Results 1 to 3 of 3

Math Help - negative exponents

  1. #1
    Junior Member
    Joined
    Mar 2009
    From
    Kentucky
    Posts
    72

    negative exponents

    Is there a mathmatical rational for the choosing of negative exponents to represent fraction reversal or was is it just a matter of convention.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570

    Negative exponents

    Hello manyarrows
    Quote Originally Posted by manyarrows View Post
    Is there a mathmatical rational for the choosing of negative exponents to represent fraction reversal or was is it just a matter of convention.
    Yes, there's every mathematical reason in the world! Like being able to continue to use the laws that apply to positive exponents. For instance, the simplest and most basic rule is:

     x^a \times x^b = x^{a+b}

    This is obviously the case for positive values of a and b, because we can simply write the meanings in full:

    x^a \times x^b = \underbrace{x \times x \times \dots \times x}_{a\,x's} \times \underbrace{x \times x \times \dots \times x}_{b\,x's} = \underbrace{x \times x \times \dots \times x}_{(a+b)\,x's}

    If we want to continue to use this law for negative powers (whatever they may mean), then we shall be forced to conclude that x^{-b} will have to be defined as \frac{1}{x^{b}}. Why? Well, if the law above continues to remain in force, then (and we'll assume for now that a>b):

     x^a \times \color{red}x^{-b} \color{black}= x^{a+(-b)} = x^{a-b}

    But we already know that x^a \times \color{red}\frac{1}{x^b}\color{black} =\frac{x^a}{x^b} = x^{a-b}, by 'cancelling' all the x's in the denominator with b of the x's in the numerator.

    And therefore we shall insist that x^{-b} and \frac{1}{x^b} mean the same thing.

    If that all seems too complicated, and you want a simpler reason, just look at the pattern of powers of 2 below, and ask yourself what happens if you continue the pattern, dividing by 2 each time:

    2^3 = 8,\, 2^2 = 4,\, 2^1 = 2,\, 2^0 = ?,\, 2^{-1} = ?,\, 2^{-2}= ?,\, ...

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,977
    Thanks
    1121
    I just want to point out that if we want that very nice property, x^ax^b= x^{a+b}, then we want x^0x^a= x^{0+a}= x^a so that we must have x^0= 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. negative exponents
    Posted in the Algebra Forum
    Replies: 3
    Last Post: January 25th 2009, 11:54 AM
  2. Negative exponents
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: December 2nd 2008, 07:09 AM
  3. Negative Exponents
    Posted in the Algebra Forum
    Replies: 7
    Last Post: February 23rd 2008, 11:06 PM
  4. Zero and negative exponents????
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 28th 2007, 09:55 AM
  5. negative fractions to negative exponents
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 24th 2006, 06:02 PM

Search Tags


/mathhelpforum @mathhelpforum