e</var><sub>8</sub>
| |
| |-
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| ! <var>e</var><sub>8</sub>
| |
| | <var>e</var><sub>8</sub>
| |
| | −<var>e</var><sub>9</sub>
| |
| | −<var>e</var><sub>10</sub>
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| | −<var>e</var><sub>11</sub>
| |
| | −<var>e</var><sub>12</sub>
| |
| | −<var>e</var><sub>13</sub>
| |
| | −<var>e</var><sub>14</sub>
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| | −<var>e</var><sub>15</sub>
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| | −1
| |
| | <var>e</var><sub>1</sub>
| |
| | <var>e</var><sub>2</sub>
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| | <var>e</var><sub>3</sub>
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| | <var>e</var><sub>4</sub>
| |
| | <var>e</var><sub>5</sub>
| |
| | <var>e</var><sub>6</sub>
| |
| | <var>e</var><sub>7</sub>
| |
| |-
| |
| ! <var>e</var><sub>9</sub>
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| | <var>e</var><sub>9</sub>
| |
| | <var>e</var><sub>8</sub>
| |
| | −<var>e</var><sub>11</sub>
| |
| | <var>e</var><sub>10</sub>
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| | −<var>e</var><sub>13</sub>
| |
| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>15</sub>
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| | −<var>e</var><sub>14</sub>
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| | −<var>e</var><sub>1</sub>
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| | −1
| |
| | −<var>e</var><sub>3</sub>
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| | <var>e</var><sub>2</sub>
| |
| | −<var>e</var><sub>5</sub>
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| | <var>e</var><sub>4</sub>
| |
| | <var>e</var><sub>7</sub>
| |
| | −<var>e</var><sub>6</sub>
| |
| |-
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| ! <var>e</var><sub>10</sub>
| |
| | <var>e</var><sub>10</sub>
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| | <var>e</var><sub>11</sub>
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| | <var>e</var><sub>8</sub>
| |
| | −<var>e</var><sub>9</sub>
| |
| | −<var>e</var><sub>14</sub>
| |
| | −<var>e</var><sub>15</sub>
| |
| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
| |
| | −<var>e</var><sub>2</sub>
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| | <var>e</var><sub>3</sub>
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| | −1
| |
| | −<var>e</var><sub>1</sub>
| |
| | −<var>e</var><sub>6</sub>
| |
| | −<var>e</var><sub>7</sub>
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| | <var>e</var><sub>4</sub>
| |
| | <var>e</var><sub>5</sub>
| |
| |-
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| ! <var>e</var><sub>11</sub>
| |
| | <var>e</var><sub>11</sub>
| |
| | −<var>e</var><sub>10</sub>
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| | <var>e</var><sub>9</sub>
| |
| | <var>e</var><sub>8</sub>
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| | −<var>e</var><sub>15</sub>
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| | <var>e</var><sub>14</sub>
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| | −<var>e</var><sub>13</sub>
| |
| | <var>e</var><sub>12</sub>
| |
| | −<var>e</var><sub>3</sub>
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| | −<var>e</var><sub>2</sub>
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| | <var>e</var><sub>1</sub>
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| | −1
| |
| | −<var>e</var><sub>7</sub>
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| | <var>e</var><sub>6</sub>
| |
| | −<var>e</var><sub>5</sub>
| |
| | <var>e</var><sub>4</sub>
| |
| |-
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| ! <var>e</var><sub>12</sub>
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| | <var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
| |
| | <var>e</var><sub>14</sub>
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| | <var>e</var><sub>15</sub>
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| | <var>e</var><sub>8</sub>
| |
| | −<var>e</var><sub>9</sub>
| |
| | −<var>e</var><sub>10</sub>
| |
| | −<var>e</var><sub>11</sub>
| |
| | −<var>e</var><sub>4</sub>
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| | <var>e</var><sub>5</sub>
| |
| | <var>e</var><sub>6</sub>
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| | <var>e</var><sub>7</sub>
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| | −1
| |
| | −<var>e</var><sub>1</sub>
| |
| | −<var>e</var><sub>2</sub>
| |
| | −<var>e</var><sub>3</sub>
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| |-
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| ! <var>e</var><sub>13</sub>
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| | <var>e</var><sub>13</sub>
| |
| | −<var>e</var><sub>12</sub>
| |
| | <var>e</var><sub>15</sub>
| |
| | −<var>e</var><sub>14</sub>
| |
| | <var>e</var><sub>9</sub>
| |
| | <var>e</var><sub>8</sub>
| |
| | <var>e</var><sub>11</sub>
| |
| | −<var>e</var><sub>10</sub>
| |
| | −<var>e</var><sub>5</sub>
| |
| | −<var>e</var><sub>4</sub>
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| | <var>e</var><sub>7</sub>
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| | −<var>e</var><sub>6</sub>
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| | <var>e</var><sub>1</sub>
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| | −1
| |
| | <var>e</var><sub>3</sub>
| |
| | −<var>e</var><sub>2</sub>
| |
| |-
| |
| ! <var>e</var><sub>14</sub>
| |
| | <var>e</var><sub>14</sub>
| |
| | −<var>e</var><sub>15</sub>
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| | −<var>e</var><sub>12</sub>
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| | <var>e</var><sub>13</sub>
| |
| | <var>e</var><sub>10</sub>
| |
| | −<var>e</var><sub>11</sub>
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| | <var>e</var><sub>8</sub>
| |
| | <var>e</var><sub>9</sub>
| |
| | −<var>e</var><sub>6</sub>
| |
| | −<var>e</var><sub>7</sub>
| |
| | −<var>e</var><sub>4</sub>
| |
| | <var>e</var><sub>5</sub>
| |
| | <var>e</var><sub>2</sub>
| |
| | −<var>e</var><sub>3</sub>
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| | −1
| |
| | <var>e</var><sub>1</sub>
| |
| |-
| |
| ! <var>e</var><sub>15</sub>
| |
| | <var>e</var><sub>15</sub>
| |
| | <var>e</var><sub>14</sub>
| |
| | −<var>e</var><sub>13</sub>
| |
| | −<var>e</var><sub>12</sub>
| |
| | <var>e</var><sub>11</sub>
| |
| | <var>e</var><sub>10</sub>
| |
| | −<var>e</var><sub>9</sub>
| |
| | <var>e</var><sub>8</sub>
| |
| | −<var>e</var><sub>7</sub>
| |
| | <var>e</var><sub>6</sub>
| |
| | −<var>e</var><sub>5</sub>
| |
| | −<var>e</var><sub>4</sub>
| |
| | <var>e</var><sub>3</sub>
| |
| | <var>e</var><sub>2</sub>
| |
| | −<var>e</var><sub>1</sub>
| |
| | −1
| |
| |}
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| ==Applications==
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| {{harvtxt|Moreno|1998}} showed that the space of norm 1 zero-divisors of the sedenions is [[homeomorphic]] to the compact form of the exceptional [[Lie group]] [[G2 (mathematics)|G<sub>2</sub>]].
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| ==See also==
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| * [[Pfister's sixteen-square identity]]
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| * [[Hypercomplex number]]
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| * [[Split-complex number]]
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| ==References==
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| *{{Citation | last1=Imaeda | first1=K. | last2=Imaeda | first2=M. | title=Sedenions: algebra and analysis | doi=10.1016/S0096-3003(99)00140-X | mr=1786945 | year=2000 | journal=Applied mathematics and computation | volume=115 | issue=2 | pages=77–88}}
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| * Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: ''C-loops: Extensions and constructions'', Journal of Algebra and its Applications 6 (2007), no. 1, 1–20. [http://arxiv.org/abs/math/0412390]
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| * Kivunge, Benard M. and Smith, Jonathan D. H: "[http://www.emis.de/journals/CMUC/pdf/cmuc0402/kivunge.pdf Subloops of sedenions]", Comment.Math.Univ.Carolinae 45,2 (2004)295–302.
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| *{{Citation | last1=Moreno | first1=Guillermo | title=The zero divisors of the Cayley–Dickson algebras over the real numbers | arxiv=q-alg/9710013 | mr=1625585 | year=1998 | journal=Sociedad Matemática Mexicana. Boletí n. Tercera Serie | volume=4 | issue=1 | pages=13–28}}
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| *{{Citation | last1=Smith | first1=Jonathan D. H. | title=A left loop on the 15-sphere | doi=10.1006/jabr.1995.1237 | mr=1345298 | year=1995 | journal=[[Journal of Algebra]] | volume=176 | issue=1 | pages=128–138}}
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| {{Number Systems}}
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| [[Category:Hypercomplex numbers]]
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| [[Category:Non-associative algebras]]
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