Results 1 to 3 of 3

Math Help - Indices for Negatives

  1. #1
    Senior Member DivideBy0's Avatar
    Joined
    Mar 2007
    From
    Melbourne, Australia
    Posts
    432

    Indices for Negatives

    Why is it that \sqrt{-1}\sqrt{-1}=-1 instead of \sqrt{(-1)(-1)}=1? What other index laws are different when involving one or more negative numbers?

    Also, according to my book, (-1)^{\frac{2}{6}} \neq (-1)^{\frac{1}{3}}, but according to my calculator, this is perfectly fine. According to it, (-1)^{\frac{2}{6}} \neq ((-1)^2)^6 and (-1)^{\frac{2}{6}} \neq ((-1)^{\frac{1}{6}})^2. This is also the case when drawing graphs on my calculator.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by DivideBy0 View Post
    Why is it that \sqrt{-1}\sqrt{-1}=-1 instead of \sqrt{(-1)(-1)}=1?
    [snip]
    "sqrt" is defined for real numbers as the positive number whose square is ..... but can't be defined that way for non-real numbers ..... The phrase the positive number no longer makes sense because the complex numbers cannot be ordered so as to make them an ordered field.

    By an ordered field I mean:

    a < b => a + c < b + c

    a < b and c > 0 => ac < bc.

    So you can't group complex numbers into positive and negative numbers. Square root, along with many other functions, has to become multi-valued .....

    A complication with that is that you can't just define i by i = \sqrt{-1} because -1 now has two square roots and there's no way to distinguish between them.

    A more formal definition is to define the complex numbers as the set of all pairs of real numbers (a, b) and deine addition and multiplication by (a, b) + (c, d) = (a + c, b + d) and (a, b) * (c, d) = (ac - bd, ad + bc) respectively. It can then be proved that all the rules of arithmetic hold, that pairs of the form (a, 0) obey the same rules as real numbers and that (0, 1) * (0, 1) = (-1, 0).

    If the symbols 1 and i are used to mean (1, 0) and (0, 1) respectively, then (a, b) can be written as (a, 0) * (1, 0) + (b, 0) * (0, 1) = a * 1 + b * i = a + ib.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by DivideBy0 View Post
    Why is it that \sqrt{-1}\sqrt{-1}=-1 instead of \sqrt{(-1)(-1)}=1? What other index laws are different when involving one or more negative numbers?
    Because the mistake is here:
    \sqrt{ab} = \sqrt{a}\sqrt{b}.

    If a,b\geq 0 then this is okay. But if a,b<0 then it CAN be wrong, like here.

    This is Mine 88th Post!!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Adding and subtracting negatives
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 6th 2011, 05:28 AM
  2. multiplying negatives
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 8th 2009, 02:02 PM
  3. Distributing negatives
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 29th 2008, 09:04 AM
  4. simple question on negatives
    Posted in the Algebra Forum
    Replies: 5
    Last Post: July 17th 2008, 09:40 AM
  5. multiplication of negatives
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 19th 2006, 03:34 AM

Search Tags


/mathhelpforum @mathhelpforum