Absolute value of imaginary numbers?

This is a discussion on Absolute value of imaginary numbers? within the A Brief History of Cprogramming.com forums, part of the Community Boards category; Originally Posted by brewbuck These numbers are quite REAL, at least so much as any other number you've ever encountered ...

  1. #16
    Registered User Stonehambey's Avatar
    Join Date
    Jan 2008
    Location
    Kent, UK
    Posts
    118
    Quote Originally Posted by brewbuck View Post
    These numbers are quite REAL, at least so much as any other number you've ever encountered is "real," and play important roles in explaining physical phenomena.
    Woah! Poor choice of words my friend. I understand the point you're trying to make, but the term real number is a well defined concept in mathematics (the set of real numbers is denoted by R)and any number with an imaginary part is most certainly NOT in the set of real numbers (denoted by C).

    All real numbers are in the set of complex numbers I should probably add (since a real number is just a complex number with b=0)

  2. #17
    and the hat of sweating
    Join Date
    Aug 2007
    Location
    Toronto, ON
    Posts
    3,545
    Quote Originally Posted by tabstop View Post
    And for that matter, there are an infinite number of norms (this is the 2-norm, but there's also the 1-norm and the 3-norm and the ... and the infinity-norm). See article.
    Maybe ancient hieroglyphics would be easier to understand?
    My brain just gets a stack overflow when I look at that page.

  3. #18
    S Sang-drax's Avatar
    Join Date
    May 2002
    Location
    Göteborg, Sweden
    Posts
    2,072
    Quote Originally Posted by cpjust View Post
    Maybe ancient hieroglyphics would be easier to understand?
    My brain just gets a stack overflow when I look at that page.
    Don't worry, because that page is irrelevant to the discussion in this thread.
    Last edited by Sang-drax : Tomorrow at 02:21 AM. Reason: Time travelling

  4. #19
    Captain Crash brewbuck's Avatar
    Join Date
    Mar 2007
    Location
    Portland, OR
    Posts
    7,239
    Quote Originally Posted by Stonehambey View Post
    Woah! Poor choice of words my friend. I understand the point you're trying to make, but the term real number is a well defined concept in mathematics (the set of real numbers is denoted by R)and any number with an imaginary part is most certainly NOT in the set of real numbers (denoted by C).

    All real numbers are in the set of complex numbers I should probably add (since a real number is just a complex number with b=0)
    I understand the basics.

    I tried for a while to find a word that meant the same as "real" in the sense I am trying to use, but could not find a good one. I decided not to waste my time finding some obscure term just because a few mathematicians decided to make everything so incredibly confusing by picking bad terminology.
    Code:
    //try
    //{
    	if (a) do { f( b); } while(1);
    	else   do { f(!b); } while(1);
    //}

  5. #20

    Join Date
    May 2005
    Posts
    1,041
    Quote Originally Posted by Sang-drax View Post
    The formula
    |a + ib| = sqrt(a^2 + b^2)
    is true by definition. There is no proof. Given this definition, the fact that the "imaginary axis is perpendicular to the real axis" follows from this.
    .....
    By the way, your question is a good one. It is important to keep track of what is defined and what is derived.
    This is really, really well said, and I also think the OP's questions are really good...it shows he's actually thinking about the topic at hand. In my experience you need to, at some point, simply accept the mathematics as valid. I also continually find myself asking 'why,' and as Sang-drax said you get to a point where the answer is 'because it's defined that way,' even though that answer sometimes seems arbitrary. Obviously this isn't always the case, but for something as 'tried and true' as imaginary numbers it is.


    I tried for a while to find a word that meant the same as "real" in the sense I am trying to use, but could not find a good one. I decided not to waste my time finding some obscure term just because a few mathematicians decided to make everything so incredibly confusing by picking bad terminology.
    Instead of saying 'real' I would've said 'useful.'

    A really good book that discusses in endless detail the connection between mathematics and reality is 'Science and Sanity' by Korzybski. There's a famous quote that goes along nicely with this:

    Quote Originally Posted by Albert Einstein!
    As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality
    In one of my math classes we would get 'slapped on the wrist' for referring to |X| as 'absolute value' rather than the more appropriate term, 'magnitude,' precisely due to this type of confusion when dealing with complex numbers.
    Last edited by BobMcGee123; 08-11-2008 at 03:39 PM.
    I'm not immature, I'm refined in the opposite direction.

  6. #21
    Cat without Hat CornedBee's Avatar
    Join Date
    Apr 2003
    Posts
    8,893
    I hated maths of all kinds at university and barely passed it, but discussions such as this one show the value of that education in the basics. We built the number systems we use formally, starting with natural numbers, then extending to whole numbers, rationals, reals and finally complex numbers - why all of them are needed and why they work the way they do.

    One important thing to remember is that whenever you extend your current model, you need to recheck all the assumptions you made about the old model - they may no longer hold for the new one. In effect, you'd be generalizing from a subset with special properties to the whole, and that's extremely dangerous.

    For example, the assumption that given |x| = |y|, x² = y² is perfectly valid for real numbers. It can be and has been proven - it trivially follows from the definition of || as sqrt(a² + b&#178 when b = 0, which is the defining trait of real numbers. When you extend to complex numbers, where b may be != 0, you need to prove the conjecture again, and this time you'll fail.
    All the buzzt!
    CornedBee

    "There is not now, nor has there ever been, nor will there ever be, any programming language in which it is the least bit difficult to write bad code."
    - Flon's Law

Page 2 of 2 FirstFirst 12
Popular pages Recent additions subscribe to a feed

Similar Threads

  1. Logical errors with seach function
    By Taka in forum C Programming
    Replies: 4
    Last Post: 09-18-2006, 05:20 AM
  2. imaginary numbers
    By Gil22 in forum C++ Programming
    Replies: 18
    Last Post: 04-17-2003, 09:39 PM
  3. Real and Imaginary numbers
    By tetra in forum C++ Programming
    Replies: 13
    Last Post: 02-03-2003, 10:49 AM
  4. the definition of a mathematical "average" or "mean"
    By DavidP in forum A Brief History of Cprogramming.com
    Replies: 7
    Last Post: 12-03-2002, 10:15 AM
  5. A (complex) question on numbers
    By Unregistered in forum C++ Programming
    Replies: 8
    Last Post: 02-03-2002, 05:38 PM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21