Page 1 of 2 12 LastLast
Results 1 to 15 of 16
Like Tree2Thanks

Thread: Complex verses Quaternion Question

  1. #1
    Newbie Flexico's Avatar
    Joined
    Dec 2016
    From
    Northern Ohio
    Posts
    4

    Complex verses Quaternion Question

    In Python, complex numbers are typed as follows: 3+4j. This kinda bugs me that they use "j" instead of the traditional "i", particularly because quaternions (4-dimensional complex numbers) use i, j, and k for the three "imaginary" directions, so this would cause some overlap. But then I got to thinking ... would using the "j" from a quaternary system be any different than using the "i"? Don't they each have the same properties relative to the real number line? I'm fairly certain they do, but my knowledge of quaternions comes almost exclusively from Youtube and Wikipedia, so I'd like to hear an actual expert's opinion.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,252
    Thanks
    2236

    Re: Complex verses Quaternion Question

    The only requirement we put on $i$ to get the complex numbers is that $i^2 = -1$

    the other two unit quaternions share this property so there would be no difference in the complex numbers generated with respect to the real axis.
    Thanks from HallsofIvy
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    6,417
    Thanks
    1675

    Re: Complex verses Quaternion Question

    Hey Flexico.

    You need to look up the multiplication tables for a quaternion to understand the algebra.

    The intuition involves making it a division algebra - meaning you can do things like do a*b/b = a [i.e. you can do divisions and they act like real numbers algebraically]. This is what the inventor of Quarternions Sir William Rowan Hamilton was trying to do when he was thinking about their construction.

    The understanding algebraically would be to understand multiplications on the unit sphere and you could write some code to do that if you are keen.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member zzephod's Avatar
    Joined
    Apr 2012
    From
    Erewhon
    Posts
    255
    Thanks
    153

    Re: Complex verses Quaternion Question

    Quote Originally Posted by Flexico View Post
    In Python, complex numbers are typed as follows: 3+4j. This kinda bugs me that they use "j" instead of the traditional "i" ....
    The use of $\bf{j}$ for the imaginary unit is the convention in engineering (probably to avoid confusion with the symbol for electrical current).

    .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,252
    Thanks
    2236

    Re: Complex verses Quaternion Question

    Quote Originally Posted by zzephod View Post
    (probably to avoid confusion with the symbol for electrical current)..
    definitely to avoid confusions with the symbol for electric current
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,971
    Thanks
    665
    Awards
    1

    Re: Complex verses Quaternion Question

    Quote Originally Posted by romsek View Post
    definitely to avoid confusions with the symbol for electric current
    Well if you Engineers would just use "I" for current like any self-respecting Physicist we wouldn't have this problem!

    -Dan
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,252
    Thanks
    2236

    Re: Complex verses Quaternion Question

    Quote Originally Posted by topsquark View Post
    Well if you Engineers would just use "I" for current like any self-respecting Physicist we wouldn't have this problem!

    -Dan
    as opposed to "i" ?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,971
    Thanks
    665
    Awards
    1

    Re: Complex verses Quaternion Question

    Quote Originally Posted by romsek View Post
    as opposed to "i" ?
    Well, yeah! I mean, geez! Currants are important enough to use a capital letter with.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,252
    Thanks
    2236

    Re: Complex verses Quaternion Question

    Quote Originally Posted by topsquark View Post
    Well, yeah! I mean, geez! Currants are important enough to use a capital letter with.

    -Dan
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie Flexico's Avatar
    Joined
    Dec 2016
    From
    Northern Ohio
    Posts
    4

    Re: Complex verses Quaternion Question

    Quote Originally Posted by romsek View Post
    The only requirement we put on $i$ to get the complex numbers is that $i^2 = -1$

    the other two unit quaternions share this property so there would be no difference in the complex numbers generated with respect to the real axis.
    Ah cool, that's what I was thinking. Thanks for the confirmation!

    Quote Originally Posted by topsquark View Post
    Well if you Engineers would just use "I" for current like any self-respecting Physicist we wouldn't have this problem!

    -Dan
    Ahahaha seems my question touched a few nerves. XD
    Also, in the Python module "Sympy" a capital "I" is used for the imaginary unit, so yay for confusion!
    Last edited by Flexico; Dec 6th 2016 at 11:36 PM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie Flexico's Avatar
    Joined
    Dec 2016
    From
    Northern Ohio
    Posts
    4

    Re: Complex verses Quaternion Question

    Quote Originally Posted by chiro View Post
    Hey Flexico.

    You need to look up the multiplication tables for a quaternion to understand the algebra.

    The intuition involves making it a division algebra - meaning you can do things like do a*b/b = a [i.e. you can do divisions and they act like real numbers algebraically]. This is what the inventor of Quarternions Sir William Rowan Hamilton was trying to do when he was thinking about their construction.

    The understanding algebraically would be to understand multiplications on the unit sphere and you could write some code to do that if you are keen.
    Yes, I have seen the multiplication table. The algebraic approach seems solid enough, though my tendency in most cases is to calculate things visually, and trying to picture the 4-D space in which quaternions exist often leaves something to be desired. The idea of multiplications being non-commutative takes some getting used to.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Senior Member zzephod's Avatar
    Joined
    Apr 2012
    From
    Erewhon
    Posts
    255
    Thanks
    153

    Re: Complex verses Quaternion Question

    For reference here is one of many discussions on stackoverflow about the imaginary unit choice in Python.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,252
    Thanks
    2236

    Re: Complex verses Quaternion Question

    [QUOTE=Flexico;911613 The idea of multiplications being non-commutative takes some getting used to.[/QUOTE]

    ever deal with matrices?
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Newbie Flexico's Avatar
    Joined
    Dec 2016
    From
    Northern Ohio
    Posts
    4

    Re: Complex verses Quaternion Question

    Quote Originally Posted by romsek View Post
    ever deal with matrices?
    Surprisingly little! I switched schools between Algebra I and Algebra II, and managed to skip the matrix lesson entirely (without realizing it at the time). I've taken several Calculus courses in college, but never really worked with matrices.

    Oh, but I have worked with the matrix data type on my graphing calculator, mostly using them as data storage when programming games. XD Never actually did operations on them though.
    Last edited by Flexico; Dec 7th 2016 at 08:23 AM.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,252
    Thanks
    2236

    Re: Complex verses Quaternion Question

    Quote Originally Posted by Flexico View Post
    Surprisingly little! I switched schools between Algebra I and Algebra II, and managed to skip the matrix lesson entirely (without realizing it at the time). I've taken several Calculus courses in college, but never really worked with matrices.
    ah. Well you'll find that they aren't commutative w/respect to multiplication either.
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Quaternion, -j * k = 0?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Sep 24th 2012, 01:34 PM
  2. Is this the Quaternion group?
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Dec 1st 2011, 07:23 AM
  3. Quaternion
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Sep 26th 2011, 06:38 PM
  4. [SOLVED] quaternion algebra
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Aug 29th 2011, 11:02 AM
  5. Quaternion Group
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Nov 4th 2009, 11:26 AM

Search Tags


/mathhelpforum @mathhelpforum