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B How are mathematical ideas discovered

  1. Nov 21, 2017 #1
    If I need to solve the equation x^2 + x = 5, I can recognize that it’s a quadratic equation, change it to x^2 + x - 5 = 0, and then plug into the quadratic formula. In general the way that math is taught, I recognize what type of problem it is, then use the techniques that I was told to use to solve that sort of problem.

    How though did people figure out this sort of stuff? If I was given that same equation, but didn’t yet know how to solve quadratic equations, I’m not even sure where one would begin to try to solve it or to find some way to solve it.
     
  2. jcsd
  3. Nov 21, 2017 #2

    fresh_42

    Staff: Mentor

    They rely on what is already known.
    https://en.wikiquote.org/wiki/Isaac_Newton

    It is often along this important pattern:
    1. Pose the question as precise as possible.
    2. List what you already have in the context.
    3. Try to figure out a path from 2. to 1.
    In the case you've mentioned, it is: We want to know the roots of ##x^2+x-5##. We already know, that ##(a+b)^2=a^2+2ab+b^2##. Now we simply try to get from one to the other:
    $$
    x^2+x-5=x^2+2\cdot x \cdot \dfrac{1}{2} + \left( \dfrac{1}{2} \right)^2 - \left( \dfrac{1}{2} \right)^2 -5 = \left( x + \dfrac{1}{2} \right)^2 - \dfrac{11}{2}
    $$
     
  4. Nov 21, 2017 #3

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Discovering how to approach a new type of problem is typically much more difficult than applying an approach that has been found before. Sometimes it is easy to reduce the new problem to a known one (e.g. the first step you made in your post), sometimes it is much more difficult.

    How to approach it: try different steps that change the problem to different forms until you find one that is easier to solve. With the quadratic equations this is typically done in class to see where the formula comes from. With cubic functions it gets much more complicated. Wikipedia has a description. Hundreds of things were tried, a few of them worked.
     
  5. Nov 23, 2017 #4

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    Sometimes such problems take a flash of insight or the perspective of a genius. As an old saying goes, if you had been born into a civilization where the wheel had not been invented, you probably wouldn't have thought of it either.
     
  6. Dec 3, 2017 #5

    Ssnow

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    Gold Member

    First of all are mathematical ideas discovered or invented ?

    Ssnow
     
  7. Dec 3, 2017 #6

    fresh_42

    Staff: Mentor

    This is a purely philosophical question and has been discussed on PF several times before, most recently here:
    https://www.physicsforums.com/threa...onsidered-a-discovery-or-an-invention.924120/

    In addition it leads to a different topic, because how to develop mathematical ideas - the OP's question - does not mean, how to label mathematical ideas. And this is it: invention or discovery is only a label and a question of taste and philosophical schools.

    Thread closed.
     
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