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B Hypercomplex number system

  1. Apr 13, 2017 #1

    I was just wondering if an extension of hypercomplex numbers like this have any use or if it would be pointless:

    The number g is defined by ##g^2=i##. Then, the powers of g from 0 to 8 (where the cycle restarts) would be 1, g, i, ig, -1, -g, -i, -ig, 1. There's a lot of interesting things I found that you can do with this. I was surprised, however, that I couldn't find a hypercomplex system like this that was established, making me wonder if this system is pointless; it does seem like something someone would come up with easily. Any thoughts?
  2. jcsd
  3. Apr 13, 2017 #2


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    There is a complex number with that property. Can you find it?

    More general: all polynomial expressions like yours have a solution. This is one of the most important properties of the complex numbers: The fundamental theorem of algebra.
  4. Apr 13, 2017 #3
    I know that ##\frac{1+i}{\sqrt{2}}## fits that definition. But, could this also be a useful hypercomplex system as well?
  5. Apr 13, 2017 #4


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    How do you define hypercomplex numbers? I've read a definition which says it's a division algebra over the reals. With this definition ##\mathbb{R}## and ##\mathbb{C}## are also hypercomplex and therefore ##\mathbb{C}[g]## as well, but ##\mathbb{C}=\mathbb{C}[g]##, so how would this help? If you read the corresponding Wikipedia entry you will find interesting objects (number systems like dual numbers, octonions, sedenions, bicomplex numbers, biquaternion numbers) and historical explanations.
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