Sedenion?

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Add a link on the top of this Sedenion page Express submission by secure payment ! <table><tr><td> <form name="APform" action="http://www.allopass.com/cb/check.php4" method="POST"> <input type="hidden" name="CDOC_ID" value="297865"> <input type="hidden" name="CSITE_ID" value="111941"> <input type="hidden" name="LG_SCRIPT" value="uk"> <table border="0" cellpadding="0" cellspacing="0" width="300"> <tr> <td colspan="3"> <img src="http://www.allopass.com/imgweb/script/uk/cb_os_top.gif" alt=""> </td> </tr> <tr> <td width="157" valign="middle" bgcolor="White"> <font face="Arial,Helvetica" color="Black" size="11" Style="font-size: 12px;"> <b>Enter your access ticket</b> </font> </td> <td width="80" bgcolor="White"> <input type="text" size="10" maxlength="10" value="Ticket" name="CODE" onFocus="if (this.form.CODE.value=='Ticket') this.form.CODE.value=''" style="BACKGROUND-COLOR: #E7E7E7; BORDER-BOTTOM: #000080 1px solid; BORDER-LEFT: #000080 1px solid; BORDER-RIGHT: #000080 1px solid; BORDER-TOP: #000080 1px solid; COLOR: #000080; CURSOR: text; FONT-FAMILY: Arial; FONT-SIZE: 10pt; FONT-WEIGHT:bold; LETTER-SPACING: normal; WIDTH:85; TEXT-ALIGN=center;"> </td> <td width="58" align="center" bgcolor="White"> <input type="button" name="APsub" value="" onClick="this.form.submit(); this.form.APsub.disabled=true;" style="border:0px;margin:0px;padding:0px;width:48px; height:18px; background:url('http://www.allopass.com/img/bt_ok.png');"> </td> </tr> <tr><td colspan="3" width="300" height="13</font><br><img src="http://www.allopass.com/img/cb_bot.gif" alt=""></td></tr> </table> <input type="hidden" name="DATAS" value=""> </form> </td><td valign= op"><table border="0" cellpadding="0" cellspacing="0"> <tr> <td> <form name="cben" action="https://secure.allopass.com/cb/subscribe.php4" method="POST" target="DisplaySub"> <input type="hidden" name="START" value="1"> <input type="hidden" name="SDOC_ID" value="297865"> <input type="hidden" name="SSITE_ID" value="111941"> <input type="hidden" name="LANG" value="L_UK"> <input type="image" src="http://www.allopass.com/imgweb/script/uk/cb_subscribe_os.gif" alt="Ticket d'accès" onClick="window.open('','DisplaySub','toolbar=0,location=0,directories=0,status=1,menubar=0,scrollbars=1,resizable=1,width=600,height=570');" border = 0> </form> </td> </tr> </table></td></tr></table>

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The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions.

Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of being power-associative.

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.

Every sedenion is a real linear combination of the unit sedenions 1, e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃, e₁₄ and e₁₅, which form a basis of the vector space of sedenions. The multiplication table of these unit sedenions looks as follows.

×

1

e1

e2

e3

e4

e5

e6

e7

e8

e9

e10

e11

e12

e13

e14

e15

1

1

e1

e2

e3

e4

e5

e6

e7

e8

e9

e10

e11

e12

e13

e14

e15

e1

e1

-1

e3

-e2

e5

-e4

-e7

e6

e9

-e8

-e11

e10

-e13

e12

e15

-e14

e2

e2

-e3

-1

e1

e6

e7

-e4

-e5

e10

e11

-e8

-e9

-e14

-e15

e12

e13

e3

e3

e2

-e1

-1

e7

-e6

e5

-e4

e11

-e10

e9

-e8

-e15

e14

-e13

e12

e4

e4

-e5

-e6

-e7

-1

e1

e2

e3

e12

e13

e14

e15

-e8

-e9

-e10

-e11

e5

e5

e4

-e7

e6

-e1

-1

-e3

e2

e13

-e12

e15

-e14

e9

-e8

e11

-e10

e6

e6

e7

e4

-e5

-e2

e3

-1

-e1

e14

-e15

-e12

e13

e10

-e11

-e8

e9

e7

e7

-e6

e5

e4

-e3

-e2

e1

-1

e15

e14

-e13

-e12

e11

e10

-e9

-e8

e8

e8

-e9

-e10

-e11

-e12

-e13

-e14

-e15

-1

e1

e2

e3

e4

e5

e6

e7

e9

e9

e8

-e11

e10

-e13

e12

e15

-e14

-e1

-1

-e3

e2

-e5

e4

e7

-e6

e10

e10

e11

e8

-e9

-e14

-e15

e12

e13

-e2

e3

-1

-e1

-e6

-e7

e4

e5

e11

e11

-e10

e9

e8

-e15

e14

-e13

e12

-e3

-e2

e1

-1

-e7

e6

-e5

e4

e12

e12

e13

e14

e15

e8

-e9

-e10

-e11

-e4

e5

e6

e7

-1

-e1

-e2

-e3

e13

e13

-e12

e15

-e14

e9

e8

e11

-e10

-e5

-e4

e7

-e6

e1

-1

e3

-e2

e14

e14

-e15

-e12

e13

e10

-e11

e8

e9

-e6

-e7

-e4

e5

e2

-e3

-1

e1

e15

e15

e14

-e13

-e12

e11

e10

-e9

e8

-e7

e6

-e5

-e4

e3

e2

-e1

-1

Further reading

• Carmody, Kevin: Circular and Hyperbolic Quaternions, Octonions and Sedenions, Applied Mathematics and Computation 28:47-72 (1988)
• Carmody, Kevin: Circular and Hyperbolic Quaternions, Octonions and Sedenions - Further results, Applied Mathematics and Computation, 84:27-47 (1997)
• Imaeda, K., Imaeda, M.: Sedenions: algebra and analysis, Applied Mathematics and Computation, 115:77-88 (2000)

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