Inequalities and Rearranging equations

  1. So i have an equation to calculate the impossibility of pair production during photon decay into two electrons and I'm having to do some momentum conservation, can't quite do it but a colleague of mine has suggested this which I don't particularly agree with some help would be appreciated.

    So we have some numbers which are constants and one we know is < 0 because its a square negative. He suggests using something like the simple example below which clearly doesn't work if you could suggest why that would be great.


    Rearrange to


    He did the same thing but with energies, where k is a constant:

    k -Ea^2/c^2 - Eb^2/c^2 - 2EaEb/c^2 = -Ei^2/c^2 < 0

    Just rearranged to:

    Ea^2/c^2 + Eb^2/c^2 + 2EaEb/c^2 -Ei^2/c^2 < 0

    Ea^2/c^2 + Eb^2/c^2 + 2EaEb/c^2 < Ei^2/c^2

    I'm not happy with this proof.
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,310
    Staff Emeritus
    Science Advisor

    Yes, -100+ 3 is equivalent to -97 which is negative.

    No. You don't say how you rearranged this but this incorrect. I suspect that you got "3= 3" by adding 100 to both sides of -100+ 3= -97 but the result is no longer "< 0". You would have to add 100 to each part to arrive at "3= 3< 100".

    "He" appears to have done two things to the first inequality:
    First, multiply by -1. But multiplying by a negative number reverses the inequality:
    -k+ Ea^2/c^3+ Eb^2/c^2+ 2EaEb/c^2= Ei^2/c^2> 0

    then add k to each part:
    Ea^2/c^3+ Eb^2/c^3+ 2EaEb/c^2= Ei^2/c^2+ k> k.

    I don't mean to be harsh but shouldn't two people who are working with "pair production" and "photon decay" be able to do basic, elementary school algebra?
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