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  • Roots of a second degree equation.

    From Richard Hachel@21:1/5 to All on Wed Jan 22 08:30:15 2025
    Roots of a quadratic equation.

    y=ax²+bx+c

    If b² >4ac then there are two roots.

    If b=4ac then there is a single root.

    If b²<4ac there are no roots.

    We can then draw as many curves as we want, as long as we want, nothing
    will change, there are no roots.

    At least in the real world.

    Let us set y=x²+1 or y=x²+4x+5; there are no roots.

    This does not exist, looking for roots in nothingness, or rabbit horns
    will not change anything. We will not find any.

    Some mathematicians will then try to find some anyway, but beyond reality, where Doctor Hachel takes possession of your computer screen and will give
    it back to you only if he wants to (I put a virus in the equation
    mentioned above).

    They call these imaginary roots, because, since they do not exist, we must imagine them. But what do they correspond to?

    It would seem, in fact, that they are not the roots of the equation, that
    is to say the place in x where the curves cross y=0 since it is impossible
    for all x, but the horizontal mirror projection passing through the vertex
    of the curve.

    For y=x²+1 then [-b(+-)sqrt(b²-4ac)]/2a --->
    (+-)sqrt(-4ac)/2
    (+-)sqrt(4i²)/2=(+-)i

    The two roots are x'=-i and x'=i (i.e. x=-1 and x=1 of the imaginary
    inverted curve).

    Let us set =x²+4x+5 which has no root, and project this curve in mirror;
    two roots appear for the mirror curve.
    [-b(+-)sqrt(b²-4ac)]/2a --->[-4(+-)sqrt(16-4*5)]/2 ---> [-4(+-)sqrt(4i²)]/2a--->(-4(+-)2i)/2
    x=-2-i
    x'=-3
    x"=-1

    But this is very interesting, but wouldn't it be worth giving right away
    the equation of the mirror curve whose vertex touches the vertex of the
    real curve by specifying that it is the imaginary horizontal mirror?

    This means that a second degree curve has two roots (or a single one) and
    that when it doesn't have any, what crosses y=0 is its imaginary mirror
    curve which will give two imaginary roots.

    R.H.

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  • From Moebius@21:1/5 to All on Wed Jan 22 11:34:11 2025
    Am 22.01.2025 um 11:28 schrieb Alan Mackenzie:

    What exactly do you mean by saying that the imaginary roots (usually
    called complex roots by mathematicians) do not exist? What attribute
    does -2 + i possess, or lack, that entitles you to attribute to it the property of non-existence? How does -2 + i differ in that respect from
    other numbers such as -1 or 42?

    The fact is, there is a vast theory of complex analysis which is
    coherent and fascinating. It is also useful in science and engineering.

    Even "worse":

    See: https://www.nature.com/articles/s41586-021-04160-4

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  • From Alan Mackenzie@21:1/5 to Richard Hachel on Wed Jan 22 10:28:11 2025
    Richard Hachel <[email protected]d> wrote:
    Roots of a quadratic equation.

    y=ax²+bx+c

    If b² >4ac then there are two roots.

    If b=4ac then there is a single root.

    If b²<4ac there are no roots.

    We can then draw as many curves as we want, as long as we want, nothing
    will change, there are no roots.

    At least in the real world.

    Let us set y=x²+1 or y=x²+4x+5; there are no roots.

    This does not exist, looking for roots in nothingness, or rabbit horns
    will not change anything. We will not find any.

    Some mathematicians will then try to find some anyway, but beyond reality, where Doctor Hachel takes possession of your computer screen and will give it back to you only if he wants to (I put a virus in the equation
    mentioned above).

    They call these imaginary roots, because, since they do not exist, we must imagine them. But what do they correspond to?

    What exactly do you mean by saying that the imaginary roots (usually
    called complex roots by mathematicians) do not exist? What attribute
    does -2 + i possess, or lack, that entitles you to attribute to it the
    property of non-existence? How does -2 + i differ in that respect from
    other numbers such as -1 or 42?

    The fact is, there is a vast theory of complex analysis which is
    coherent and fascinating. It is also useful in science and engineering.

    [ .... ]

    R.H.

    --
    Alan Mackenzie (Nuremberg, Germany).

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