The Psychotic Line – 3rd dimension of the Real Line

We have the Real Line, from negative infinity on one end to positive infinity on the other. Then we have the Imaginary Line, where we rotate numbers on the Real Line around to obtain imaginary numbers (or complex numbers). So what’s the natural logical progression?

Meet the Psychotic Line, with delusional numbers. As expected, special cases of delusional numbers collapse to either a complex number or real number, by simply setting the delusional component to zero.

The delusional part, j, shall be defined as
j^2 = -i
where i is the unit pure imaginary number.

Thus, j^4 = (-i)^2 = (-1)^2 * i^2 = -1

A typical delusional number is written as
d = a + bi + cj
(d stands for delusional, how coincidentally fortunate!)

Where complex numbers require rotation of 360 degrees to span the full complex plane, delusional numbers only require 180 degrees. Simply study spherical coordinates to understand why (part of the effort is already done by rotation from complex numbers). Once one can leap from the real world to the imaginary world, it takes half the energy to jump to the psychotic world.

One should study the psychotic line, delusional numbers and their properties, for they (possibly) hold the secret to untapped human cerebral abilities, interstellar travel, and maybe even a longer answer to the Ultimate Question of Life, the Universe, and Everything. I wish you luck.

PS: This was written in jest. You’re supposed to laugh.

  1. Cambone

    The name “delusional” is pretty cool. =)
    As pointed in the end, it’s just for fun, as all delusional numbers must be complex, accordingly to that definition. 😉

    j is +/- (1-i)/sqrt(2), which is complex.
    So the space has the same 2 dimensions over R. The base is {1, i}.

  2. Vincent Tan

    Thanks! Delusional numbers sound… delusional. Then again, we have imaginary numbers, so there…

    And I haven’t realised
    j = +/- (1-i)/sqrt(2)
    yet… you’re good…

  3. Eric

    The complex plane is algebraically closed. Therefore the solutions to any algebraic equation will themselves be complex, which is why what Cambone said works. He just happened to solve the particular equation that you presented.

    Other solutions to the equation are e^(i*3*Pi/4) and e^(-i*Pi/4).
    (Actually, those are the same solutions, just presented in a different format. You can convert between the two using DeMoivre’s formula: e^(i*t) = cos t + i*sin t.)

  4. Vincent Tan

    Hey thanks Eric for the De Moivre’s version. Now *that*, I really didn’t think of… 🙂

  5. Vincent

    Hey Tommi,

    Quaternions will extend the real and imaginary line. From my research (that was just done hastily), they exist in R^4.

    Actually, I have no idea what space my psychotic line resides. It should be in R^3. I was just messing around with the terms and mixing their English and mathematical meanings.

    Hmm… I didn’t know much about quaternions… I only knew they were useful in game programming for the gimbal lock in viewing scenes.

    Thanks for letting me know about the quaternion thing.

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